Some Asymptotic Results on Multiple Testing under Weak Dependence

Some Asymptotic Results on Multiple T esting under W eak Dep endence Sw arnadeep Datta ∗ 1 and Monitirtha Dey † 2 1 Inter disciplinary Statistic al R ese ar ch Unit, Indian Statistic al Institute, Kolkata, India 2 Institute for Statistics, University of Br emen, Br emen, Germany Abstract This paper studies the means-testing problem under w eakly correlated Normal setups. Although quite common in genomic applications, test pro cedures ha ving exact FWER con trol under suc h dep endence structures are nonexistent. W e explore the asymptotic b eha viors of the classical Bonferroni (when adjusted suitably) and the Sidak pro cedure; and sho w that b oth of these control FWER at the desired lev el exactly as the num b er of h yp otheses approaches infinity . W e deriv e analogous limiting results on the generalized family-wise error rate and p ow er. Simulation studies depict the asymptotic exactness of the pro cedures empirically . 1 In tro duction Large-scale m ultiple testing problems under dep endence are a staple of mo dern statistical science ( Dic khaus et al. , 2026 ; Finner et al. , 2007 , 2009 ). Dep endence among observ ations or test statistics has b een a springb oard for the theoretical and applied dev elopmen ts of sim ultaneous statistical inference ( Dickhaus , 2008 , 2014 ). Often, there exists a w eak dependence among the observ ations of in terest. Prosc han and Shaw ( 2011 ) men tion that the correlation b et w een tw o single nucleotide polymorphisms (SNP) is generally b elieved to decrease with genomic distance. Sev eral authors ha v e argued that for the large num b ers of markers typically used for a genome- wide asso ciation study (GW AS), the test statistics are w eakly correlated due to this largely lo cal presence of correlation b etw een SNPs ( Dey and Bhandari , 2025 ; Storey and Tibshirani , 2003 ). There ha v e b een a few works on weakly dep endent structures with resp ect to asymptotic false disco v ery rate (FDR) con trol. F arcomeni ( 2007 ) shows that a certain degree of dep endence is allow ed among the test statistics when the n umber of tests is large, without an y need for a correction to the traditional pro cedures. Storey et al. ( 2003 ) show that linear step-up (LSU) and plug-in LSU tests control the FDR asymptotically under weak dep endence, assuming that the prop ortion of rejected n ull hypotheses is asymptotically larger than zero in some sense. Gon tsc haruk and Finner ( 2013 ) sho w that w eak dep endence is not sufficient for FDR control if the proportion of rejected n ulls conv erges to zero with p ositive probabilit y . Ho w ev er, similar theoretical results on asymptotic familywise error rate (FWER) control are scarce. Das and Bhandari ( 2025 ) explore ho w closely the FWER of Bonferroni metho d resem bles its b ehavior ∗ sw arnadeep datta0122@gmail.com † monitirthadey3@gmail.com , mdey@uni-bremen.de 1 under indep endence and introduces an asymptotic correction factor to improv e accuracy in nearly indep endent cases. A series of recen t w orks viz. Dey and Bhandari ( 2023 , 2024 , 2025 ); Dey ( 2024 ) has elucidated that the Bonferroni metho d and the class of stepwise multiple testing pro cedures (MTP) hav e asymptotically zero family-wise error rate (FWER) under correlated Normal scenarios where the infimum of correlations is considered to b e strictly p ositive. How ever, their limiting results do not help in devising an MTP that is asymptotically exact under equicorrelation, i.e., which has limiting FWER exactly equal to α under equicorrelation. Inspired b y such problems, the authors propose a simple single-step MTP that asymptotically con trols the FWER at the desired level exactly under the equicorrelated multiv ariate Gaussian setup in Datta and Dey ( 2026 ). Ho w ev er, suc h results guaranteeing asymptotic exact FWER con trol are nonexisten t for weakly dep endent scenarios. Also, asymptotic b ehaviors of Bonferroni-type pro cedures hav e not yet b een studied under general weak dep endence structures. This work fills b oth these gaps b y elucidating that b oth the Sidak procedure and the classical Bonferroni metho d (with a little adjustmen t) hav e asymptotic FWER exactly equal to α under w eakly dep endent scenarios. This w ork is organized as follo ws. Section 2 intr o duces the testing framew ork formally . Section 3 presen ts the main results on asymptotically exact FWER con trol in the one-sided setting. Section 4 inv estigates the p ow er properties. Section 5 studies the same problem in the t wo-sided setting. Section 6 depicts the empirical performance of the pro cedures through sim ulations. Section 7 concludes the pap er. The pro ofs are deferred to the app endix. 2 Preliminaries Throughout this pap er, ϕ and Φ denote the p.d.f and c.d.f of N (0 , 1) distribution resp ectiv ely . Let I denote the set { 1 , 2 , . . . , n } . Consider n observ ations X i ∼ N ( µ i , 1) , i ∈ I . The elements of the co v ariance matrix Σ n are given by C or r ( X i , X j ) = ρ ij , with ρ ij ∈ ( − 1 , 1) ∀ i  = j. Let ρ m = sup 1 ≤ i ≤ n − m | ρ i i + m | and also γ = sup n ≥ 1 ρ n < 1. W e assume that Σ n satisfies the follo wing we ak dep endenc e c ondition : ρ m = o  1 log m  ∀ 1 ≤ m ≤ n as n gro ws. (1) In sections 3 and 4 , w e consider the one-sided testing problem H 0 i : µ i = 0 vs H 1 i : µ i > 0 , i ∈ I . W e focus on the corresponding both-sided problem H 0 i : µ i = 0 vs H 1 i : µ i  = 0 , i ∈ I in section 6 . In both settings, the global n ull H 0 = T n i =1 H 0 i asserts that eac h µ i is zero. This paper aims to obtain v alid MTPs for which FWER con verges to the desired lev el, under the w eak dep endence structure depicted in ( 1 ). 2 3 Main Results Throughout this work, let I 0 and I 1 resp ectiv ely denote the set of indices of true and false n ulls. Let n 0 and n 1 (= n − n 0 ) b e the cardinalities of these t wo sets, resp ectiv ely . Also, unless otherwise mentioned, by the phrase we akly dep endent Gaussian se quenc e , w e mean an y Gaussian sequence satisfying ( 1 ). W e consider the following t w o testing pro cedures: T est Pro cedure 1 (Adjusted Bonferroni) . F or e ach i ∈ I , r eje ct H 0 i if X i > c B on ( n, α ) : = Φ − 1 (1 − − log (1 − α ) n ) . T est Procedure 2 (Sidak) . F or e ach i ∈ I , r eje ct H 0 i if X i > c S id ( n, α ) : = Φ − 1 ((1 − α ) 1 /n ) . W e first provide the follo wing result on the asymptotic distribution of the k -th largest v alue for a weakly dep endent standard Gaussian sequence: Theorem 3.1. L et { X n } b e a we akly dep endent standar d Gaussian se quenc e. Supp ose { u n } b e a se quenc e and { d n } b e a p ositive inte ger se quenc e of n such that d n → ∞ and d n (1 − Φ( u n )) → τ as n → ∞ for some τ ∈ [0 , ∞ ] . F or 1 ≤ k ≤ d n , let M k d n b e the k -th lar gest among X 1 , X 2 , . . . , X d n . Then, for e ach fixe d k , we have lim n →∞ P n M k d n ≤ u n o =      e − τ P k − 1 s =0 τ s s ! if τ ∈ (0 , ∞ ) , 1 if τ = 0 , 0 if τ = ∞ . (2) Let i 1 < i 2 < · · · < i n 0 b e the indices in I 0 . Then, we re-notate X i ’s as Z j = X i j , j = 1 , 2 , . . . , n 0 . The weak dep endence structure gives C or r ( Z i , Z j ) ≤ ρ | i − j | . Hence, as n (and thereb y n 0 ) gro ws, C or r ( Z 1 , Z m ) ≤ ρ m − 1 = o ( 1 log( m − 1) ) satisfying the w eak dependence structure for { Z m } as w ell. No w, FWER of any pro cedure with common right-sided cutoff τ n can b e expressed as: F W E R ( n, τ n , α, Σ n ) = P n 0 [ i =1 { Z i > τ n } ! = 1 − P n 0 \ i =1 { Z i ≤ τ n } ! = 1 − P ( M n 0 ≤ τ n ) , where M n 0 = max 1 ≤ i ≤ n 0 Z i . (3) Note that for any α ∈ (0 , 1), (1 − α ) 1 n = exp( log(1 − α ) n ) = 1 − − log(1 − α ) n + O ( 1 n 2 ) = 1 − − log(1 − α ) − O ( 1 n ) n . (4) Hence, we can express b oth c B on ( n, α ) and c S id ( n, α ) as c n ( α ) = Φ − 1 (1 − − log(1 − α ) − o (1) n ). W e note that n (1 − Φ( c n ( α ))) − → − log (1 − α ) as n → ∞ . T aking u n = c n ( α ), d n = n 0 and k = 1 in Theorem 3.1 , w e obtain the quintessen tial result of this w ork. Theorem 3.2. Under the we akly dep endent standar d Gaussian setting with c orr elation matrix Σ n , b oth the adjuste d Bonferr oni pr o c e dur e( T est Pr o c e dur e 1 ) and the Sidak pr o c e dur e( T est Pr o c e dur e 2 ) ar e asymptotic al ly exact, i.e., lim n →∞ F W E R ( n, c B on , α, Σ n ) = lim n →∞ F W E R ( n, c S id , α, Σ n ) = α, under any c onfigur ation of true and false nul l hyp otheses satisfying lim n →∞ n 0 /n = 1 . 3 Remark 1. L et p 0 := lim n →∞ n 0 /n . Notably, the FWER of T est Pr o c e dur e 1 for this we ak dep endenc e setup, under any c onfigur ation of true and false nul l hyp otheses, do es not ne c essarily c onver ge to tar get α for any p 0 > 0 . The or em 3.2 states that this c onver genc e is valid for p 0 = 1 . If p 0 is known a priori, then one may suitably twe ak the c ommon cutoffs as mentione d in T est Pr o c e dur e 1 and T est Pr o c e dur e 2 to guar ante e exact c onver genc e. Inde e d, the pr o c e dur es utilizing the cutoffs c B on ( α, p 0 ) : = Φ − 1 (1 − − log (1 − α ) np 0 ) and c S id ( α, p 0 ) : = Φ − 1 ((1 − α ) 1 /np 0 ) have their FWERs c onver ging to α . Remark 2. L et ν b e some c onstant such that 0 < ν < 1 − γ 1+ γ , γ = sup n ≥ 1 ρ n < 1 and also let γ n = sup m ≥ n ρ m . Then the r ate of c onver genc e in The or em 3.2 c an b e given as | F W E R ( n, c n ( α ) , α, Σ n ) − α | ≤ l · R n for some l > 0 as n → ∞ wher e R n = max  n 1+ ν − 2 1+ γ (log n ) 1 1+ γ , γ [ n ν ] log n ν , 1 − n 0 n , 1 n  . Her e c n ( α ) ∈ { c B on ( n, α ) , c S id ( n, α ) } . Remark 3. F or the L ehmann-R omano pr o c e dur e ( L ehmann and R omano , 2005 ), k -FWER ( n, α, Σ n ) = P Σ n  X i > Φ − 1 (1 − kα/n ) for at le ast k i ’s ∈ I 0  = 1 − P Σ n  M k n 0 ≤ Φ − 1 (1 − kα/n )  [ M k n 0 : k ’th maximum among Z 1 , . . . , Z n 0 . ] Considering u n = Φ − 1 (1 − kα/n ) and d n = n 0 in The or em 3.1 dir e ctly gives the asymptotic k -FWER c ontr ol for the L ehmann-R omano pr o c e dur e under the we akly dep endent standar d Gaussian setting with c orr elation matrix Σ n , i.e., k -FWER ( n, α, Σ n ) − → 1 − e − kα k − 1 X s =0 ( k α ) s s ! as n → ∞ . 4 P o w er Analysis The simultaneous inference literature has sev eral notions of p ow er ( Dudoit and Laan , 2008 ). In this work, w e shall work with A nyPwr , whic h is defined as the probabilit y of making at least one true rejection. W e ha v e X i ∼ N ( µ i , 1), µ i > 0 for i ∈ I 1 , where I 1 denotes the set of indices of the originally false n ulls. Hence, |I 1 | = n 1 . The follo wing result describ es the asymptotic p o w ers of T est Procedure 1 and T est Pro cedure 2 . Theorem 4.1. Supp ose n 1 → ∞ and n 1 n → p 1 ∈ (0 , 1] as n → ∞ . Then, under the we akly c orr elate d Gaussian setting, for b oth T est Pr o c e dur e 1 and T est Pr o c e dur e 2 , one has lim n →∞ Any P wr = 1 if lim n 1 →∞ √ 2 log n 1 µ n 1 < 1 wher e µ n 1 = max { µ i , i ∈ I 1 } . W e discuss no w the asymptotic of p o wer for a more general setup of the non-null means. The following result would b e crucial for that. Prop osition 1. Supp ose, for any β > 0 , β n = β + o (1) > 0 and c β n , n = Φ − 1 (1 − β n /n ) . Also supp ose { d n } n ≥ 1 is a se quenc e of n which diver ges to ∞ as n → ∞ . Then, if for some t ∈ R , lim n →∞ d n n · e − t √ 2 log n = ∞ , one has lim n →∞ d n · (1 − Φ ( c β , n + t )) = ∞ 4 Let i 1 < i 2 < · · · < i n 1 b e the indices in I 1 . Then w e re-notate X i ’s as Y j = X i j , and also, throughout this section, w e w ould use µ j to denote µ i j , j = 1 , 2 , . . . , n 1 . Eviden tly , C or r ( Y i , Y j ) ≤ ρ | i − j | and as n and thereb y n 1 gro ws, C or r ( Y 1 , Y m ) ≤ ρ m − 1 = o ( 1 log( m − 1) ). One ma y write Y j = Z j + µ j , Z j ∼ N (0 , 1) ∀ j = 1 , 2 , . . . , n 1 . Clearly , C or r ( Y i , Y j ) = C or r ( Z i , Z j ) ∀ i, j = 1 , 2 , . . . , n 1 . No w, for any testing procedure with common righ t-sided cutoff τ n , we hav e the following inequalit y . 1 − Any P w r = P ( n 1 \ i =1 { Z i ≤ τ n − µ i } ) ≤ P ( n 1 \ i =1 { Z i ≤ τ n − µ } ) [assuming µ = lim n →∞ min i ∈I 1 µ i > 0] = P ( Z ( n 1 ) ≤ τ n − µ ) [where Z ( n 1 ) = max i =1 , 2 , ..., n 1 Z i ] . (5) W e put d n = n 1 , t = − µ and β n = − log(1 − α ) − o (1). Prop osition 1 implies that lim n →∞ d n · (1 − Φ ( c β , n + t )) = ∞ . Applying Theorem 3.1 on u n = c n ( α ) − µ and k = 1, w e get the following result viding ( 5 ). Theorem 4.2. Supp ose lim n →∞ min i ∈I 1 µ i = µ > 0 . Then, for b oth T est Pr o c e dur e 1 and T est Pr o c e dur e 2 , one has lim n →∞ Any P wr = 1 if lim n →∞ n 1 n e µ √ 2 log n = ∞ . 5 Extension to Both-sided T esting Up to this point, w e ha v e considered only the one-sided testing problem, i.e., H 1 i : µ i > 0 , i ∈ I . Ho w ev er, one often encounters b oth-sided testing situations: H 0 i : µ i = 0 vs H 1 i : µ i  = 0 , i ∈ I . W e shall denote the FWER, k -FWER, and An yPwr in this setting b y F W E R B S , k - F W E R B S and Any P w r B S resp ectiv ely . F or this problem, w e consider the corresp onding testing pro cedures. T est Procedure 3 (Adjusted Bonferroni) . F or any c ovarianc e matrix (or c orr elation matrix) Σ n satisfying the we ak dep endenc e c ondition ( 1 ) , taking the c ommon cutoff { c B on ( n, α ) } n ≥ 1 as define d in T est Pr o c e dur e 1 , we r eje ct H 0 i if | X i | > c B on (2 n, α ) for e ach i ∈ { 1 , 2 , . . . , n } . T est Pro cedure 4 (Sidak) . F or any c ovarianc e matrix (or c orr elation matrix) Σ n satisfying the we ak dep endenc e c ondition ( 1 ) , taking the c ommon cutoff { c S id ( n, α ) } n ≥ 1 as define d in T est Pr o c e dur e 2 , we r eje ct H 0 i if | X i | > c S id (2 n, α ) for e ach i ∈ { 1 , 2 , . . . , n } . Analogous to section 3 , we first provide the result on asymptotic distribution of the k -th largest among the absolute v alues of a w eakly dep endent standard Gaussian sequence. Theorem 5.1. Consider the assumptions as in The or em 3.1 . Supp ose L k d n b e the k -th lar gest among | X 1 | , | X 2 | , . . . , | X d n | for 1 ≤ k ≤ d n . F or e ach fixe d k , we have lim n →∞ P n L k d n ≤ u n o =      e − 2 τ P k − 1 s =0 (2 τ ) s s ! if τ ∈ (0 , ∞ ) , 1 if τ = 0 , 0 if τ = ∞ . (6) 5 Consider the rearrangements and re-notations of X i ’s corresp onding to the originally ture n ulls men tioned in section 3 . The FWER of any pro cedure with common left-sided and righ t- sided cutoffs − τ n and τ n , resp ectively , can b e expressed in the follo wing form: F W E R B S ( n, τ n , α, Σ n ) = P n 0 [ i =1 {| Z i | > τ n } ! = 1 − P n 0 \ i =1 {| Z i | ≤ τ n } ! = 1 − P ( L n 0 ≤ τ n ) , where L n 0 = max 1 ≤ i ≤ n 0 | Z i | . (7) Then, taking u n = c 2 n ( α ) = Φ − 1 (1 − − log(1 − α ) − o (1) 2 n ), d n = n 0 and k = 1 in Theorem 5.1 , w e get the following result for the tw o-sided testing problem. Theorem 5.2. F or b oth T est Pr o c e dur e 3 and T est Pr o c e dur e 4 for the b oth-side d pr oblem, under gener al c orr elate d Gaussian setting with Σ n satisfying the we ak dep endenc e c ondition 1 , lim n →∞ F W E R B S ( n, c B on , α, Σ n ) = lim n →∞ F W E R B S ( n, c S id , α, Σ n ) = α, under any c onfigur ation of true and false nul l hyp otheses for which lim n →∞ n 0 /n = 1 . Remark 4. L et R n b e as define d in R emark 2 . Then the r ate of c onver genc e in The or em 5.2 is also given as | F W E R B S ( n, c n ( α ) , α, Σ n ) − α | ≤ l · R n for some l > 0 as n → ∞ . Remark 5. F or the L ehmann-R omano pr o c e dur e ( L ehmann and R omano , 2005 ) for this two- side d testing pr oblem, k -FWER B S ( n, α, Σ n ) = P Σ n  | X i | > Φ − 1 (1 − k α/ 2 n ) for at le ast k i ’s ∈ I 0  = 1 − P Σ n  L k n 0 ≤ Φ − 1 (1 − k α/ 2 n )  h L k n 0 : k ’th maximum among | Z 1 | , . . . , | Z n 0 | as define d in Se ction 3 . i Considering u n = Φ − 1 (1 − k α/ 2 n ) and d n = n 0 in The or em 5.1 dir e ctly gives the asymptotic k -FWER c ontr ol for the L ehmann-R omano pr o c e dur e under the we akly dep endent standar d Gaussian setting with c orr elation matrix Σ n , i.e., k - F W E R B S ( n, α, Σ n ) − → 1 − e − kα k − 1 X s =0 ( k α ) s s ! as n → ∞ . F ollowing the one-sided case, we derive analogous asymptotic p ow er results for the presen t t w o-sided testing problem. The follo wing t w o theorems depict the asymptotics. Theorem 5.3. Supp ose n 1 → ∞ and n 1 n → p 1 ∈ (0 , 1] as n → ∞ . Then, under the we akly c orr elate d Gaussian setting, for b oth T est Pr o c e dur e 3 and T est Pr o c e dur e 4 , lim n →∞ Any P wr B S = 1 if lim n 1 →∞ √ 2 log n 1 µ ( n 1 ) < 1 , wher e µ ( n 1 ) = max i ∈I ∞ | µ i | . Theorem 5.4. Supp ose lim n →∞ min i ∈I 1 | µ i | = µ > 0 . Then, if lim n →∞ n 1 n e µ √ 2 log n = ∞ , lim n →∞ Any P wr B S = 1 , for b oth T est Pr o c e dur e 3 and T est Pr o c e dur e 4 . 6 6 Sim ulation Studies The FWER of a single-step m ultiple testing pro cedure (using the cut-off c n for each of the h yp otheses) for the equicorrelated Normal setup under the global n ull is given b y ( Dey , 2025 ): F W E R H 0 = 1 − E  Φ n  c n + √ ρZ √ 1 − ρ  , where Z ∼ N (0 , 1) . (8) Here ρ is the common correlation. This essentially elucidates the ease of sim ulating from a equicorrelated Normal setup. Simulating Normal random v ariables ha ving more general dep endencies might b e extremely computationally intensiv e (dep ending on the correlation structure) b ecause of the presence of a large, and muc h more complex cov ariance matrix. Here w e shall w ork with product correlation structures: ∀ i  = j, ρ ij = λ i λ j , where λ i ∈ ( − 1 , 1) for eac h i . T ong ( 1980 ) refers to this as structur e ℓ . Now, suppose X 1 , X 2 , . . . , X n are standard Normal v ariables having C or r ( X i , X j ) = ρ ij = λ i λ j for all i  = j , λ i ∈ ( − 1 , 1). Equiv alently , one can write X i = λ i Z + q 1 − λ 2 i ϵ i for all i = 1 , 2 , . . . , n , where Z , ϵ 1 , ϵ 2 , . . . , ϵ n are indep endent N (0 , 1) v ariables. One can generalize ( 8 ) to this case as follo ws. Consider a single-step MTP using common righ t-sided cutoff τ n . The FWER of this pro cedure is: F W E R H 0 = 1 − P H 0 n \ i =1 { X i ≤ τ n } ! = 1 − P H 0 n \ i =1  λ i Z + q 1 − λ 2 i ϵ i ≤ τ n  ! = 1 − P H 0   n \ i =1 { ϵ i ≤ τ n − λ i Z q 1 − λ 2 i }   = 1 − E Z   n Y i =1 Φ   τ n − λ i Z q 1 − λ 2 i     , where Z ∼ N (0 , 1) . (9) Similarly , for a single-step MTP using common both-sided cutoffs − τ n and τ n , the FWER of this pro cedure is F W E R H 0 = 1 − E Z   n Y i =1    Φ   τ n − λ i Z q 1 − λ 2 i   − Φ   − τ n − λ i Z q 1 − λ 2 i        , where Z ∼ N (0 , 1) . (10) 6.1 Sim ulation Sc heme F or our purpose, w e need to sim ulate FWER under global n ull for m ultiv ariate standard Normal distribution under weak dep endence structure, ( 1 ). F or this, we utilize the notion of product correlation. So, w e find real n umbers λ 1 , λ 2 , . . . , λ n , each in (0 , 1) suc h that ρ ij ≥ 0 and the w eak dep endence condition ( 1 ) is satisfied. In particular, if w e take λ i = ( λ for i = 1 , 1 (log i ) 1+ δ for i = 2 , 3 , . . . , n, for some λ > 0 and δ > 0 , (11) 7 then for this pro duct correlation structure the weak dependency condition is satisfied. Hence, considering the pro duct correlation structure defined in ( 11 ), w e sim ulate FWER for b oth T est Pro cedure 1 and T est Pro cedure 2 using ( 9 ) in the follo wing wa y : F or given ( n, α ), we compute c B on ( n, α ) = Φ − 1 (1 − − log(1 − α ) n ) and c S id ( n, α ) = Φ − 1  (1 − α ) 1 /n  . Then, b y choosing some λ > 0 and δ > 0, w e compute λ 1 , λ 2 , . . . , λ n according to ( 11 ). Now, we generate 10000 indep endent observ ations from N (0 , 1) (these are the Z v ariables) and for each of these observ ations, we corresp ondingly compute Φ  c n ( α ) − λ i Z √ 1 − λ 2 i  for all n λ i ’ s and tak e the pro duct of them to obtain Q n i =1 Φ  c n ( α ) − λ i Z √ 1 − λ 2 i  . Then taking the mean of this quan tit y o ver all 10000 Z observ ations and subtracting this mean from 1, we obtain the sim ulated FWER under global null for this setting. Analogous steps are carried out for the b oth-sided case. W e rep eat this whole pro cess for several com binations of ( n, α, δ ). T ables are provid ed whic h present the estimates of FWER (under H 0 ) for b oth the testing pro cedures, under a w eakly dep enden t m ultiv ariate Normal setup satisfying the pro duct correlation structure, as defined in previous section. F or b oth the procedures, there are t wo tables for t w o choices of α , viz., 0.1 and 0.05, considering differen t v alues of n and δ in eac h table. 6.2 Sim ulation Results 6.2.1 Adjusted Bonferroni (One-sided) T able 1: Results for α = 0 . 10 δ = 0 . 1 δ = 0 . 25 δ = 0 . 5 δ = 0 . 75 δ = 1 n = 2500 0.09887 0.09883 0.09989 0.09985 0.09981 n = 5000 0.09765 0.09979 0.09981 0.09993 0.09980 n = 7500 0.09949 0.09938 0.09960 0.09991 0.09992 n = 10000 0.09927 0.09892 0.10012 0.10007 0.10007 T able 2: Results for α = 0 . 05 δ = 0 . 1 δ = 0 . 25 δ = 0 . 5 δ = 0 . 75 δ = 1 n = 2500 0.04971 0.04972 0.04979 0.05000 0.05003 n = 5000 0.04986 0.04989 0.05008 0.04984 0.05007 n = 7500 0.04928 0.05018 0.04994 0.04998 0.05003 n = 10000 0.04983 0.04980 0.05011 0.05002 0.04999 8 6.2.2 Sidak (One-sided) T able 3: Results for α = 0 . 10 δ = 0 . 1 δ = 0 . 25 δ = 0 . 5 δ = 0 . 75 δ = 1 n = 2500 0.09888 0.09884 0.09989 0.09986 0.09981 n = 5000 0.09765 0.09979 0.09981 0.09993 0.09980 n = 7500 0.09949 0.09938 0.09961 0.09992 0.09992 n = 10000 0.09927 0.09892 0.10012 0.10007 0.10007 T able 4: Results for α = 0 . 05 δ = 0 . 1 δ = 0 . 25 δ = 0 . 5 δ = 0 . 75 δ = 1 n = 2500 0.04971 0.04972 0.04979 0.05000 0.05003 n = 5000 0.04986 0.04989 0.05008 0.04984 0.05007 n = 7500 0.04928 0.05018 0.04994 0.04998 0.05003 n = 10000 0.04983 0.04980 0.05011 0.05002 0.04999 6.2.3 Adjusted Bonferroni (Tw o-sided) T able 5: Results for α = 0 . 10 δ = 0 . 1 δ = 0 . 25 δ = 0 . 5 δ = 0 . 75 δ = 1 n = 2500 0.09972 0.09989 0.09994 0.09995 0.09999 n = 5000 0.09980 0.10018 0.09996 0.10002 0.09998 n = 7500 0.10012 0.10001 0.09994 0.09997 0.09998 n = 10000 0.10011 0.09991 0.10002 0.09999 0.10000 T able 6: Results for α = 0 . 05 δ = 0 . 1 δ = 0 . 25 δ = 0 . 5 δ = 0 . 75 δ = 1 n = 2500 0.04993 0.04996 0.05002 0.05007 0.05003 n = 5000 0.04988 0.04999 0.04997 0.04999 0.05000 n = 7500 0.05004 0.05000 0.04997 0.05001 0.04999 n = 10000 0.05004 0.04999 0.05000 0.04999 0.05000 9 6.2.4 Sidak (Two-sided) T able 7: Results for α = 0 . 10 δ = 0 . 1 δ = 0 . 25 δ = 0 . 5 δ = 0 . 75 δ = 1 n = 2500 0.09972 0.09989 0.09994 0.09995 0.09999 n = 5000 0.09980 0.10018 0.09996 0.10002 0.09998 n = 7500 0.10012 0.10001 0.09994 0.09997 0.09998 n = 10000 0.10011 0.09991 0.10002 0.09999 0.10000 T able 8: Results for α = 0 . 05 δ = 0 . 1 δ = 0 . 25 δ = 0 . 5 δ = 0 . 75 δ = 1 n = 2500 0.04993 0.04996 0.05002 0.05007 0.05003 n = 5000 0.04988 0.04999 0.04997 0.05000 0.05000 n = 7500 0.05004 0.05000 0.04997 0.05001 0.04999 n = 10000 0.05004 0.04999 0.05000 0.04999 0.05000 The sim ulation results convincingly illustrate that under this type of w eakly dep endent correlation structure, for a large n um b er of hypotheses, the FWER under global null is extremely close to the target v alues α for all δ > 0 for b oth pro cedures. 7 Concluding Remarks Explicit ev aluation of FWER or k -FWER requires the knowledge of joint distribution of test statistics under null h yp otheses. While this computation is simple under indep endence, this b ecomes in tractable under dep endence ( Dey and Bhandari , 2025 ). W e revisit here the classical testing problem of Normal means under w eak dependence. W e fo cus on an adjusted v ersion of Bonferroni procedure and the Sidak pro cedure. W e show that b oth of them control FWER at the desired level exactly as the n um b er of hypotheses approac hes infinit y . T ow ards obtaining these results, we establish several new probabilistic asymptotic results that might b e insightful in probability theory to o. Sev eral works hav e fo cused on asymptotic FWER in multiple testing ( Das and Bhandari , 2025 ; Prosc han and Shaw , 2011 ). The premise of this work is muc h more general from the asp ects of dep endence structures, general configurations of hypotheses, and p ow er considerations. There are several in triguing extensions worth exploring. One interesting problem would b e to study the limits of stepwise pro cedures under such weakly dep enden t structures. Another c hallenge is to study these from a Ba y esian p oint of view ( Bogdan et al. , 2011 ). References M. Bogdan, A. Chakrabarti, F. F rommlet, and J. K. Ghosh. Asymptotic Bay es-optimality under sparsit y of some multiple testing procedures. The Annals of Statistics , 39(3):1551 – 1579, 2011. URL https://doi.org/10.1214/10- AOS869 . 10 H. Cramer. Mathematic al Metho ds of Statistics . Princeton Universit y Press, 1946. N. Das and S. K. Bhandari. FWER for normal distribution in nearly indep enden t setup. Statistics & Pr ob ability L etters , 219:110340, 2025. URL https://www.sciencedirect.com/ science/article/pii/S0167715224003092 . S. Datta and M. Dey . An asymptotically exact m ultiple testing pro cedure under dep endence. Statistics & Pr ob ability L etters , 230:110609, 2026. URL https://www.sciencedirect.com/ science/article/pii/S0167715225002548 . M. Dey . On limiting b ehaviors of step wise multiple testing procedures. Statistic al Pap ers , 65: 5691–5717, 2024. URL https://doi.org/10.1007/s00362- 024- 01613- 6 . M. Dey . Asymptotics in multiple hypotheses testing under dep endence: b eyond normalit y . Statistic al Pap ers , 66(7), 2025. URL https://doi.org/10.1007/s00362- 025- 01770- 2 . M. Dey and S. K. Bhandari. FWER go es to zero for correlated normal. Statistics & Pr ob ability L etters , 193:109700, 2023. URL https://www.sciencedirect.com/science/article/pii/ S0167715222002139 . M. Dey and S. K. Bhandari. Bounds on generalized family-wise error rates for normal distributions. Statistic al Pap ers , 65:2313–2326, 2024. URL https://doi.org/10.1007/ s00362- 023- 01487- 0 . M. Dey and S. K. Bhandari. Some results on generalized familywise error rate con trolling pro cedures under dep endence. Stat , 14(3):e70088, 2025. URL https://onlinelibrary. wiley.com/doi/abs/10.1002/sta4.70088 . T. Dic khaus. F alse Disc overy R ate and Asymptotics, PhD Thesis . Heinrich-Heine-Univ ersit y D ¨ usseldorf, 2008. T. Dic khaus. Simultane ous Statistic al Infer enc e With Applic ations in the Life Scienc es . Springer, Heidelb erg, 2014. T. Dickhaus, R. Heller, A. T. Hoang, and Y. Rinott. A pro cedure for multiple testing of partial conjunction hypotheses based on a hazard rate inequalit y . Bernoul li , 32(1):274–298, 2026. S. Dudoit and Mark J. Laan. Multiple T esting Pr o c e dur es with Applic ations to Genomics . Springer Series in Statistics. Springer, 2008. A. F arcomeni. Some results on the con trol of the false disco v ery rate under dep endence. Sc andinavian Journal of Statistics , 34(2):275–297, 2007. URL https://onlinelibrary. wiley.com/doi/abs/10.1111/j.1467- 9469.2006.00530.x . H. Finner, T. Dickhaus, and M. Roters. Dep endency and false disco very rate: Asymptotics. The Annals of Statistics , 35(4):1432–1455, 2007. URL https://www.jstor.org/stable/ 25464546 . H. Finner, T. Dic khaus, and M. Roters. On the false discov ery rate and an asymptotically optimal rejection curv e. The A nnals of Statistics , 37(2):596–618, 2009. URL https://doi. org/10.1214/07- AOS569 . V. Gon tsc haruk and H. Finner. Asymptotic fdr control under weak dependence: A coun terexample. Statistics & Pr ob ability L etters , 83(8):1888–1893, 2013. URL https: //www.sciencedirect.com/science/article/pii/S0167715213001466 . 11 J. H ¨ usler. Asymptotic approximation of crossing probabilities of random sequences. Zeitschrift f¨ ur Wahrscheinlichkeitsthe orie und V erwandte Gebiete , 63(2):257–270, 1983. URL https: //doi.org/10.1007/BF00538965 . M. R. Leadbetter, G. Lindgren, and H. Ro otz´ en. Extr emes and R elate d Pr op erties of R andom Se quenc es and Pr o c esses . Springer Series in Statistics, 1983. URL https://doi.org/10. 1007/978- 1- 4612- 5449- 2 . E. L. Lehmann and J. P . Romano. Generalizations of the familywise error rate. The Annals of Statistics , 33(3):1138 – 1154, 2005. URL https://doi.org/10.1214/009053605000000084 . M. A. Proschan and P . A. Sha w. Asymptotics of Bonferroni for dep endent normal test statistics. Statistics & Pr ob ability L etters , 81(7):739–748, 2011. URL https://www. sciencedirect.com/science/article/pii/S0167715210003329 . Statistics in Biological and Medical Sciences. J. D. Storey and R. Tibshirani. Statistical significance for genomewide studies. Pr o c e e dings of the National A c ademy of Scienc es , 100(16):9440–9445, 2003. URL https://www.pnas.org/ doi/abs/10.1073/pnas.1530509100 . J. D. Storey , J. E. T a ylor, and D. Siegm und. Strong control, conserv ativ e p oin t estimation and sim ultaneous conserv ative consistency of false discov ery rates: A unified approach. Journal of the R oyal Statistic al So ciety Series B: Statistic al Metho dolo gy , 66(1):187–205, 2003. URL https://doi.org/10.1111/j.1467- 9868.2004.00439.x . Y. L. T ong. Pr ob ability Ine qualities in Multivariate Distributions . Probabilit y and Mathematical Statistics. Academic Press, 1980. App endix A Pro ofs of theoretical results men tioned in Section 3 A.1 Pro of of Theorem 3.1 W e at first prov e the result for τ ∈ (0 , ∞ ) and then extend to the extreme cases. Note that the correlation condition (1.1) in H¨ usler ( 1983 ) is equiv alen t to the correlation structure of the weakly dep enden t standard Gaussian sequence { X n } . Let { v n } b e a sequence of n such that n (1 − Φ( v n )) → τ ∈ (0 , ∞ ) as n → ∞ . Th us, condition A of H ¨ usler ( 1983 ) for the random sequence { X n } and the cutoff sequence { v n } clearly holds. Let N n := # { i ≤ n : X i > v n } . Then, utilizing Theorem 4.1 and Theorem 3.5 of H ¨ usler ( 1983 ), we get N n d − → P oisson( τ ) as n → ∞ . Therefore, for eac h fixed k ≥ 1, P  M k n ≤ v n  = P ( N n ≤ k − 1) = k − 1 X s =0 P ( N n = s ) 12 − → e − τ k − 1 X s =0 τ s s ! as n → ∞ . (12) So, we are only required to sho w P  M k d n ≤ u n  − P  M k d n ≤ v d n  → 0 as n → ∞ . | P  M k d n ≤ u n  − P  M k d n ≤ v d n  | = P  min { u n , v d n } ≤ M k d n ≤ max { u n , v d n }  ≤ P d n [ i =1 { min { u n , v d n } ≤ X i ≤ max { u n , v d n }} ! ≤ | d n (Φ( u n ) − Φ( v d n )) | =     d n (1 − Φ( v d n )) − d n (1 − Φ( u n ))     − → 0 as n → ∞ . (13) [ ∵ lim n →∞ d n (1 − Φ( v d n )) = lim n →∞ d n (1 − Φ( u n )) = θ τ . ] The extreme cases are remaining to b e sho wn. So, now we take { d n } and { u n } such that d n (1 − Φ( u n )) → 0 as n → ∞ . Supp ose, for each τ ′ ∈ (0 , 1), we hav e a sequence { w n } ( w n = w n ( τ ′ )) so that d n (1 − Φ( w n )) → τ ′ as n → ∞ . Then, using ( 12 ) and ( 13 ) we hav e, P { M k d n ≤ w n } → e − τ ′ P k − 1 s =0 τ ′ s s ! . No w if lim n →∞ d n (1 − Φ( u n )) = 0, we must ha v e u n > w n for sufficiently large n , so that lim sup n →∞ P { M k d n ≤ u n } ≥ lim n →∞ P { M k d n ≤ w n } = e − τ ′ k − 1 X s =0 τ ′ s s ! . Since this holds for arbitrary τ ′ > 0, it follows that P { M k d n ≤ u n } → 1 as n → ∞ . The corresp onding result for the case lim n →∞ d n (1 − Φ( u n )) = ∞ follo ws similarly , completing the proof. A.2 Pro of of Remark 2 W e no w analyze the con vergence rates of the limiting results of Theorem 3.2 . W e consider the same re-notations and setup as mentioned in section 3 of the main pap er. Let c n denote an y of c B on ( n, α ) or c S id ( n, α ). No w, | F W E R − α | = | P ( M n 0 ≤ c n ) − (1 − α ) | ≤ | P ( M n 0 ≤ c n ) − Φ n 0 ( c n ) | + | Φ n 0 ( c n ) − (1 − α ) | = I 1 + I 2 (sa y) . (14) Using Corollary 4.2.4 of Leadbetter et al. ( 1983 ), I 1 ≤ K · X 1 ≤ i 0, Φ − 1  1 − β n  = p 2 log n − log log n + log 4 π + log β 2 √ 2 log n + O  1 log n  . (19) Hence, using the expansion (1 − α ) 1 n = 1 − − log(1 − α ) − O ( 1 n ) n and ( 19 ), we can write, for all sufficien tly large n , Φ − 1 ((1 − α ) 1 n ) = p 2 log n − log log n + log 4 π + log( − log (1 − α ) − O ( 1 n )) 2 √ 2 log n + O  1 log n  . (20) No w, c B on ( n, α ) = Φ − 1  1 − − ln(1 − α ) n  ∼ Φ − 1  1 − − ln(1 − α ) p 1 n 1  ∼ p 2 ln n 1 (using ( 19 )) . One also has c S id ( n, α ) = Φ − 1  (1 − α ) 1 n  ∼ Φ − 1  [(1 − α ) p 1 ] 1 n 1  ∼ p 2 ln n 1 (using ( 20 )) . Since lim n →∞ √ 2 log n 1 µ n 1 < 1, the rest follo ws using ( 18 ). B.2 Pro of of Prop osition 1 Using equation ( 19 ), c 2 β n , n = 2 log n − log log n − log 4 π − log β n + o (1) . (21) Hence, 1 √ 2 π · e − 1 2 c 2 β n , n · e − c β n , n · t √ 2 log n ∼ e − 1 2 (2 log n − log log n − log 4 π − log β n ) √ 2 π · √ 2 log n · e − t ( c β n , n − √ 2 log n ) · e − t √ 2 log n (using ( 21 )) ∼ 1 n · √ log n · 2 √ π · √ β √ 2 π · √ 2 log n · e − t √ 2 log n (since c β n , n − p 2 log n → 0 from ( 19 )) ∼ √ β · e − t √ 2 log n n . (22) 15 No w, d n · (1 − Φ ( c β , n + t )) ∼ d n · ϕ ( c β , n + t ) c β , n + t ∼ d n · e − t 2 / 2 √ 2 π · e − 1 2 c 2 β , n · e − c β , n t √ 2 log n = p β · e − t 2 / 2 · d n · e − t √ 2 log n n (utilizing ( 22 )). The rest is obvious. C Pro ofs of theoretical results men tioned in Section 5 The following results are imp erativ ely necessary in proving Theorem 5.1 : Lemma C.1 (Lemma 11.1.2 of Leadbetter et al. ( 1983 )) . L et ξ 1 , , ξ 2 , . . . , ξ n b e standar d normal variables with c ovarianc e matrix Λ 1 = (Λ 1 ij ) and η 1 , η 2 , . . . , η n b e standar d normal variables with c ovarianc e matrix Λ 0 = (Λ 0 ij ) . L et λ ij = max( | Λ 1 ij | , | Λ 0 ij | ) . F urther, let u = ( u 1 , . . . , u n ) and v = ( v 1 , . . . , v n ) b e ve ctors of r e al numb ers and write w = min ( | u 1 | , . . . , | u n | , | v 1 | , . . . , | v n | ) . Then, P {− v j < ξ j ≤ u j for j = 1 , 2 , . . . , n } − P {− v j < η j ≤ u j for j = 1 , 2 , . . . , n } ≤ 4 2 π X 1 ≤ i v n , | X j | > v n ) ≤ (1 − Φ ⋆ ( v n )) 2 + K · | ρ ij | · exp  − v n 1 + | ρ ij |  . (26) No w, let r ≤ n be some integer and I b e a subset of { 1 , 2 , . . . , n } suc h that | I | := # { i ∈ I } ≤ n/r . Then, using ( 26 ) w e hav e, X i v n , | X j | > v n ) ≤ X i v n , | X j | > v n ) = 0, verifying condition D ′ of H ¨ usler ( 1983 ). Th us, by Theorem 3.5 of H ¨ usler ( 1983 ), letting N ⋆ n = # { i ≤ n : | X i | > v n } , we get N n d − → P oisson(2 τ ) as n → ∞ . Therefore, for eac h fixed k ≥ 1, P  L k n ≤ v n  = P ( N ⋆ n ≤ k − 1) = k − 1 X s =0 P ( N ⋆ n = s ) − → e − 2 τ k − 1 X s =0 (2 τ ) s s ! as n → ∞ . (28) So, we are only required to sho w P  L k d n ≤ u n  − P  L k d n ≤ v d n  → 0 as n → ∞ .    P  L k d n ≤ u n  − P  L k d n ≤ v d n     = P  min { u n , v d n } ≤ L k d n ≤ max { u n , v d n }  ≤ P d n [ i =1 { min { u n , v d n } ≤ | X i | ≤ max { u n , v d n }} ! ≤   d n (Φ ⋆ ( u n ) − Φ ⋆ ( v d n ))   17 = 2     d n (1 − Φ( v d n )) − d n (1 − Φ( u n ))     − → 0 as n → ∞ . [ ∵ lim n →∞ d n (1 − Φ( v d n )) = lim n →∞ d n (1 − Φ( u n )) = θ τ . ] No w, the result for the extreme cases, i.e., when lim n →∞ d n (1 − Φ( u n )) = 0 and lim n →∞ d n (1 − Φ( u n )) = ∞ follows similarly as in the pro of of Theorem 3.1 . This completes the pro of. C.2 Pro of of Remark 4 | F W E R − α | = | P ( | Z i | ≤ c 2 n , i = 1(1) n 0 ) − (1 − α ) | ≤ | P ( | Z i | ≤ c 2 n , i = 1(1) n 0 ) − (Φ ( c 2 n ) − Φ ( − c 2 n )) n 0 | + | (Φ ( c 2 n ) − Φ ( − c 2 n )) n 0 − (1 − α ) | . No w, [Φ( c 2 n ( α )) − Φ( − c 2 n ( α ))] n 0 = [2Φ( c 2 d n ( α )) − 1] n 0 = [2  1 − − log(1 − α ) − o (1) 2 n  − 1] n 0 =  1 − − log(1 − α ) − o (1) n  n 0 . (29) Using equations ( 24 ) and ( 29 ), the rest follo ws along similar lines to the pro of of Remark 2 . C.3 Pro of of Theorem 5.3 F or the b oth-sided testing problem, we hav e Any P wr B S = 1 − P   \ i ∈I 1 {| X i | ≤ c 2 n ( α ) }   = 1 − P   \ i ∈I 1 {− c 2 n ( α ) − µ i ≤ Z i ≤ c 2 n ( α ) − µ i }   , Z i = X i − µ i ∼ N (0 , 1) , ≥ 1 − P  − c 2 n ( α ) − µ ( n 1 ) ≤ Z ≤ c 2 n ( α ) − µ ( n 1 )  , Z ∼ N (0 , 1) , = 1 − [Φ  c 2 n ( α ) − µ ( n 1 )  − Φ  − c 2 n ( α ) − µ ( n 1 )  ] → 1 as n → ∞ , under giv en condition. C.4 Pro of of Theorem 5.4 Let I 11 and I 12 denote the index sets with corresp onding X i ’s ha ving p ositiv e and negativ e means, resp ectively . Let U i = ( Z i for i ∈ I 11 , − Z i for i ∈ I 12 . Then, 1 − Any P w r B S 18 = P   \ i ∈I 1 {− c 2 n ( α ) − µ i ≤ Z i ≤ c 2 n ( α ) − µ i }   , Z i = X i − µ i ∼ N (0 , 1) , = P   \ i ∈I 11 {− c 2 n ( α ) − µ i ≤ Z i ≤ c 2 n ( α ) − µ i } \ \ j ∈I 12 {− c 2 n ( α ) − µ j ≤ Z j ≤ c 2 n ( α ) − µ j }   ≤ P   \ i ∈I 11 { Z i ≤ c 2 n ( α ) − µ i } \ \ j ∈I 12 {− c 2 n ( α ) − µ j ≤ Z j }   ≤ P   \ i ∈I 11 { Z i ≤ c 2 n ( α ) − µ } \ \ j ∈I 12 {− c 2 n ( α ) + µ ≤ Z j }   = P n 1 \ i =1 { U i ≤ c 2 n ( α ) − µ } ! → 0 as n → ∞ , using Prop osition 1 and Theorem 3.1 , completing the pro of. 19