Attribution of Spurious Factors from High-Dimensional Functional Time Series

A ttribution of Spurious F actors from High-Dimensional F unctional Time Series A dam Nie 1 , Y anrong Y ang 1 , Han Lin Shang 2 , Yi He 3 1 Researc h School of Finance, A ctuarial Studies and Statistics, The Australian National Univ ersity , A ustralia, adam.nie@anu.edu.au , yanrong.yang@anu.edu.au 2 Departmen t of Actuarial Studies and Business Analytics, Macquarie Universit y , Australia, hanlin.shang@mq.edu.au 3 Sc ho ol of Mathematical Sciences, Eastern Institute of T ec hnology , Ningbo, China, yihe@eitech.edu.cn Abstract This article explores a general factor structure for high-dimensional nonstationary functional time series, encompassing a wide range of factor models studied in the existing literature. W e in vestigate the asymptotic sp ectral b eha viors of the sample cov ariance op erator under this general data structure. A nov el fundamen tal suﬃcient condition, formulated in terms of a newly in tro duced eﬀectiv e rank tailored to this setup, is established under whic h empirical eigen-analysis yields spurious results, rendering sample eigenv alues and eigen vectors unreliable for accurately reco vering the underlying factor structure. This generalizes the results of Onatski and W ang [ 2021 ] from typical high-dimensional time series (HDTS) to the more intricate functional framew ork. The newly deﬁned eﬀectiv e rank is rigorously analyzed through a decomp osition of the eﬀects attributable to functional factor loadings and functional factors. Con trary to the ﬁndings in the HDTS setting, empirical eigen-analysis of mo dels with only a small num b er of strong non-stationary factors may still produce spurious limits in the functional framework. Therefore, additional caution is w arranted when applying co v ariance-based statistical metho ds to p oten tially nonstationary functional data. Simulation studies are p erformed to determine conditions under which spurious limits o ccur. Real data analysis on age-sp eciﬁc mortality rate data from m ultiple lo cations is conducted for evidence of spurious factors induced b y empirical eigen-analysis. Keyw ords: F actor Structure; High-dimensional F unctional time series; Non-stationarity; Sample Co v ariance Op erator; Spurious Analysis. 1 In tro duction F unctional data analysis concerns random observ ations that take v alues in spaces of functions, t ypically square-in tegrable function spaces ov er compact interv als. Suc h data ha v e gained substan tial atten tion o ver the past few decades, b oth among theoreticians and researc hers in applied ﬁelds [see, e.g., W ang et al. , 2016 , K oner and Staicu , 2023 , Ramsa y and Silverman , 1997 ]. T o address the inheren t inﬁnite-dimensional nature of functional data, dimension-reduction tec hniques—such as functional principal comp onen t analysis and functional factor mo dels—are widely employ ed [ Hall and Hosseini-Nasab , 2006 , Benko et al. , 2009 , Shang , 2014 , Hall et al. , 2006 ]. F unctional time series constitute a sp ecial class of functional data in which the observed random functions exhibit serial dep endence. In the univ ariate functional time series setting, a substan tial 1 b ody of literature has extended classical time series metho dologies to the functional framew ork [ Hörmann and K okoszka , 2012 , 2010 , Hyndman and Shang , 2009 , Aue et al. , 2015 , Horváth et al. , 2014 , see, e.g., and references therein]. The analysis of multiple functional time series, particularly in high-dimensional regimes where the n umber of series is comparable to or exceeds the sample size, has received gro wing atten tion; see, for example, Gao et al. [ 2019 ], Jiménez-V arón et al. [ 2024 ], Hallin et al. [ 2023 ], Chang et al. [ 2025 , 2024 ], Guo et al. [ 2026 ], Leng et al. [ 2026 ]. A cen tral diﬃculty in this setting is the co existence of multiple sources of dep endence: within-curv e dep endence across the functional domain, serial dep endence o ver time, and cross-sectional dep endence across series. Each comp onen t may b e high-dimensional. F or instance, the Japanese sub-national mortalit y data analysed in Jiménez-V arón et al. [ 2024 ] comprise of 46 functional time series observed o ver 50 y ears, with eac h ann ual observ ation recorded on a grid of 110 p oin ts. Dimension-reduction metho ds for m ultiv ariate and high-dimensional functional time series include extensions of functional principal comp onen t analysis [see, e.g., Gao et al. , 2019 , Chiou et al. , 2014 , Happ and Greven , 2018 , Di et al. , 2009 , Zapata et al. , 2022 ] and functional factor mo dels [see, e.g., Jiménez-V arón et al. , 2024 , Hallin et al. , 2023 , Chang et al. , 2025 , 2024 , Guo et al. , 2026 , Leng et al. , 2026 ]. These approac hes appro ximate the data by lo w-rank represen tations and are eﬀective when the underlying data-generating mec hanism is itself appro ximately lo w-rank. Man y of the classical metho ds discussed ab o ve are primarily developed under the assumption of stationarit y . In recent y ears, ho wev er, increasing atten tion has b een devoted to the non-stationary setting [see, e.g., v an Delft and Eichler , 2018 , v an Delft and Dette , 2021 , Chang et al. , 2016 , Li et al. , 2023 , Horváth et al. , 2014 ]. F or non-functional data, it is w ell established that directly applying PCA or factor analysis to high-dimensional non-stationary time series can lead to spurious conclusions. In such settings, temp oral dep endence may dominate cross-sectional dep endence, even when strong common factors are present. As a result, the leading eigen v alues obtained from the standard PCA pro cedure primarily reﬂect serial dep endence rather than genuine cross-sectional correlation. Consequen tly , the extracted comp onents may fail to represent the underlying factor structure and instead capture p ersisten t time dynamics. This issue w as rigorously studied by Onatski and W ang [ 2021 ], who sho wed that standard PCA may pro duce spurious factors when applied to high-dimensional integrated time series lac king a gen uine lo w-rank structure. Even in the absence of true common factors, the leading sample principal comp onents can explain a large fraction of the total v ariance, leading to erroneous inference. They further established that, under high-dimensional asymptotics, the sample eigen v alues and eigen vectors con verge to functionals of a Wiener pro cess, irresp ectiv e of the underlying cov ariance structure, indicating that strong temp oral dep endence can domin ate the sp ectral b eha vior. Building on this framew ork, He and Zhang [ 2024 ] extended the analysis to mo dels with more general forms of temp oral dep endence b eyond the unit ro ot case. They also deriv ed the asymptotic distributions of the sample eigen v alues, which enable formal tests to distinguish genuine factors from spurious ones. T o mitigate the o ccurrence of spurious factors in practice, Zhang et al. [ 2025 ] prop osed a data-adaptiv e metho d to iden tify the underlying structure of high-dimensional time series b efore conducting eigen-analysis. The present pap er in vestigates the emergence of spurious phenomena in high-dimensional functional time series, with an extra functional dimension. W e consider a general high-dimensional functional factor mo del with non-stationary factors that encompasses sev eral existing framew orks in the literature [see, e.g., Hallin et al. , 2023 , T a v ak oli et al. , 2023 , Guo et al. , 2026 , Leng et al. , 2026 ]. In parallel with results from the high-dimensional literature, under mild rank-type conditions on the cov ariance op erator of the pro cess, the empirical eigenv alues and eigen-functions conv erge to a spurious limit that fails to represen t the genuine cov ariance structure. As a result, metho dologies 2 relying on functional principal component analysis can pro duce inconsisten t or misleading conclusions in the presence of non-stationarit y . W e generalize the core results of Onatski and W ang [ 2021 ] to the setting of high-dimensional functional time series (see Theorem 3.8 ). The inﬁnite-dimensional nature of functional observ ations in tro duces substantial tec hnical c hallenges and gives rise to asymptotic b ehavior that is qualitativ ely diﬀeren t from that observ ed in ﬁnite-dimensional settings. In particular, the conclusions of our main theorems diverge signiﬁcantly from those of Onatski and W ang [ 2021 ], highligh ting structural features that are intrinsic to functional data. W e elab orate on these distinctions b elo w. Onatski and W ang [ 2021 ] introduce a suﬃcient condition for the emergence of a spurious limit based on the notion of an “eﬀective rank,” whic h characterizes the exten t to whic h the mo del can b e w ell appro ximated by a low-rank structure. They show that mo dels with a large eﬀectiv e rank are prone to exhibiting spurious limits. In our setting, how ev er, the concept of eﬀective rank becomes more challenging due to an additional la yer of dependence inherent in functional data. F or high-dimensional (or m ultiv ariate) functional time series, the cov ariance op erator enco des b oth cross-sectional dep endence across co ordinates and dep endence along the functional domain. Disen tangling these tw o sources of dep endence requires additional technical work in order to deriv e results that admit a clear interpretation; see Theorem 3.11 and Corollary 3.15 . F urthermore, the existing theoretical framew ork do es not accommo date mo dels with a ﬁxed n umber of strong factors. In con trast, this restriction is relaxed in our setting due to the intrinsic high dimensionality of functional data. Speciﬁcally , Theorem 3.11 establishes that ev en mo dels driv en b y only a small num b er of strong non-stationary factors may con verge to the spurious limits c haracterized in Theorem 3.8 . This constitutes a substan tial departure from the settings considered in Onatski and W ang [ 2021 ] and He and Zhang [ 2024 ], where a ﬁnite num b er of strong factors are treated as gen uine and thus do not conv erge to spurious limits. Our ﬁndings indicate that, in the functional framew ork, the presence of an apparen tly well-deﬁned factor structure do es not safeguard against spurious eigenstructure in practice. Consequently , particular caution is warran ted when applying cov ariance-based metho ds to non-stationary functional time series, even in the presence of strong factors. The remainder of the pap er is organized as follo ws. Section 2.1 in tro duces the model under consideration, and Section 2.2 presen ts several illustrativ e examples together with their connections to the existing literature. Section 2.3 deﬁnes the principal mathematical ob jects of interest, namely a class of sample co v ariance op erators, and introduces the notation and preliminary results required for the developmen t of our theory . Section 3.1 states and discusses the main assumptions underlying our theoretical analysis, while the primary results are presented in Section 3 . F urther discussion of the k ey suﬃcient condition, Assumption 3.3 , is pro vided in Section 3.3 , where we also establish additional theoretical results that clarify the circumstances under which spurious limits arise. Section 4 rep orts simulation results and examines their implications in ligh t of the theory . In Section 5 , we apply our metho dology to empirical datasets. Finally , Section 6 summarizes the main ﬁndings and prop oses a conjecture motiv ated b y sev eral notable simulation patterns for future inv estigation. Finally , w e will in tro duce some notations for the rest of the pap er. Throughout the pap er, for a separable Hilb ert space H , we write ⟨· , ·⟩ H and ∥·∥ H for its inner pro duct and the asso ciated norm. W e will routinely drop the subscript H and simply write ⟨· , ·⟩ and ∥·∥ when the context is clear. F or elemen ts f , g on H , we write f ⊗ g for the b ounded linear op erator h 7→ f ⟨ h, g ⟩ . F or a v ector x = ( x 1 , . . . , x n ) ∈ R n , we write ∥ x ∥ 0 = |{ i ≤ n, x i  = 0 }| for the n umber of non-zero co ordinates of x and ∥ x ∥ p = ( P i ≤ n x p i ) 1 /p for the usual ℓ p norm of x . F or p ∈ N , w e write H p = ⊕ p k =1 H for the external direct sum of p copies of H endo wed with the usual inner pro duct. Elemen ts of H p will b e denoted as f = ( f 1 , . . . , f p ) where f i ∈ H for 1 ≤ i ≤ p . F or a bounded linear operator A on H , w e write ∥ A ∥ and A ∗ for the op erator norm and the adjoint of A resp ectiv ely . The rank, the 3 trace, the nuclear norm and the Hilb ert-Sc hmidt norm of A will b e denoted as ∥ A ∥ 0 , ⟨ A ⟩ , ∥ A ∥ 1 and ∥ A ∥ 2 resp ectiv ely , whenev er they are ﬁnite. F or Hilbert-Schmidt op erators A, B on H , we write ⟨ A, B ⟩ = ⟨ A ∗ B ⟩ for the natural inner pro duct in the Hilb ert space of Hilb ert-Sc hmidt op erators on H . F or sequences ( a n ) n ∈ N and ( b n ) n ∈ N , we write a n ≲ b n if a n ≤ cb n for all n for some p ositiv e constan t c indep enden t of n . W e write a n ∼ b n if a n ≲ b n and b n ≲ a n . W e write a n ≪ b n or a n = o ( b n ) if a n /b n → 0 as n → ∞ and a n ≫ b n if b n ≪ a n . Finally , we use O p and o p to denote the usual notions of sto c hastic compactness and conv ergence in probabilit y . 2 Setup: Mo del and Assumptions 2.1 A General F unctional F actor Mo del Let H = L 2 ( I ) b e the Hilb ert space of square in tegrable functions on some compact interv al I and H p = ⊕ p k =1 H b e the external direct sum of p copies of H endo wed with the us ual inner pro duct. W e consider an integrated p -dimensional functional time series X = ( X 1 , . . . , X T ) of length T taking v alues on H p . W e assume X follo ws the functional factor mo del b elo w X t ( u ) = (Ψ F t )( u ) + ζ t ( u ) , u ∈ I , (1) where F t is an integrated functional time series on H K for some K < p , the function Ψ is a b ounded linear op erator from H K to H p , and ( ζ t ) t is a stationary functional time series on H p with mean zero and cov ariance op erator C ζ . It is natural to view ( 1 ) as a functional factor mo del with non-stationary factors F t and factor loading op erator Ψ . This functional factor mo del takes a general form in the sense of b oth factors and loadings b eing functions, whic h co vers common factor styles in relev an t literature Hallin et al. [ 2023 ], Guo et al. [ 2026 ] and Leng et al. [ 2026 ]. Analogous to Onatski and W ang [ 2021 ], the functional factor ( F t ) t is assumed to b e a random w alk on H K of the form F t = P t s =1 ϵ s where ϵ t = ( ϵ 1 t , . . . , ϵ K t ) ⊤ and ( ϵ kt ) kt is a collection of indep enden t random functions in H with mean zero and co v ariance op erator C ϵ . The linear op erator Ψ is assumed to b e a matrix of in tegral op erators, i.e., there exists a collection (Ψ ik ) i ≤ p,k ≤ K of in tegral op erators suc h that (Ψ ϵ t )( u ) i = K X k =1 Z I Ψ ik ( u, v ) ϵ kt ( v ) dv , u ∈ I , where Ψ ik ( · , · ) denotes the in tegral kernel of the in tegral op erator Ψ ik . This structure is also adopted b y Leng et al. [ 2026 ]. W e will write Ψ( · , · ) as the matrix of k ernel functions (Ψ ik ( · , · )) ik and write the ab o ve equation in the matrix form (Ψ ϵ )( u ) = Z I Ψ( u, v ) ϵ ( v ) dv , u ∈ I . A dditional tec hnical assumptions on the mo del are discussed in Section 3 . 2.2 Mo del Generality Our framework is built up on a highly general functional factor structure in whic h b oth the factors F and the factor loadings Ψ are functional ob jects. This formulation is conceptually related to the dual functional factor mo del prop osed b y Leng et al. [ 2026 ]; how ev er, their analysis is restricted to stationary time series. The fully functional factor structure sp eciﬁed in ( 1 ) encompasses a broad class of existing models as sp ecial cases. W e no w illustrate this generalit y by presenting sev eral concrete examples and discussing their connections to the literature. 4 Example 2.1. (F unctional Loadings). Supp ose the factors F kt tak es the form of F kt ( · ) = ˜ F kt g ( · ) for some deterministic function g ∈ H and real random v ariables ( ˜ F kt ) kt . Set ˜ Ψ( · ) = Z Ψ( · , v ) g ( v ) dv, then our factor structure in ( 1 ) reduces to the functional factor mo del of Hallin et al. [ 2023 ] and T av ak oli et al. [ 2023 ] with factors ( ˜ F kt ) kt and factor loading op erator ˜ Ψ . In this case the factor loading is a functional ob ject while the factors is giv en b y a multiv ariate time series on R K . Example 2.2. (F unctional F actors). Supp ose the factor loadings are giv en b y Ψ ik ( u, v ) = δ ( u − v ) ˜ Ψ ik where ˜ Ψ = ( ˜ Ψ ik ) ik is a p × K real matrix and δ is the Dirac δ function at zero. Then we hav e (Ψ F t )( u ) = Z I δ ( u − v ) ˜ Ψ F t ( v ) dv = ˜ Ψ F t ( u ) and our factor structure in ( 1 ) reduces to the functional factor mo del recen tly prop osed by Guo et al. [ 2026 ]. In this case the factors are functional ob jects while the factor loadings are given b y a p × K real matrix. W e note that, under the factor structure in Example 2.1 , b oth the cross-sectional dep endence and the dep endence along the functional domain are en tirely induced by the factor loadings. In con trast, under the structure in Example 2.2 , the cross-sectional dep endence is determined by the factor loading matrix, whereas the dep endence along the functional domain is join tly driv en by the laten t factors and the loading matrix. In our general framew ork, b oth t yp es of dep endence emerge from the interaction b et w een the factors and the factor loadings, as will b ecome evident from the main theoretical results. The fully functional factor structure is also closely related to the study of non-stationarity and coin tegration in functional time series, topics that hav e attracted increasing attention in the recent literature [see, e.g., Beare et al. , 2017 , Chang et al. , 2016 , and references therein]. W e present a simpliﬁed example b elo w to illustrate these connections. Example 2.3. (Coin tegrated F unctional Time Series). Supp ose p = K = 1 and Ψ is a b ounded linear op erator on H of ﬁnite rank. W riting ∆ X t = Ψ ϵ t + ∆ ζ t , our mo del in ( 1 ) b ecomes the the Beveridge-Nelson decomp osition of the coin tegrated functional time series studied in Beare et al. [ 2017 ] and Chang et al. [ 2016 ]. The rank of Ψ here represen ts the dimension of the attractor space, i.e. the v ectors h in the kernel of Ψ are the directions suc h that the pro jection ⟨ h, X t ⟩ is of I (0) . Our mo del can b e extended to b e compatible with recent results that study in tegrated time series of order I ( d ) where d can b e fractional [see Li et al. , 2023 , Beare and Seo , 2020 ]. W e do not pursue suc h extensions in our current w ork. 5 Finally , although our mo del is formulated with the factors F ev olving as a random w alk on H , the framework in fact accommo dates a broad class of temp oral dep endence structures, in the spirit of Onatski and W ang [ 2021 ]. In particular, the sp eciﬁcation in ( 1 ) can b e in terpreted as the Beveridge–Nelson decomp osition of I (1) linear pro cess on H exhibiting more general short-run dynamics. As demonstrated b y our main results, the random-walk comp onen t of the factors asymptotically dominates the b eha vior of the sample cov ariance op erator, while mild deviations from pure random- w alk dynamics hav e asymptotically negligible eﬀects. W e conclude this section with an additional example illustrating the range of temp oral dep endence structures encompassed by our framework. Example 2.4. (Multiv ariate F unctional Time Series). Supp ose the factors ( F 1 , . . . , F T ) ⊆ H K follo w the mo del F t ( u ) = N X n =1 Y n t ϕ n ( u ) , where ( ϕ n ) is a complete orthonormal set of functions on H and for each n , the v ector-v alued co eﬃcien ts ( Y n t , t = 1 , . . . , T ) follo w a K -dimensional vector ARIMA mo del of the form Φ n ( L )(1 − L )( Y n t − Y n 0 ) = Θ n ( L ) ϵ n t where Φ n and Θ n are the matrix v alued autoregressiv e and moving av erage p olynomials. Assuming causalit y of the ARIMA pro cesses and writing Ψ n ( L ) := Φ n ( L ) − 1 Θ n ( L ) , the m ultiv ariate Bev eridge- Nelson decomp osition of Y n t is given b y Y n t = Ψ n (1) X s ≤ t ϵ n s + ζ n t , where ( ζ n t ) t is a stationary pro cess given b y ζ n t = − P ∞ k =0 P ∞ i = k +1 Ψ i ϵ n t . No w, let Ψ : H K → H K b e a linear op erator given by Ψ( aϕ n ) := Ψ n (1) aϕ n where a is an arbitrary v ector in R K . The factor pro cess can then b e written as F t ( u ) = Ψ X s ≤ t N X n =1 ϵ n s ϕ n ( u ) + N X n =1 ζ n t ϕ n ( u ) . This representation sho ws that, the multiv ariate functional time series F t ( u ) is also a sp ecial case of the functional factor mo del in ( 1 ). In light of Examples 2.1 – 2.4 , the prop osed functional factor mo del accommo dates a ric h v ariety of cross-sectional and temp oral dep endence structures. 2.3 The Sample Cov ariance Op erator Matrix The analysis of the mo del ( 1 ) and the estimation of factors and factor loadings typically rely on the eigen-analysis of the sample co v ariance op erator 1 T P T t =1 [ X t ⊗ X t ] on H p . In Euclidean space, it is common practice to leverage on the duality b et w een the column space and the ro w space of a matrix and consider the T × T Gram matrix instead. In the context of functional data, this estimation approac h has also b een tak en in existing work [ Benk o et al. , 2009 , Leng et al. , 2026 , T a v ak oli et al. , 2023 ] when estimating factors and factor loadings. W e provide a formal justiﬁcation of this duality in Lemma A.5 in the App endix. 6 Let X ( u ) := 1 T P T t =1 X t ( u ) b e the sample mean. By Lemma A.5 , the sample cov ariance op erator of X has the same non-zero sp ectrum as the Gram matrix b S = ( b S st ) given b y b S st : = 1 p p X i =1 ⟨ X is − X i , X it − X i ⟩ (2) Let M denote the orthogonal projection onto the orthogonal complement of the T -dimensional v ector of ones, i.e. M := I T − T − 1 1 T 1 ′ T . Then w e ma y write b S = 1 p Z U ( X ( u ) − X ( u )) ′ ( X ( u ) − X ( u )) du = 1 p Z U M X ( u ) ′ X ( u ) M du As will b e shown, in cases where spurious b eha viors arise, this Gram matrix is dominated b y the Gram matrix of the non-stationary part of the original data, whic h w e will denote as ˜ S : = 1 p Z U M (Ψ F )( u ) ′ (Ψ F )( u ) M du. (3) W rite Θ for the T × T upp er triangular matrix with en tries equal to 1 so that, based on the random w alk structure on F ( u ) , (Ψ F )( u ) = (Ψ ϵ )( u )Θ = Z I Ψ( u, v ) ϵ ( v )Θ dv . By F ubini’s theorem, the Gram matrix ˜ S can b e written as ˜ S = 1 p M Θ ′ Z I  Z I ϵ ( v ) ′ Ψ( u, v ) ′ dv Z I Ψ( u, w ) ϵ ( w ) dw  du Θ M = 1 p M Θ ′ Z Z I 2 ϵ ( v ) ′  Z I Ψ( u, v ) ′ Ψ( u, w ) du  ϵ ( w ) dvdw Θ M =: 1 p M Θ ′ W Θ M , where W denotes the T × T random matrix given by W := Z Z I 2 ϵ ( v ) ′ Ω( v , w ) ϵ ( w ) dv dw , (4) and Ω( v , w ) is the K × K matrix giv en b y Ω( v , w ) := Z I Ψ( u, v ) ′ Ψ( u, w ) du, v, w ∈ I . (5) Note that the transp ose of Ω is giv en by Ω( v , w ) ′ = R I Ψ( u, w ) ′ Ψ( u, v ) du = Ω( w , v ) , i.e. for an y i, j w e ha ve Ω j i = Ω ∗ ij . Proposition 2.6 b elow shows that Ω is a self-adjoint op erator on H K . In the analysis of the Gram op erator ˜ S , we will often encounter certain comp ositions b et w een matrices of op erators. F or conv enience, we provide a new deﬁnition on the op erator pro duct and state some of its basic prop erties. Deﬁnition 2.5. (A New Op erator-Matrices Pro duct) Let C 1 and C 2 b e b ounded linear op erators on H and Ω = (Ω ij ) b e a K × K matrix of b ounded linear op erators on H for some ﬁxed K . W e deﬁne C 1 Ω C 2 := ( C 1 Ω ij C 2 ) ij to b e the K × K matrix of op erators where C 1 Ω ij C 2 is the usual comp osition b et ween op erators. The op erator C 1 Ω C 2 acts on H K b y ( C 1 Ω C 2 f ) i := K X j =1 C 1 Ω ij C 2 f j , ( f j ) K j =1 ∈ H K . 7 W e ﬁrst state some linear algebraic prop erties for these types of op erators. The pro ofs can b e found in Section A . Prop osition 2.6. L et C 1 and C 2 b e given as in Deﬁnition 2.5 , and Ω is deﬁne d in ( 5 ). Then 1. The adjoint of Ω is given by (Ω ∗ ) ij = Ω ∗ j i , henc e Ω is self-adjoint if and only if Ω ∗ ij = Ω j i . F urthermor e, the adjoint of C 1 Ω C 2 is given by ( C 1 Ω C 2 ) ∗ = C ∗ 2 Ω ∗ C ∗ 1 . 2. Supp ose C 1 Ω ii C 2 is a tr ac e-class op er ator on H for al l i = 1 , . . . , K . Then C 1 Ω C 2 is a tr ac e class op er ator on H K and its tr ac e, denote d as ⟨ C 1 Ω C 2 ⟩ , is given by ⟨ C 1 Ω C 2 ⟩ = K X i =1 ⟨ C 1 Ω ii C 2 ⟩ . 3. Supp ose the op er ator C 1 Ω ij C 2 is Hilb ert-Schmidt on H for al l i, j = 1 , . . . , K . Then the op er ator C 1 Ω C 2 is Hilb ert-Schmidt on H K and ∥ C 1 Ω C 2 ∥ 2 2 := ⟨ C 1 Ω C 2 C ∗ 2 Ω ∗ C ∗ 1 ⟩ = X ij ∥ C 1 Ω ij C 2 ∥ 2 2 . Remark 2.7. W e in tro duce this new form of pro duct to pro vide a con venien t representation of the cov ariance structure for a large collection of functional time series. In typical settings, C 1 and C 2 corresp ond to the co v ariance op erators of the latent factors, whereas Ω captures the functional dep endence present in the factor loadings. The in teraction b et ween these tw o sources of dep endence is describ ed through the pro duct deﬁned ab o v e. 3 Asymptotic Theory for Spurious Phenomenon 3.1 Assumptions The following set of assumptions are made throughout the pap er. Assumption 3.1 (Momen ts Condition) . Assume p → ∞ as T → ∞ . F urthermore, supp ose that 1. The pro cess ϵ := ( ϵ it ) it is a K × T matrix of indep enden t random elemen ts on H with E [ ϵ it ] = 0 and cov ariance op erator C ϵ := E [ ϵ it ⊗ ϵ it ] for all i, t . 2. The fourth momen t of ϵ it is uniformly b ounded in the sense that κ ∗ 4 := sup i,t sup u,v ,α,β E [ ϵ it ( u ) ϵ it ( v ) ϵ it ( α ) ϵ it ( β )] C ϵ ( u, v ) C ϵ ( α, β ) < ∞ . Remark 3.2. The condition p → ∞ places the analysis in a high-dimensional regime, while no restriction is imp osed on the relativ e gro wth rates of p and T . Assumption 3.1 . 1 ensures that the laten t factors ϵ follo w a cov ariance stationary pro cess with K indep enden t comp onen ts. The requirement that the elements of ϵ b e indep enden t across t = 1 , 2 , . . . , T ma y app ear restrictive. How ev er, this assumption can b e readily relaxed to accommo date common time-series structures suc h as ARMA mo dels; see Example 2.4 in Section 2.2 . The presen t 8 form ulation of the factor structure is adopted for con venience, as it clearly distinguishes the I (1) and I (0) comp onen ts of the mo del, which is the main fo cus of this article. Assumption 3.1 . 2 simply ensures that the latent factors ha ve uniformly b ounded 4 th momen t. The app earance of the cov ariance function C ϵ in the denominator takes into accoun t the fact that ϵ it is an elemen t of a Hilb ert space and has a non trivial co v ariance structure. The following assumption on the co v ariance op erators C ϵ and Ω outlines the key condition for the emergence of spurious b eha viours in sample cov ariance-based eigen-analysis. Assumption 3.3 (Eﬀective Rank) . As p, T → ∞ , th e eﬀectiv e rank, based on the op erators C ϵ and Ω , satisﬁes R := ⟨ C 1 / 2 ϵ Ω C 1 / 2 ϵ ⟩ / ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ → ∞ . (6) Remark 3.4. The ratio R is the eﬀectiv e rank of the operator C 1 / 2 ϵ Ω C 1 / 2 ϵ , a quantit y widely used in the high dimensional p robabilit y literature [ Koltc hinskii and Lounici , 2016 , Rudelson and V ershynin , 2010 , W ainwrigh t , 2019 ]. A large eﬀective rank indicates that the op erator cannot b e w ell appro ximated b y lo w-rank op erators in the op erator norm. It is straightforw ard to see that the op erator C := C 1 / 2 ϵ Ω C 1 / 2 ϵ , as deﬁned in Deﬁnition 2.5 , is a self-adjoin t, non-negative deﬁnite op erator on H K . This implies ∥C ∥ 2 ≤ ∥C ∥ 2 2 ≤ ⟨C ⟩∥C ∥ , from whic h we obtain the estimates ⟨C ⟩ / ∥C ∥ 2 ≤ ⟨C ⟩ 2 / ∥C ∥ 2 2 ≤ ⟨C ⟩ 2 / ∥C ∥ 2 . Therefore, Assumption 3.3 is equiv alent to the condition ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 ≪ ⟨ C 1 / 2 ϵ Ω C 1 / 2 ϵ ⟩ . W e no w oﬀer several remarks on the interpretation of Assumption 3.3 . Remark 3.5. In the non-functional settings considered by Onatski and W ang [ 2021 ] and He and Zhang [ 2023 ], the eﬀective rank R is b ounded b elo w b y the num b er of non-stationary factors K . Consequen tly , condition ( 6 ) is ensured under the stronger but in terpretable assumption that K → ∞ . This is also the framework within which Onatski and W ang [ 2021 ] and He and Zhang [ 2024 ] develop the asymptotic theory of spurious phenomena. W e w ork directly under the in trinsic condition ( 6 ) in the functional setting. One reason is that the lo wer b ound R ≳ K no longer holds due to the presence of the operator C ϵ . In fact, condition ( 6 ) reﬂects b oth the cross-sectional and the functional dep endence structures in the data; see Section 3.3 for further discussion. This represen ts a k ey distinction betw een the presen t pap er and the existing literature. W e show that mo dels with K ∼ p factors may exhibit no spurious b eha viour, while mo dels with only a few factors ma y displa y pronounced spurious eﬀects. Consequen tly , even mo dels with a small num b er of genuine non-stationary factors, eigen-analysis based on the sample co v ariance can pro duce spurious results in the functional setting. Extra caution is therefore required when applying eigen-analysis to p oten tially non-stationary functional time series. It is worth noting that, in Section 3.3 , we deriv e more in terpretable results by establishing upp er and low er b ounds for R in common time series mo dels. These results provide a clearer insigh t into when spurious b eha viour arises and how it relates to K . Finally , the following assumption gives a lenient b ound on the size of the error time series ζ . Assumption 3.6 (Error Comp onen ts) . Assume that the error term ( ζ t , t = 1 , . . . , T ) is a weakly stationary time series on H p with mean zero and cov ariance op erator C ζ = E [ ζ t ⊗ ζ t ] ∈ H p satisfying ⟨ C ζ ⟩ ≪ T ⟨ C ϵ Ω ⟩ , T → ∞ . (7) 9 Remark 3.7. Condition ( 7 ) ensures that ζ is negligible asymptotically compared to the non- stationary component of the mo del. T o illustrate the scop e of this condition, supp ose that the comp onen ts ζ 1 t , . . . , ζ pt of ζ t ha ve uniformly b ounded co v ariance op erators, in whic h case condition ( 7 ) is equiv alent to ⟨ C ϵ Ω ⟩ ≫ pT − 1 . Dra wing a parallel with the high dimensional factor mo delling literature, it is natural to assume that the strength of the factors increases with p , i.e. ∥ Ω ∥ ∼ p 1 − δ for some δ ∈ [0 , 1] [see Lam and Y ao , 2012 , F an et al. , 2021 ]. Since C ϵ is a trace class op erator, condition ( 7 ) reduces to p δ T − 1 = o (1) , which clearly holds for a wide range of high-dimensional factor mo dels. In particular, condition ( 7 ) is trivially satisﬁed when the factors are “strong” in the sense of δ = 0 . 3.2 Asymptotic Theory W e are now ready to state the main result of our w ork. Theorem 3.8 (Asymptotic Sp ectral Limits) . L et ( b λ t ) T t =1 b e the eigenvalues of the Gr am matrix b S arr ange d in non-asc ending or der and ( b u n ) b e the c orr esp onding eigenve ctors. Supp ose A ssumptions 3.1 - 3.6 hold, then for any ﬁxe d natur al numb er k , as p, T → ∞ , we have (i) the sample eigenve ctor b u k satisﬁes |⟨ b u k , d k ⟩| = 1 + o p (1) wher e d k ∈ R T with the t -th c o or dinate e qual to d kt = p 2 /T cos( π k t/T ) ; (ii) the sample eigenvalue b λ k satisﬁes b λ k = T 2 k 2 π 2 p ⟨ C ϵ Ω ⟩ (1 + o p (1)); (iii) the p er c entage of varianc e explaine d by b λ k satisﬁes b λ k P j b λ j = 6 ( k π ) 2 + o p (1) . Remark 3.9 (Spurious Eigen v alues) . Theorem 3.8 establishes the asymptotic limits of the sample eigen v alues and eigenv ectors as T → ∞ . P art ( ii ) sho ws that the largest sample eigen v alues b λ k con verge to deterministic limits. Sev eral observ ations follow from this result. First, the true co v ariance structure of the mo del, c haracterized by C ϵ and Ω , enters the limit of b λ k only through a tracial quantit y . In particular, the limit of b λ k dep ends on the index k solely through the factor k − 2 and is not related to the k -th eigenv alue of C 1 / 2 ϵ Ω C 1 / 2 ϵ . Consequen tly , the sample eigenv alues b λ k fail to capture meaningful information ab out the cross-sectional dep endence structure of the mo del. Second, for an y ﬁxed k , the empirical eigenv alue b λ k is of the order T 2 p − 1 ⟨ C ϵ Ω ⟩ , regardless of the actual num b er of factors present in the mo del. In particular, although the empirical eigenv alues app ear to decay rapidly at the rate k − 2 , an y inference on the n um b er of factors based on these eigen v alues would b e misleading. Finally , recall that in the factor-mo deling literature the term strong factors typically refers to the regime | C ϵ Ω | ∼ p . Ho wev er, even under this assumption, the leading empirical eigenv alue b λ k 10 remains of order T 2 , which is substan tially larger than the strength of the factors. This indicates that the non-stationarit y in the mo del dominates the p opulation co v ariance structure, even in the presence of strong factors. P art ( iii ) of the theorem pro vides the asymptotic limits of the prop ortion of v ariance explained b y eac h sample eigen v alue. These limits are deterministic and indep enden t of the true factor structure, represen ted by C ϵ and Ω . Consequen tly , the scree plot carries no useful information ab out factor strength, and an y inference based on it w ould b e unreliable. Remark 3.10 (Spurious Eigenfunctions) . Part ( i ) of the theorem states that the estimated eigen vectors b u k , which are used as factor estimates in certain mo dels [see T a v ak oli et al. , 2023 , Leng et al. , 2026 ], con verge in probability to a deterministic trigonometric function d k . Imp ortan tly , this limit dep ends only on the index k and is indep enden t of the p opulation cov ariance structure of the mo del. Cons equen tly , factors estimated in this manner are spurious, as they b ear no relation to the true factors ϵ or to the factor loadings Ψ . The app earance of trigonometric functions in the limit is not surprising in this conte xt. These functions coincide with the eigenfunctions arising in the Karhunen–Loève expansion of a demeaned standard Wiener pro cess, which itself app ears as the limit of a suitably scaled random w alk. As discussed abov e, in the spurious regime the co v ariance structure is dominated b y the non-stationarit y of the mo del rather than b y the cross-sectional dep endence. As a result, the estimated factors eﬀectiv ely reco ver the eigenfunctions of the Wiener pro cess instead of the true latent factors. 3.3 Separable Eﬀects from F actors and F actor Loadings Assumption 3.3 pro vides a suﬃcient condition for the emergence of the spurious limit, in a similar sense to that considered in the preceding works of Onatski and W ang [ 2021 ] and He and Zhang [ 2024 ]. The key quan tity is the eﬀective rank ⟨C ⟩ / ||C || 2 , deﬁned as the ratio b et w een the trace and the Hilb ert–Sc hmidt norm of the op erator C := C 1 / 2 ϵ Ω C 1 / 2 ϵ . This quantit y reﬂects the structures of b oth common factors and factor loadings in the mo del. Under some regular assumptions, w e further study concrete b ounds for the eﬀective rank R . This, in turn, enables us to form ulate more interpretable conditions for Theorem 3.8 . In details, b y Mercer’s theorem, the op erator C ϵ has the sp ectral decomp osition giv en b y C ϵ = ∞ X n =1 c n ϕ n ⊗ ϕ n (8) where ( ϕ k ) k is a complete orthonormal basis for H and ( c n ) n is a non-negative sequence in ℓ 1 . Supp ose that the factor loading op erators (Ψ ik ) are giv en b y Ψ ik = ∞ X n =1 a nik ϕ n ⊗ ϕ n (9) for some collection of real num b ers ( a nik ) nik satisfying sup n P i,k a 2 nik < ∞ . F or n ∈ N , deﬁne the matrices A n = ( a nik ) ik and B n = A n A ′ n . Essen tially , c n B n is the co v ariance matrix of the factor part of the mo del Ψ ϵ t pro jected comp onen t-wise to the one dimensional closed linear subspace of H spanned by the function ϕ n . Under this sp eciﬁc structure ( 8 ) and ( 9 ) for common factors and factor loadings, resp ectiv ely , w e can formulate the following t wo-sided estimates on the eﬀective rank R := ⟨C ⟩ / ∥C ∥ 2 . 11 Theorem 3.11 (Separable Bounds on Eﬀective Rank) . L et C ϵ and Ψ b e given as in ( 8 ) and ( 9 ) . Supp ose that the matric es ( B n , n ∈ N ) satisfy ∥ B n ∥ 2 ≲ ∥ B m ∥ 2 uniformly in n, m ∈ N . Then we have the upp er b ound R := ⟨C ⟩ ∥C ∥ 2 ≲ 1 ∥ C ϵ ∥ 2 sup n ∈ N ⟨ B n ⟩ ∥ B n ∥ 2 ≤ 1 ∥ C ϵ ∥ 2 sup n ∈ N ∥ B n ∥ 1 / 2 0 , (10) and the lower b ound R := ⟨C ⟩ ∥C ∥ 2 ≳ 1 ∥ C ϵ ∥ 2  1 + sup n c n ⟨ B n ⟩ ∥ B n ∥ 2  ≳ 1 ∥ C ϵ ∥ 2  1 + sup n c n α ( B n ) 1 / 2 ∥ B n ∥ 1 / 2 0  , (11) wher e α ( B n ) denotes the r atio b etwe en the le ast non-zer o eigenvalues and the lar gest eigenvalue of B n . The omitte d c onstants in the ab ove estimates ar e al l uniform in T . Remark 3.12. Theorem 3.11 derives upp er and low er b ounds for the eﬀective rank R , allowing the contributions of the factor loadings, captured by the sequence of matrices ( B n , n ∈ N ) , to b e distinguished from those of the common factors, represented b y the op erator C ϵ . The term ⟨ B n ⟩ 2 / ∥ B n ∥ 2 2 is analogous to the ratio in Assumption A3 of Onatski and W ang [ 2021 ]. It is b ounded ab ov e b y rank ( B n ) and b elow by rank ( B n ) scaled by the eigenv alue gap α ( B n ) . In tuitively , this quan tity measures the eﬀectiv e rank of the pro jected data on to the direction ϕ n . The quantit y ∥ C ϵ ∥ − 1 2 can b e interpreted as an eﬀective rank of C ϵ , reﬂecting the deca y rate of the eigen v alues { c n } . When the eigenv alues decay rapidly , this quan tity is close to 1 , whereas slo w decay leads it to approach the actual rank of C ϵ , which may b e inﬁnite. F or example, if c n = q − 1 1 n ≤ q for some ﬁxed q > 0 , then ∥ C ϵ ∥ − 1 2 = √ q , equal to the square ro ot of the rank of C ϵ . In contrast, if c n = 2 − n for all n , then ∥ C ϵ ∥ − 1 2 = √ 3 even though the rank of C ϵ is inﬁnite. T o facilitate the discussion, it is helpful to deﬁne the following t wo diﬀeren t regimes for C ϵ . Deﬁnition 3.13 (Lo calization and Delo calization) . The op erator C ϵ is said to b e “lo calized” if ∥ C ϵ ∥ 2 is uniformly b ounded a wa y from zero as T → ∞ , and “delo calized” if ∥ C ϵ ∥ 2 → 0 as T → ∞ . Remark 3.14. The terminology we used here is partly taken from the random matrix literature, see for example [ Johnstone and P aul , 2018 , R udelson and V ershynin , 2015 ]. The sample eigen vectors of large dimensional random matrices t ypically exhibit tw o t yp es of asymptotic b eha viours dep ending on the true eigen v alues and the presence or the lack of spikes. In the de-lo calized regime, the sample eigen vector tends to a limit where the entries are all of a similar size, muc h similar to our condition ∥ C ϵ ∥ 2 → 0 . The delo calized regime is when the factors are generated b y an increasing num b er of basis functions with comparable w eights as p → ∞ . A simple y et very general example of this can b e seen b y setting c n := ( aT ) − δ 1 n ≤ ( aT ) δ for some ﬁxed a > 0 and δ ∈ (0 , 1] in the sp ectral decomp osition of C ϵ . The localized regime refers to when the factors are generated by an arbitrary num b er of basis functions, but the weigh ts ( c n ) rapidly deca y to zero. In this setting ∥ C ϵ ∥ − 1 2 is b ounded even though the rank of C ϵ can b e inﬁnite. As previously discussed in Section 3.1 , the suﬃcien t condition for the main theorem of Onatski and W ang [ 2021 ] implies that the n umber of sto c hastic trends K in the data must diverge as T → ∞ . As a consequence, their main theorem necessarily exclude mo dels with a ﬁnite num ber of strong non-stationary factors. This observ ation is analogous to the upp er b ounds in ( 10 ) in our case. In particular, under the lo calized regime deﬁned ab o v e, condition ( 6 ) still implies K → ∞ . Ho w ever, this is no longer the case under the delo calized regime. Condition ( 6 ) can in fact b e satisﬁed by mo dels with a ﬁnite num b er of strong factors, as long as those factors are generated b y an increasing 12 n umber of basis functions with comparable weigh ts as T → ∞ . This is an imp ortan t distinction since it implies that functional time series with few strong non-stationary factors can exhibit spurious b eha viour as w ell. W e will also illustrate this through simulations in Section 4 . F or ease of referencing we summarize the ab o v e discussion into the following theorem. Theorem 3.15. Under the assumptions of The or em 3.11 , the spurious c ondition in ( 6 ) holds if either of the fol lowing c onditions is satisﬁe d Condition (a). C ϵ is delo c alize d; Condition (b). c n ⟨ B n ⟩ / ∥ B n ∥ 2 diver ges for some n . Remark 3.16. In fact, from ( 11 ) w e can conclude that condition ( 6 ) holds in the d elocalized regime when all the B n ’s are comparable in ∥·∥ 2 , regardless of their actual rank. This is a signiﬁcan t departure from the conclusions in Onatski and W ang [ 2021 ] for the non-functional case, since it sho ws concretely that a mo del with single ( K = 1 ) integrated functional factor can indeed tend to the spurious limit. In the lo calized regime, a suﬃcient condition for ( 6 ) is the divergence of the quantit y c n ⟨ B n ⟩ / ∥ B n ∥ for some n . By the second low er b ound in ( 11 ) , this is satisﬁed whenev er the rank of B n div erges as T → ∞ , and the quantities c n and α ( B n ) do not deca y to o fast. This concretely sho ws that a mo del with large eﬀective rank in some direction ϕ n with non-diminishing weigh ts tends to the spurious limit, which is in line with the ﬁndings of Onatski and W ang [ 2021 ]. 4 Sim ulation Study 4.1 Setup W e start with a description of the data generating pro cess. Recall from ( 1 ) the mo del X it ( u ) = t X s =1 K X k =1 (Ψ ik ϵ ks )( u ) + ζ it ( u ) , i = 1 , . . . , p, t = 1 , . . . , T , where ( ϵ kt ) kt are indep enden t random functions on H with mean zero and cov ariance op erator C ϵ and ( ζ it ) it is a weakly stationary functional times series on H p with zero mean and cov ariance C ζ . F or simplicit y , we fo cus on the settings discussed in Section 3.3 and Theorem 3.11 . Fix q > 0 and let ( ϕ n ) q n =1 b e a set of F ourier basis functions on H . W e set C ϵ = ∞ X n =1 c n ϕ n ⊗ ϕ n , (12) where c n is a sequence of non-negative real n umbers summing to one, to b e sp eciﬁed b elo w. Similar to Leng et al. [ 2026 ] and Guo et al. [ 2026 ], w e generate the factor pro cess ϵ b y setting ϵ kt ( u ) := q X n =1 Z n kt ϕ n ( u ) , where ( Z n kt ) is a collection of i.i.d. Gaussian v ariables with mean zero and v ariance v ar ( Z n kt ) = c n for all k , t . F or the n oise time series ζ , w e simply set ζ it ( u ) := q X n =1 W n it ϕ n ( u ) , 13 where ( W n it ) is a collection of i.i.d. Gaussian v ariables with mean zero and v ariance equal to v ar( W n it ) = 2 − n for all i, t . F ollo wing ( 9 ), w e generate the factor loadings { Ψ ik } by setting Ψ ik ( u, v ) = q X n =1 a n ik ϕ n ( u ) ϕ n ( v ) , (13) where for eac h n = 1 , . . . , q , the p × K matrix A n := { a n ik } i,k is to b e sp eciﬁed b elo w. F or the mo del parameters, we set T = 200 , p = 100 and q = 20 . Recall from Assumption 3.3 and Theorem 3.8 that the sample co v ariance structure tends to a spurious limit whenever the ratio R = ⟨ C 1 / 2 ϵ Ω C 1 / 2 ϵ ⟩ 2 ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 2 . div erges as p → ∞ . F rom Theorem 3.11 , it is clear that whether R div erges is determined by the parameters K , the eigenv alues ( c n ) n in ( 12 ) and the matrices ( A n ) n in ( 13 ) . W e will p erform sim ulations with v arious choices of these parameters describ ed b elo w. F or the n umber of factors w e set K ∈ { 50 , 10 , 2 } . In He and Zhang [ 2023 ] and Onatski and W ang [ 2021 ], the K = 2 setting represents as a mo del with genuine factors and is shown to b e far from the spurious limit, while the K = 50 setting represents a mo del with no genuine factors for which spurious b eha viours o ccur. W e will see that this is not necessarily the case with functional data. F or the loading matrices ( A n , n ∈ N ) , we will consider the high rank setting where A n ∼ G p,K (14) indep enden tly for all n ∈ N , where G p,K denotes a p × K matrix with i.i.d. standard Gaussian random v ariables, and the low-eﬀectiv e rank setting where A n ∼ p 1 / 2 U p · diag ( { 2 − n/ 2 } K n =1 , 0 , . . . , 0) · U ′ K (15) indep enden tly for all n ∈ N , with U N denoting an N × N random orthogonal matrix uniformly sampled from S N − 1 for N ∈ { p, K } . In the high-rank setting ( 14 ), each A n has full column rank with high probability , while in the low-eﬀectiv e rank setting ( 15 ), each A n has low eﬀective rank with ⟨ A n A ′ n ⟩ 2 ∥ A n A ′ n ∥ 2 2 = 3 (1 − 2 − K ) 2 1 − 4 − K ≈ 3 ≪ K. F or the cov ariance op erator C ϵ , we consider the follo wing three scenarios b y setting Delo calized : c n = q − 1 1 n ≤ q , (16) Lo calized : c n = 2 − n 1 n ≤ q , (17) Lo calized : c n = 2 − 1 1 n ≤ 2 . (18) In the ﬁrst setting ( 16 ), each basis function ϕ n has equal weigh t and we ha ve ∥ C ϵ ∥ 2 2 = q − 1 = 0 . 05 whic h is v ery close to zero. This represent the scenario where the op erator C ϵ is delocalized as deﬁned in Deﬁnition 3.13 . In the second setting ( 17 ), the eigenv alues rapidly decay to zero and we ha ve ∥ C ϵ ∥ 2 2 = 1 − 4 − q 3 ≈ 1 / 3 . This represen ts the situation where C ϵ is lo calized. Finally , a more extreme version of this is found in the last setting ( 18 ), where C ϵ is completely lo calized on tw o basis functionsand w e ha ve ∥ C ϵ ∥ 2 2 = 0 . 5 . 14 T able 1: Summary of Simulation Settings c n ∥ C ϵ ∥ 2 2 A n ⟨ B n ⟩ 2 / ∥ B n ∥ 2 2 R Lo calized Setting 1 q − 1 1 n ≤ q q − 1 full rank ∼ K ∼ √ q K Setting 2 q − 1 1 n ≤ q q − 1 lo w rank ≈ 3 ∼ √ q Delo calized Setting 3 2 − n 1 n ≤ q 1 3 (1 − 4 − q ) full rank ∼ K ∼ √ K Setting 4 2 − n 1 n ≤ q 1 3 (1 − 4 − q ) lo w rank ≈ 3 ∼ 1 Setting 5 2 − 1 1 n ≤ 2 0 . 5 full rank ∼ K ∼ √ K Setting 6 2 − 1 1 n ≤ 2 0 . 5 lo w rank ≈ 3 ∼ 1 In the simulations, w e consider com binations of the tw o settings for A and the three settings for C ϵ describ ed ab o v e. T ab le 1 summarizes these combinations of choices for A n and C ϵ . W e also list the asymptotic orders of the eﬀective ranks of eac h setting, which are obtained from Theorem 2. F or eac h setting in T able 1 , w e simulate X from the ab ov e data generating pro cess with K ∈ { 50 , 10 , 2 } and compute the leading eigenv alues and eigenv ectors of the sample Gram matrix deﬁned in ( 2 ) . W e plot the top ﬁve sample eigenv ectors for K ∈ { 50 , 10 , 2 } (red, green and blue lines resp ectiv ely) against the spurious limit (black line) as sho wn in Theorem 3.8 . W e also plot the prop ortion of v ariance explained b y eac h eigen v alue with the same color co ding. 4.2 Results W e start with the setting that shows ob vious spurious behaviors. Figure 1 plots the sample eigen vectors of Setting 1 where C ϵ is in the delo calized regime and A n is of high rank. It can b e seen that all sample eigenv ectors as w ell as the sample eigenv alues closely resemble the theoretical spurious limits. Figure 1: Setting 1 - C ϵ delo calized, loading matrices are of full-column rank This b eha viour is in agreement with our theoretical results, since Theorem 3.15 states that spurious limits m ust app ear whenever C ϵ is delo calized. The results under Setting 2 (i.e. C ϵ delo calized with A n ha ving lo w-eﬀective rank) are visually v ery similar to Figure 1 under Setting 1 and are hence omitted. Setting 2 is designed to sho w that these limits may arise even in the presence 15 of only K = 2 factors and when A n p ossesses low-eﬀectiv e rank, representing a clear departure from the ﬁndings of He and Zhang [ 2024 ] and Onatski and W ang [ 2021 ]. This underscores a key diﬀerence b et w een HDFTS and HDTS, namely that the proliferation of basis functions in functional represen tations can itself in tro duce an additional source of dimensionality . Figure 2 plots the results for Setting 3 where the eigenv alues of C ϵ are fast deca ying and the rank of eac h A n is equal to K . As can b e seen from Figure 2 , the sample eigenstructure in Setting 3 closely resem bles the spurious limit for K = 50 , how ever, as K decreases, visible deviations from the spurious limit can b e observ ed. This is to b e exp ected, since for Setting 3, the ﬁrst upp er b ound in ( 10 ) is b ounded f rom ab o v e b y a m ultiple of K . By Theorem 3.11 this implies that condition ( 6 ) is not satisﬁed when K is ﬁnite. The results for this setting is in line with the ﬁndings of Onatski and W ang [ 2021 ] and He and Zhang [ 2024 ], where a small v alue of K represen ts the presence of genuine factors. Figure 2: Setting 3 - C ϵ lo calized, loading matrices are of full-column rank Figure 3 plots the results for Setting 4 where C ϵ has decaying eigenv alues and A n has low-eﬀectiv e rank. It can b e seen that the sample eigenstructure b egins to visibly deviate from the spurious limit, regardless of the v alue of K . Similar to Setting 3, this can b e explained by the upp er b ounds in ( 10 ) and Theorem 3.11 . Note that in particular, having divergen t K do es not seem to guarantee the presence of the spurious limit. Figures 4 and 5 show the results for Settings 5 and 6, where C ϵ is completely lo calized on t wo basis functions, and A n has full rank and lo w-eﬀective rank resp ectiv ely . In Figure 4 , the case K = 2 signiﬁcantly deviates from the spurious limit. This eﬀect is similar to Setting 3 but is visibly more pronounced, since the eﬀective rank of C ϵ , which is an imp ortan t quan tity due to Theorem 3.11 , is even smaller in Setting 5. In Setting 6, the sample eigen-structures all deviate from the spurious limit regardless of the v alue K , similar to Setting 4. The deviations are more visibly pronounced compared to Setting 4 due to the decrease in the rank of A n . 5 Empirical applications W e consider tw o age-sp eciﬁc mortalit y rate datasets as an empirical illustration of our results. Our ﬁrst data set contains age-sp eciﬁc and gender-sp eciﬁc mortality rates obtained from the Human Mortalit y Database. The dataset consists of mortality rates for p = 32 countries observed from 1960 16 Figure 3: Setting 4 - C ϵ lo calized, loading matrices are of lo w-eﬀective rank Figure 4: Setting 5 - C ϵ is of rank 2, loading matrices are of full-column rank to 2013 ( T = 54 ). Since exp osures and death counts are sparse at high ages, w e aggregate data with age ov er 100 into a single observ ation. The ra w data is transformed in to functional data via smo othed using the demography pac kage in R and the logarithm of the mortalit y rates are computed. This dataset w as considered in T ang et al. [ 2022 ] where a no vel clustering algorithm is prop osed based on functional principal comp onen t analysis. F rom the smo othed log mortalit y rates, we computed the sample cov ariance matrix as deﬁned in ( 2 ) and computed its eigenv alues and eigenv ectors. Figure 6 plots the ﬁrst four eigen vectors against the spurious limits as found in Theorem 3.8 . As can b e seen, the sample eigenv ectors closely resem ble the trigonometric curv es in the spurious limit. This suggests that the mortalit y rates are lik ely non-stationary and that the n umber of non-stationary comp onen ts in the data, as measured b y the eﬀective rank R ∼ √ q K of the mo del, ma y b e large. A large v alue of R implies three p ossible scenarios: Case 1 (large q and small K ): a large n umber of common factors ( K ) are shared across lo cations; Case 2 (large K and small q ): a large num ber of common factors ( q ) contribute to the functional v ariation; and Case 3 (mo derately large q and K ): both the cross-lo cation and functional 17 Figure 5: Setting 6 - C ϵ is of rank 2, loading matrices are of lo w-eﬀective rank Figure 6: Sample eigen vectors of the cov ariance matrix of the male, female and total age-sp eciﬁc mortalit y rates of 32 countries from 1960 to 2013 18 dimension are driven by a relatively large n umber of common factors. Case 1 is consistent with ﬁndings in the mortality literature [ Gao et al. , 2019 , Zhang et al. , 2025 , Guo et al. , 2026 ], which suggest that only a small num b er of common factors exist across lo cations. Cases 2 and 3 represent new ﬁndings for multi-location mortality data. This raises concerns ab out the v alidity of principal comp onen t analysis based metho ds when analyzing this dataset without diﬀerencing. F or comparison, we also consider the Japanese sub-national mortality rates observ ed for p = 47 prefectures ov er 1975-2022 ( T = 48 ). Since the mortality rates are recorded from diﬀerent prefectures in the same country , it is reasonable to exp ect a higher lev el of homogeneity within the across the 47 cov ariates. If the cross-sectional dep endence (factor strength) is strong enough, i.e. if the mo del has low eﬀectiv e rank in the sense of condition ( 6 ), it is p ossible to observe gen uine factors present in the mo del instead of the spurious limit. Figure 7: Sample eigen vectors of the cov ariance matrix of the male, female and total mortality rates of 47 Japanese prefectures from 1975 to 2022 Similar to the multi-coun try data set men tioned abov e, w e aggregate ages ab ov e 100 b efore smo othing and taking the logarithm. Figure 7 shows the sample eigen v ectors of the cov ariance matrix deﬁned in ( 2 ) plotted against the spurious limits of Theorem 3.8 . The eigenv ectors still resem ble the spurious limits, but the eﬀects are not as visually pronounced as in the m ulti-country dataset. The second and third sample eigenv ectors show signiﬁcant deviation from the trigonometric limits at the year 2011. A p ossible explanation is the T ohoku Earthquak e and T sunami that to ok 19 place in March of 2011 whic h results in nearly 20,000 deaths [ Nakahara and Ic hikaw a , 2013 ]. In this case, the eigen vectors could b e thought of as genuine factors that con tains relev ant information on the dataset and instead of b eing lab eled as spurious. Nevertheless, the o verall resem blance b et w een the sample eigenv ectors and the trigonometric limits suggests that additional care should b e tak en when dealing with the non-stationary nature of the dataset. 6 Conclusion and F uture W ork W e study the asymptotic sp ectral b eha viour of high-dimensional functional factor mo dels with non-stationary factors. Under a rank-type condition on the co v ariance op erator, we sho w that con ven tional functional principal comp onent analysis can pro duce spurious results, extending Onatski and W ang [ 2021 ] to the functional setting. In particular, the sample eigen v alues and eigenfunctions con verge to limits unrelated to the true cov ariance structure, so that leading comp onen ts ma y explain a large prop ortion of the v ariation ev en in the absence of gen uine factors. Moreo ver, due to the intrinsic high dimensionalit y of functional data, such spurious b eha viour may arise ev en with a single non-stationary factor, in contrast to Onatski and W ang [ 2021 ] and He and Zhang [ 2024 ]. Sim ulation studies illustrate the theory and iden tify regimes in whic h spurious b eha viour o ccurs. An empirical analysis of t wo m ultiv ariate age-sp eciﬁc mortality datasets further demonstrates the phenomenon under standard functional PCA. Our simulations reveal asymptotic features of the sample eigen vectors that are not explained by existing theory . F ormalizing these observ ations and establishing the asso ciated conjecture is left for future work, whic h w ould pro vide practical to ols for detecting spurious b ehaviour and for identifying k ey prop erties of common factors, including their eﬀective dimensionality . Another direction is to study the asymptotic distribution of the sample eigenv alues, in the spirit of He and Zhang [ 2024 ]. Suc h results w ould enable formal inference for testing the presence of high-rank comp onen ts and assessing the emergence of spurious b eha viour in functional factor mo dels. 20 Supplemen tary Material to “A ttribution of Spurious F actors from High-Dimensional F unctional Time Series” This material consists of t wo parts: the ﬁrst part provides pro ofs of theoretical results in the main pap er; and the second part prop oses a new conjecture on empirical eigenv ectors. A Pro ofs A.1 Preliminary results W e ﬁrst give the pro of to Prop osition 2.6 . of Pr op osition 2.6 . F or f = ( f i ) and g = ( g i ) in H K , we ha ve ⟨ Ω f , g ⟩ = X i ⟨ X j Ω ij f j , g i ⟩ = X j ⟨ f j , X i Ω ∗ ij g i ⟩ = ⟨ f , (Ω ∗ j i ) ij g ⟩ . Similarly , w e ha ve ⟨ C 1 Ω C 2 f , g ⟩ = X i ⟨ C 1 X j Ω ij C 2 f j , g i ⟩ = X j ⟨ f j , X i C ∗ 2 Ω ∗ ij C ∗ 1 g i ⟩ whic h sho ws ( C 1 Ω C 2 ) ∗ = C ∗ 2 Ω ∗ C ∗ . Let ( e n ) n b e an orthonormal basis for H . Let ( ˜ e n ) n b e a set of vectors in H K giv en b y ˜ e ( n − 1) K + m := (0 , . . . , e n , . . . , 0) where n = 1 , . . . , m = 1 , . . . , K and e n app ears in the m -th co- ordinate. Clearly ( ˜ e n ) is an orthonormal basis of H K . Then w e ha ve X n ⟨ C 1 Ω C 2 ˜ e n , ˜ e n ⟩ = X n X ij ⟨ C 1 Ω ij C 2 ˜ e j n , ˜ e i n ⟩ , where ˜ e i n denotes the i th co-ordinate of ˜ e n . By deﬁnition, at least one of of ˜ e i n and ˜ e j n is 0 whenever i  = j . Therefore X n ⟨ C 1 Ω C 2 ˜ e n , ˜ e n ⟩ = X i X n ⟨ C 1 Ω ii C 2 e n , e n ⟩ = X i ⟨ C 1 Ω ii C 2 ⟩ , whic h giv es the trace of C Ω . Finally , w e h a v e ∥ C 1 Ω C 2 ∥ 2 F = ⟨ C 1 Ω C 2 C ∗ 2 Ω ∗ C ∗ 1 ⟩ = X i ⟨ C 1 (Ω C 2 C ∗ 2 Ω ∗ ) ii C ∗ 1 ⟩ , where the last equality follows from the second claim of this Prop osition. Since (Ω ∗ ) j i = Ω ∗ ij b y part 1 of this Prop osition, we ha ve ∥ C 1 Ω C 2 ∥ 2 F = X ij ⟨ C 1 Ω ij C 2 C ∗ 2 (Ω ∗ ) j i C ∗ 1 ⟩ = X ij ⟨ C 1 Ω ij C 2 C ∗ 2 Ω ∗ ij C ∗ 1 ⟩ = X ij ∥ C 1 Ω ij C 2 ∥ 2 F where ∥ C 1 Ω ij C 2 ∥ F denotes the Hilb ert-Sc hmidt norm of C 1 Ω ij C 2 on H . The follo wing lemma is the k ey technical result underpinning Theorem 3.8 . It computes moments of certain bilinear forms asso ciated with the random matrix W app earing in ( 4 ). 21 Lemma A.1. L et W b e the T × T r andom matrix deﬁne d in ( 4 ) . Under A ssumption 1 , for any deterministic ve ctors a, b ∈ R T , we have E [ a ′ W b ] = ⟨ a, b ⟩⟨ C ϵ Ω ⟩ , (19) F urthermor e, for any deterministic ve ctors a, b, c, d ∈ R T , we have    co v( a ′ W b, c ′ W d ) − ⟨ a, c ⟩⟨ b, d ⟩∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 F −⟨ a, d ⟩⟨ b, c ⟩∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 F    ≤ 2 κ ∗ 4 X t a t b t c t d t ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 F , wher e a t , b t , c t and d t ar e the t -th c o or dinate of a, b, c and d r esp e ctively. Pr o of. F rom the deﬁnition of W we may write a ′ W b = K X i,j =1 Z Z ⟨ a, ϵ i · ( v ) ⟩⟨ b, ϵ j · ( v ) ⟩ Ω ij ( v , w ) dv dw , where ϵ i · ( v ) ∈ R T is the i -th column of the matrix ϵ ′ ( v ) . Since ( ϵ it ) it is a matrix of uncorrelated elemen ts b y assumptions, we ha ve E [ a ′ W b ] = K X i,j =1 Z Z T X s,t =1 E [ a s ϵ is ( v ) b t ϵ j t ( w )]Ω ij ( v , w ) dv dw = T X t =1 a t b t K X i =1 Z Z E [ ϵ it ( v ) ϵ it ( w )]Ω ii ( v , w ) dv dw (20) = ⟨ a, b ⟩ K X i =1 Z Z C ϵ ( v , w )Ω ii ( v , w ) dv dw . By Prop osition 2.6 w e obtain E [ a ′ W b ] = K X i =1 ⟨ C ϵ Ω ii ⟩ = ⟨ a, b ⟩⟨ C ϵ Ω ⟩ . This gives the equality in ( 19 ). F or second momen ts, w e write E [ a ′ W b, c ′ W d ] = K X i,j,k,l =1 Z Z Z Z A ij kl ( v , w, α , β )Ω ij ( v , w )Ω kl ( α, β ) dv dw dαdβ , where we ha ve deﬁned A ij kl ( v , w, α , β ) : = E [ ⟨ a, ϵ i · ( v ) ⟩⟨ b, ϵ j · ( w ) ⟩⟨ c, ϵ k · ( α ) ⟩⟨ d, ϵ l · ( β ) ⟩ ] . W e will ﬁrst compute A ij kl and P ij kl Ω ij ( v , w )Ω kl ( α, β ) for all the p ossible partitions of index set { i, j, k , l } . F or ease of notation w e will write ˜ Ω := X i Ω ii ( v , w )Ω ii ( α, β ) . 22 On the set { i = j  = k = l } , w e hav e A iikk ( v , w, α , β ) = E [ ⟨ a, ϵ i · ( v ) ⟩⟨ b, ϵ i · ( w ) ⟩ ] E [ ⟨ c, ϵ k · ( α ) ⟩⟨ d, ϵ k · ( β ) ⟩ ] = ⟨ a, b ⟩⟨ c, d ⟩ C ϵ ( v , w ) C ϵ ( α, β ) , where the last line follo ws from similar computations as in ( 20 ), and X i = j  = k = l Ω ij ( v , w )Ω kl ( α, β ) = X i  = k Ω ii ( v , w )Ω kk ( α, β ) = tr Ω( v , w )tr Ω( α, β ) − ˜ Ω . Similarly , on the set { i = k  = j = l } , w e ha ve A ij ij ( v , w, α , β ) = E [ ⟨ a, ϵ i · ( v ) ⟩⟨ c, ϵ i · ( α ) ⟩⟩ ] E [ ⟨ b, ϵ j · ( w ) ⟩⟨ d, ϵ j · ( β ) ⟩ ] = ⟨ a, c ⟩⟨ b, d ⟩ C ϵ ( v , α ) C ϵ ( w , β ) , and X i = k  = j = l Ω ij ( v , w )Ω kl ( α, β ) = X i  = j Ω ij ( v , w )Ω ij ( α, β ) = tr Ω( v , w ) ′ Ω( α, β ) − ˜ Ω . On the set { i = l  = j = k } , w e ha ve A ij j i ( v , w, α , β ) = E [ ⟨ a, ϵ i · ( v ) ⟩⟨ d, ϵ i · ( β ) ⟩⟩ ] E [ ⟨ b, ϵ j · ( w ) ⟩⟨ c, ϵ j · ( α ) ⟩ ] = ⟨ a, d ⟩⟨ b, c ⟩ C ϵ ( v , β ) C ϵ ( w , α ) , and X i = l  = j = k Ω ij ( v , w )Ω kl ( α, β ) = X i  = j Ω ij ( v , w )Ω j i ( α, β ) = tr Ω( v , w )Ω( α, β ) − ˜ Ω . In cases where exactly three of the four indices coincide we ha ve A i,j,k,l ≡ 0 since E [ ϵ ] = 0 . Finally on the diagonal { i = j = k = l } , w e ha ve A iiii ( v , w, α , β ) = E [ ⟨ a, ϵ i · ( v ) ⟩⟨ b, ϵ i · ( w ) ⟩⟨ c, ϵ i · ( α ) ⟩⟨ d, ϵ i · ( β ) ⟩ ] and X i = j = k = l Ω ij ( v , w )Ω kl ( α, β ) = X i Ω ii ( v , w )Ω ii ( α, β ) = ˜ Ω . Expanding the expression for A iiii and partitioning the index set in a similar w ay , we get A iiii ( v , w, α , β ) = X s,t,m,n E [ a s b t c m d n ϵ is ( v ) ϵ it ( w ) ϵ im ( α ) ϵ in ( β )] =   X s = t  = m = n + X s = m  = t = n + X s = n  = t = m + X s = t = m = n   E [ a s b t c m d n ϵ is ( v ) ϵ it ( w ) ϵ im ( α ) ϵ in ( β )] , 23 since E [ ϵ is ϵ it ϵ im ϵ in ] = 0 on sets of exactly three coinciding indices. Therefore A iiii ( v , w, α , β ) = ⟨ a, b ⟩⟨ c, d ⟩ C ϵ ( v , w ) C ϵ ( α, β ) + ⟨ a, c ⟩⟨ b, d ⟩ C ϵ ( v , α ) C ϵ ( w , β ) + ⟨ a, d ⟩⟨ b, c ⟩ C ϵ ( v , β ) C ϵ ( w , α ) + X t κ it ( v , w, α , β ) a t b t c t d t where we ha ve deﬁned κ it ( v , w, α , β ) := E [ ϵ it ( v ) ϵ it ( w ) ϵ it ( α ) ϵ it ( β ) − 3] . Recall from ( 19 ) that E [ a ′ W b ] = ⟨ a, b ⟩⟨ C ϵ Ω ⟩ , E [ c ′ W d ] = ⟨ c, d ⟩⟨ C ϵ Ω ⟩ . After collecting the ab o ve terms and simplifying, w e note that all terms con taining the quantit y ˜ Ω cancel out and we obtain co v( a ′ W b, c ′ W d ) = ⟨ a, c ⟩⟨ b, d ⟩ Z I 4 C ϵ ( v , α ) C ϵ ( w , β )tr Ω( v , w ) ′ Ω( α, β ) dv dw dαdβ + ⟨ a, d ⟩⟨ b, c ⟩ Z I 4 C ϵ ( v , β ) C ϵ ( w , α )tr Ω( v , w )Ω( α, β ) dv dw dαdβ + X t a t b t c t d t X i Z I 4 κ it ( v , w, α , β )Ω ii ( v , w )Ω ii ( α, β ) dv dw dαdβ . Recall that C ϵ is self-adjoint. By deﬁnition of Ω and Prop osition 2.6 , we ha ve Ω ij ( u, v ) = Ω ∗ j i ( u, v ) = Ω j i ( v , u ) . (21) By linearity , the ﬁrst term in the ab o v e expression simpliﬁes to Z C ϵ ( v , α ) C ϵ ( w , β )tr Ω( v , w ) ′ Ω( α, β ) dv dw dαdβ = X i,j Z I 4 C ϵ ( β , w )Ω ij ( w , v ) C ϵ ( v , α )Ω j i ( α, β ) dv dw dαdβ = X i,j ⟨ C ϵ Ω ij C ϵ Ω j i ⟩ Using ( 21 ) and the self-adjointness of C ϵ again, we obtain X i,j ⟨ C ϵ Ω ij C ϵ Ω j i ⟩ = X i,j ⟨ C 1 / 2 ϵ Ω ij C 1 / 2 ϵ C 1 / 2 ϵ Ω ∗ ij C 1 / 2 ϵ ⟩ = X i,j ∥ C 1 / 2 ϵ Ω ij C 1 / 2 ϵ ∥ 2 F = ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 F . Similarly we ha ve Z I 4 C ϵ ( v , β ) C ϵ ( w , α )tr Ω( v , w )Ω( α, β ) dv dw dαdβ = ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 F . 24 Therefore we ha ve co v( a ′ W b, c ′ W d ) − ( ⟨ a, c ⟩⟨ b, d ⟩ + ⟨ a, d ⟩⟨ b, c ⟩ ) ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 F = X t a t b t c t d t X i Z I 4 κ it ( v , w, α , β )Ω ii ( v , w )Ω ii ( α, β ) dv dw dαdβ and it remains to b ound the last term. By Assumption 3.1 . 2 , w e ha ve      X i Z I 4 κ it ( v , w, α , β )Ω ii ( v , w )Ω ii ( α, β ) dv dw dαdβ      ≤ X i Z I 4 C ϵ ( β , v )Ω ii ( v , w ) C ϵ ( w , α )Ω ii ( α, β ) dv dw dαdβ = X i ⟨ C ϵ Ω ii C ϵ Ω ii ⟩ . By Prop osition 2.6 ( 2 ), w e ha ve X i ⟨ C ϵ Ω ii C ϵ Ω ii ⟩ = X i ∥ C 1 / 2 ϵ Ω ii C 1 / 2 ϵ ∥ 2 F ≤ X i,j ∥ C 1 / 2 ϵ Ω ij C 1 / 2 ϵ ∥ 2 F = ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 F , where the last equality follo ws from Prop osition 2.6 ( 3 ). The following lemma gives the sp ectral decomp osition of the matrix M Θ ′ . Lemma A.2. The matrix M Θ ′ admits a singular value de c omp osition M Θ ′ = T X t =1 σ t w t v ′ t , wher e σ t = (2 sin( tπ 2 T )) − 1 and w tn = − r 2 T cos  ( n − 1 / 2) π t T  , v tn = r 2 T sin  ( n − 1) π t T  for t < T , and σ T = 0 , w T n = 1 / √ T and v T n = 1 { n =1 } . F urthermor e, we have σ n ∼ T for any ﬁxe d n as T → ∞ . Pr o of. The sp ectral decomp osition follo ws from Lemma 5 of Onatski and W ang [ 2021 ]. F or the last estimate, note that sin( nπ 2 T ) ≈ nπ 2 T as T → ∞ , so σ n ∼ T . Using Lemma A.1 and Lemma A.2 , we ma y derive the follo wing concentration bound s for certain quadratics forms of W . Lemma A.3. L et ( σ i ) i , ( v i ) i b e as deﬁne d in L emma A.2 . Under A ssumptions 3.1 - 3.6 , for any i ≤ j ≤ T , we have the fol lowing estimate on the quadr atic form of W : v ′ i W v j = δ ij ⟨ C ϵ Ω ⟩ + ⟨ C ϵ Ω ⟩ o p (1) . F urthermor e, we have j X k = i σ 2 k v ′ k W v k = ⟨ C ϵ Ω ⟩ j X k = i σ 2 k + ⟨ C ϵ Ω ⟩ o p ( T 2 ) . 25 Pr o of. Since ⟨ v i , v j ⟩ = δ ij , by Lemma A.1 w e ha ve E [ v ′ i W v j ] = δ ij ⟨ C ϵ Ω ⟩ . F urthermore, v ar( v ′ i W v j ) = (1 + δ ij ) ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 F + O (2 κ ∗ 4 X t v 2 it v 2 j t ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 F ) = O ( ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ 2 F ) , where the last estimate holds since P t v 2 it v 2 j t ≤ P s,t v 2 it v 2 j s = 1 . By Chebyshev’s inequality v ′ i W v j = δ ij ⟨ C ϵ Ω ⟩ + O p ( ∥ C 1 / 2 ϵ Ω C 1 / 2 ϵ ∥ F ) and the ﬁrst claim follows from Assumption 3.3 . F or the second estimate, recall from Lemma A.2 that σ − 1 k ∼ sin ( k /T ) as T → ∞ . Clearly this implies σ k ∼ T . Since i, j are ﬁxed, this implies P j k = i σ 2 k = O ( T 2 ) and the second claim follows. The following algebraic identit y from Lemma 3.3 of Yin et al. [ 2014 ] gives a geomatric interpre- tation on the ratio b et w een ℓ 1 and ℓ 2 norms of a vector. Lemma A.4 (Lemma 3.3 of Yin et al. [ 2014 ]) . L et x ∈ R n . Then 2 α ( x ) 1 / 2 1 + α ( x ) ∥ x ∥ 1 / 2 0 ≤ ∥ x ∥ 1 ∥ x ∥ 2 ≤ ∥ x ∥ 1 / 2 0 wher e ∥ x ∥ 0 := |{ i ≤ n, x i  = 0 }| and α ( x ) is given by α ( x ) := inf i ≤ n,x i  =0 x i sup i ≤ n,x i  =0 x i . Finally , Lemma A.5 is a linear algebraic result used to compute the sp ectrums of certain op erators. Lemma A.5. L et H b e a Hilb ert sp ac e with inner pr o duct ⟨· , ·⟩ and p, q , T b e p ositive inte gers. Supp ose f 1 , . . . , f T and g 1 , . . . , g T ar e elements of H p with c omp onents f t = ( f 1 t , . . . , f pt ) and g t = ( g 1 t , . . . , g pt ) r esp e ctively. L et A = ( A st ) b e the T × T matrix given by A st = p X i =1 ⟨ f is , g it ⟩ , s, t = 1 , . . . , T and A b e the c omp act op er ator on H p given by A = T X t =1 f t ⊗ g t . Then A and A shar e the same non-zer o singular values. F urthermor e, we have ∥ A ∥ 2 ≤ ∥ A f ∥∥ A g ∥ , wher e A f and A g ar e the matric es whose entries ar e given by ( A f ) st = p X i =1 ⟨ f is , f it ⟩ , ( A g ) st = p X i =1 ⟨ g is , g it ⟩ . 26 Pr o of. Let F : R T → H p b e the linear op erator giv en b y F u = T X t =1 u t f t , u = ( u 1 , . . . , u T ) ′ ∈ R T . F or h = ( h 1 , . . . , h p ) ∈ H p , since ⟨ h, F u ⟩ = p X i =1 T X t =1 ⟨ h i , u t f it ⟩ = T X t =1 u t ⟨ h, f t ⟩ , the adjoint of F is given by F ∗ : H p → R T F ∗ h = ( ⟨ h, f 1 ⟩ , . . . , ⟨ h, f T ⟩ ) ′ , h ∈ H p . Similarly , deﬁne G : H p → R T b y Gu = T X t =1 u t g t , u ∈ R T whose adjoint is given b y G ∗ h = ( ⟨ h, g 1 ⟩ , . . . , ⟨ h, g T ⟩ ) ′ , h ∈ H p . Let ( e t ) t =1 ,...,T b e the standard basis of R T , then w e ha ve ⟨ e s , F ∗ Ge t ⟩ = ⟨ F e s , Ge t ⟩ = ⟨ f s , g t ⟩ = A st the matrix A can b e written as A = F ∗ G . Similarly , we ha ve A = F G ∗ since ⟨ h 1 , F G ∗ h 2 ⟩ = ⟨ F ∗ h 1 , G ∗ h 2 ⟩ = T X t =1 ⟨ h 1 , f t ⟩⟨ h 2 , g t ⟩ = ⟨ h 1 , A h 2 ⟩ for h 1 , h 2 ∈ H p . It remains to observe that A = F ∗ G shares the same non-zero singular v alues as A ∗ = G ∗ F whic h shares the same non-zero singular v alues as A = F G ∗ . This giv es the ﬁrst claim. The second claim follows from the inequalit y ∥ A ∥ 2 ≤ ∥ F ∥ 2 ∥ G ∥ 2 = ∥ F F ∗ ∥∥ GG ∗ ∥ and the fact that A f = F F ∗ and A g = GG ∗ . A.2 Pro of of the main results of The or em 3.8 . First we show that show that under Assumption 3.1 - 3.6 , Theorem 3.8 holds for the matrix b S whenev er it holds for the cov ariance matrix of the non-stationary part of the mo del ˜ S , i.e. the p erturbation of the stationary error term ζ is asymptotically negligible. W e will show b elo w that the sp ectral gaps of ˜ S are asymptotically of size | λ k ( ˜ S ) − λ k +1 ( ˜ S ) | ∼ ∥ ˜ S ∥ ∼ T 2 p − 1 ⟨ C ϵ Ω ⟩ . (22) 27 By standard p erturbation theory [ Kato , 2013 ], to pro ve that the p erturbation of ζ is asymptotically negligible, it suﬃces to sho w that ∥ b S − ˜ S ∥ = o p ( T 2 p − 1 ⟨ C ϵ Ω ⟩ ) . This is what we show now. Since X ( u ) := (Ψ F )( u ) + ζ ( u ) , by the triangle inequality we hav e ∥ b S − ˜ S ∥ ≲     1 p Z I M ζ ( u ) ′ ζ ( u ) M du     +     1 p Z I M (Ψ F )( u ) ′ ζ ( u ) M du     . (23) where M denotes the orthogonal pro jection onto the orthogonal complemen t of the T -dimensional v ector of ones, i.e. M = I T − T − 1 1 T 1 ′ T . W e will use the fact that ∥ M ∥ = 1 throughout the rest of the pro of without men tion. F or the ﬁrst term in ( 23 ), by Lemma A.5 , w e ha ve     1 p Z I M ζ ( u ) ′ ζ ( u ) M du     ≤ 1 p      T X t =1 ζ t ⊗ ζ t      ≤ 1 p T X t =1 ∥ ζ t ∥ 2 H p . Since E [ ∥ ζ t ∥ 2 ] = ⟨ C ζ ⟩ , by Mark ov’s inequalit y w e ha ve     1 p Z I M ζ ( u ) ′ ζ ( u ) M du     = O p ( T p − 1 ⟨ C ζ ⟩ ) = o p ( T 2 p − 1 ⟨ C ϵ Ω ⟩ ) , (24) where the last estimate follows from Assumption 3.6 . Similarly , by the second claim of Lemma A.5 , the second term in ( 23 ) is b ounded by     1 p Z I M (Ψ F )( u ) ′ ζ ( u ) M du     ≤     1 p Z I M (Ψ F )( u ) ′ (Ψ F )( u ) M du     1 / 2     1 p Z I M ζ ( u ) ′ ζ ( u ) M du     1 / 2 = ∥ ˜ S ∥ 1 / 2     1 p Z I M ζ ( u ) ′ ζ ( u ) M du     1 / 2 = o p ( T 2 p − 1 ⟨ C ϵ Ω ⟩ ) , where the last estimate follo ws from ( 24 ) and the b ound ∥ ˜ S ∥ = O p ( T 2 p − 1 ⟨ C ϵ Ω ⟩ ) (25) whic h will b e sho wn b elo w. F rom this w e conclude that ∥ ˜ S − b S ∥ = o p ( T 2 p − 1 ⟨ C ϵ Ω ⟩ ) which shows that the p erturbation of ζ is asymptotically negligible. It remains to sho w that Theorem 3.8 holds for the cov ariance op erator ˜ S , which also pro ves the t wo estimated ( 22 ) and ( 25 ) w e used ab o ve. This pro of is adapted from the arguments in Section A.3.1 of Onatski and W ang [ 2021 ]. W e sketc h the k ey comp onen ts of the argument adapted to the functional setting of our mo del. Let ˜ λ n , ˜ u n , n = 1 , . . . , T b e the eigen v alues of ˜ S arranged in non-ascending order and the corresp onding eigen vectors. W e will fo cus on the largest eigen v alues ˜ λ 1 and the corresp onding eigen vector ˜ u 1 ; the general case can b e obtained by mathematical induction. W e omit the details of the induction argumen t, since it can b e found on Onatski and W ang [ 2021 , p.610-611]. Recall from Lemma A.2 that the set { w n , n = 1 , . . . , T } forms an orthonormal basis of R T and w T is prop ortional to a v ector of ones. Since ˜ u 1 is orthogonal to w T b y construction, we hav e the represen tation ˜ u 1 = T − 1 X t =1 α t w t 28 for some set of w eights ( α t ) . Note that since ( w t ) is an orthonormal set, we ha ve 1 = ∥ ˜ u 1 ∥ 2 = ∥ T − 1 X t =1 α t w t ∥ = T − 1 X t =1 α 2 t ∥ w t ∥ 2 = T − 1 X t =1 α 2 t . By the deﬁnitions of ˜ λ 1 , ˜ u 1 , ˜ S we hav e ˜ λ 1 = ˜ u ′ 1 ˜ S ˜ u 1 = T − 1 X s,t =1 α s α t w ′ s ˜ S w t . Since ˜ S = p − 1 M Θ ′ W Θ M where M Θ ′ = P t σ t w t v ′ t b y Lemma A.2 , w e ha ve p ˜ λ 1 = T − 1 X s,t =1 α s α t ˜ σ s ˜ σ t v ′ s W v t . The quadratic form v ′ s W v t can b e estimated using Lemma A.3 , ho wev er, the sum of T − 1 suc h terms has to b e dealt with additional care since T → ∞ . Using a truncation argument similar to the one found in Onatski and W ang [ 2021 , p.607-p.609], w e ma y sho w that p b λ 1 ≤ ⟨ C ϵ Ω ⟩ T − 1 X t =1 α 2 t ˜ σ 2 t + o p ( T 2 ) ! . Since ˜ σ 1 is the leading eigenv alue, and P T − 1 t =1 α 2 t = 1 , we ha ve T − 1 X t =1 α 2 t ˜ σ 2 t = α 2 1 ˜ σ 2 1 + T − 1 X t =2 α 2 t ˜ σ 2 t ≤ α 2 1 ˜ σ 2 1 + (1 − α 2 1 ) ˜ σ 2 2 , whic h giv es p b λ 1 ≤ α 2 1 ˜ σ 2 1 ⟨ C ϵ Ω ⟩ + (1 − α 2 1 ) ˜ σ 2 2 ⟨ C ϵ Ω ⟩ + o p ( T 2 ) ⟨ C ϵ Ω ⟩ . On the other hand, since ˜ λ 1 is the leading eigenv alue, we ha ve p ˜ λ 1 ≥ pw ′ 1 ˜ Σ w 1 = σ 1 v ′ 1 W v 1 = σ 2 1 ⟨ C ϵ Ω ⟩ + o p ( T 2 ) ⟨ C ϵ Ω ⟩ . Com bining the upp er and lo wer b ounds and simplifying, w e obtain ( σ 2 1 − σ 2 2 )(1 − α 2 1 ) ⟨ C ϵ Ω ⟩ ≤ o p ( T 2 ) ⟨ C ϵ Ω ⟩ . Using the fact that σ 2 1 − σ 2 2 ∼ T 2 , we obtain 1 − α 2 1 = o p (1) , which implies α 2 1 = ( ˜ u ′ 1 w 1 ) 2 → 1 in probabilit y as T → ∞ . This giv es the ﬁrst claim. Using the upp er and low er b ounds again, w e ma y obtain | p ˜ λ 1 − σ 2 ⟨ C ϵ Ω ⟩| ≤ | 1 − α 2 1 | ( σ 2 1 + σ 2 2 ) ⟨ C ϵ Ω ⟩ + o p ( T 2 ) ⟨ C ϵ Ω ⟩ = o p ( T 2 ) ⟨ C ϵ Ω ⟩ , since σ 2 1 ∼ T 2 /π 2 and 1 − α 2 1 = o p (1) . This giv es ˜ λ 1 = T 2 π 2 p ⟨ C ϵ Ω ⟩ (1 + o p (1)) , whic h is the asymptotic limit of the largest empirical eigen v alue. This also implies the b ounds ( 22 ) and ( 25 ) . Finally , the last claim follows from the same arguments as p.609-610 of Onatski and W ang [ 2021 ], and w e omit the details. 29 of The or em 3.11 . Using ( 5 ) and ( 9 ), the op erator Ω can b e written as Ω kl ( v , w ) = p X i =1 Z I Ψ ik ( u, v )Ψ il ( u, w ) du = p X i =1 ∞ X n =1 a nik a nil ϕ n ( v ) ϕ n ( w ) = ∞ X n =1 ( A ′ n A n ) kl ϕ n ( v ) ϕ n ( w ) . Since C 1 / 2 ϵ = P n c 1 / 2 n ϕ n ⊗ ϕ n , we ha ve ( C 1 / 2 ϵ Ω kl C 1 / 2 ϵ )( u, v ) = Z I C 1 / 2 ϵ ( u, α )Ω kl ( α, β ) C 1 / 2 ϵ ( β , v ) dαdβ = ∞ X n =1 c n ( A ′ n A n ) kl ϕ n ( u ) ϕ n ( v ) . By Prop osition 2.6 , the trace of C is given by ⟨C ⟩ = K X k =1 ⟨ C ϵ Ω kk ⟩ = K X k =1 ∞ X n =1 c n ( A ′ n A n ) kk = ∞ X n =1 c n ⟨ B n ⟩ . Similarly , the Hilb ert-Sc hmidt norm of C is given b y ∥C ∥ 2 2 = X kl ∥ C 1 / 2 ϵ Ω kl C 1 / 2 ϵ ∥ 2 = ∞ X n =1 X kl c 2 n ( A ′ n A n ) 2 kl = ∞ X n =1 c 2 n ∥ B n ∥ 2 2 . Since 1 = ⟨ C ϵ ⟩ = P n c n and ∥ C ϵ ∥ 2 2 = P n c 2 n , we ha ve the following upp erbound ⟨C ⟩ 2 ∥C ∥ 2 2 = lim sup q →∞ ( P q n =1 c n ⟨ B n ⟩ ) 2 P q n =1 c 2 n ∥ B n ∥ 2 2 ≤ inf Q ∈ N sup q >Q max n ≤ q ⟨ B n ⟩ 2 min n ≤ q ∥ B n ∥ 2 2 ( P q n =1 c n ) 2 P q n =1 c 2 n ≤ 1 ∥ C ϵ ∥ 2 2 inf Q ∈ N sup q >Q max n ≤ q ⟨ B n ⟩ 2 min n ≤ q ∥ B n ∥ 2 2 . Let k q b e the index suc h that the maxin um in the n umerator is attained, i.e. k q := arg max n ≤ q ⟨ B n ⟩ . Since ∥ B n ∥ 2 ≳ ∥ B k q ∥ 2 uniformly in n and q by assumption, we hav e max n ≤ q ⟨ B n ⟩ min n ≤ q ∥ B n ∥ 2 ≲ ⟨ B k q ⟩ ∥ B k q ∥ 2 ≤ max n ≤ q ⟨ B n ⟩ ∥ B n ∥ 2 , whic h giv es the ﬁrst upp er b ound ⟨C ⟩ ∥C ∥ 2 ≲ 1 ∥ C ϵ ∥ 2 inf Q ∈ N sup q >Q max n ≤ q ⟨ B n ⟩ ∥ B n ∥ 2 ≤ 1 ∥ C ϵ ∥ 2 sup n ⟨ B n ⟩ ∥ B n ∥ 2 . 30 By the Cauc h y-Sch w arz inequality we hav e ⟨ B n ⟩ ≤ ∥ B n ∥ 1 / 2 0 ∥ B n ∥ 2 and the second upp er b ound follo ws from the ﬁrst upp er b ound. Similar to the ab ov e computations, using ∥ B n ∥ 2 ∼ ∥ B m ∥ 2 w e obtain a low er b ound of the form ⟨C ⟩ ∥C ∥ 2 ≥ 1 ∥ C ϵ ∥ 2 sup Q ∈ N inf q >Q min n ≤ q ⟨ B n ⟩ max n ≤ q ∥ B n ∥ 2 ≳ 1 ∥ C ϵ ∥ 2 sup Q ∈ N inf q >Q min n ≤ q ⟨ B n ⟩ ∥ B n ∥ 2 . Since ⟨ B n ⟩ ≥ ∥ B n ∥ 2 , we obtain the ﬁrst half of the ﬁrst low er b ound ⟨C ⟩ ∥C ∥ 2 ≳ 1 ∥ C ϵ ∥ 2 . F or the second half, using the non-negativity of the summand, we ha ve ( P q n =1 c n ⟨ B n ⟩ ) 2 P q n =1 c 2 n ∥ B n ∥ 2 2 ≥ P q n =1 c 2 n ⟨ B n ⟩ 2 P q n =1 c 2 n ∥ B n ∥ 2 2 = c 2 m ⟨ B m ⟩ 2 + P q n  = m c 2 n ⟨ B n ⟩ 2 P q n =1 c 2 n ∥ B n ∥ 2 2 , where m ≤ q is arbitrarily c hosen. Since ∥ B n ∥ 2 ∼ ∥ B m ∥ 2 uniformly for all n , w e ha ve ( P q n =1 c n ⟨ B n ⟩ ) 2 P q n =1 c 2 n ∥ B n ∥ 2 2 ≳ c 2 m ⟨ B m ⟩ 2 + P q n  = m c 2 n ⟨ B n ⟩ 2 ∥ C ϵ ∥ 2 2 ∥ B m ∥ 2 2 ≥ 1 ∥ C ϵ ∥ 2 2 c 2 m ⟨ B m ⟩ 2 ∥ B m ∥ 2 2 . T aking the limit in q yields the ﬁrst low er b ound. Finally , using Lemma A.4 , w e get ⟨ B m ⟩ ∥ B m ∥ 2 ≳ ∥ B m ∥ 1 / 2 0 α ( B m ) 1 / 2 whic h completes the pro of. B A conjecture on the asymptotics of sample eigen v ectors Lastly , we end the pap er with some in teresting observ ations and a conjecture, which forms the basis of some future researc h. W e observe that for the case K = 2 in Setting V (Figure 4 ), the ﬁfth sample eigenv ector seemingly b eha ves in a signiﬁcantly diﬀerent w ay to the ﬁrst four. More sp eciﬁcally , the ﬁrst four sample eigen vectors v agues resembles the spurious limit and are relativ ely smo oth, while the ﬁfth eigenv ector resembles a white noise pro cess which has rough sample paths. The mo del in this setting is generated b y tw o basis functions with equal weigh t, and along the direction of eac h basis function, the non-stationary factor is loaded by a matrix A n of rank t wo. Heuristically , the non-stataionary comp onen t of the mo del spans a subspace of dimension four, while the orthogonal complement of this subspace is spanned by the stationary comp onen ts of the mo del. This is similar to the concept of the attractor and cointegrating space that has recen tly garnered m uch attention in the functional time series literature [see Chang et al. , 2016 , Beare et al. , 2017 , Li et al. , 2023 ]. In this line of work, the dimension of the non-stationary space is determined by mo del parameters, and the asymptotic b eha viour of the eigenstructure diﬀers on the stationary and non-stationary subspaces. It is therefore natural to conjecture that for mo dels with C ϵ and A n b eing of ﬁnite rank, only the ﬁrst ∥ C ϵ ∥ 0 ∥ A n ∥ 0 eigen vectors will tend to the spurious limit, while the rest b eha ve lik e white noise. This is b ecause the limits in Theorem 3.8 arise as eigenfunctions of Wiener pro cesses, which are the 31 Figure 8: An eample where C ϵ is of rank 2 and loading matrices are of rank 3 scaling limits of the non-stationary part of the mo del. This is conﬁrmed empirically b y sim ulations under a wide range of diﬀerent settings, w e include a simple example to illustrate what t ypically o ccurs. The top eight subﬁgures of Figure 8 plot the sample eigen vectors of a mo del with K = 10 , C ϵ follo wing Setting (C) and A n = G p × 3 × G 3 × K , where G m,n denotes a m × n matrix of i.i.d. standard Gaussians. Clearly the ﬁrst six eigen vectors b eha v e diﬀeren tly than the last t wo, whic h is in line with our conjecture since here the “eﬀective rank” of the mo del is equal to ∥ C ϵ ∥ 0 ∥ A n ∥ 0 = 6 . The b ottom eight ﬁgures plot the s ample ACF of each of the sample eigen vectors as a time series of length 200. 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Attribution of Spurious Factors from High-Dimensional Functional Time Series

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