Quenched large deviations for randomly weighted geodesic random walks

Quenc hed large deviations for randomly w eigh ted geo desic random w alks Rik V ersendaal ∗ F ebruary 20, 2026 Abstract W e consider w eighted geo desic random w alks in a complete Riemannian manifold ( M , g ) . W e sho w that for almost all sequences of weigh ts (with respect to a suitable measure), these w eighted geo desic random walks satisfy , when suitably scaled, a large deviation principle with a univ ersal rate function. This extends the results from [3], where this w as shown for the real- v alued case. It turns out the argument is also v alid for general vector spaces. This allo ws us to use the methodology of [9], in which large deviations for geo desic random w alks are obtained from large deviation estimates for asso ciated random walks in tangent spaces. K ey wor ds: Geodesic random w alks, weigh ted random w alks, large deviations, Cramér’s theorem, sto c hastic pro cesses in manifolds 2020 Mathematics Subje ct Classific ation: 60F10, 60G50, 60D05 . 1 In tro duction Let X 1 , X 2 , . . . b e a sequence of indep enden t, iden tically distributed random v ariables in R and consider the random walk S n = P n i =1 X i . A classical result in large deviation theory is Cramér’s theorem ([2, Theorem 2.2.3]), which roughly states that P  1 n S n ≈ x  ≈ e − nI ( x ) where I ( x ) = sup t ∈ R { tx − Λ( t )) } . with Λ( t ) = log E ( e tX 1 ) . In [3], large deviations w ere studied for weigh ted random w alks. More precisely , given unit vectors θ n ∈ R n , consider the random v ariables W θ n n := 1 √ n n X i =1 θ n i X i . It is then shown that for almost all sequences { θ n } n of w eights (with resp ect to the measure σ as defined in Section 2.1.2), we ha ve P  W θ n n ≈ x  ≈ e − nI w ( x ) . ∗ Delft Institute of Applied Mathematics, TU Delft, Netherlands; r.versendaal@tudelft.nl . 1 Here, I w ( x ) = sup t ∈ R { tx − Ψ( t ) } , where Ψ( t ) = E (Λ( tZ )) with Z ∼ N (0 , 1) . In particular, the rate function I w is indep endent of the w eights θ n . This shows that for most weigh ts, the asso ciated w eighted random w alks hav e the same large deviations. Moreo ver, we see that the rate function in Cramér’s theorem is different from this univ ersal rate function I w . As argued in [3], this shows that Cramér’s theorem is in a sense ’atypical’ . Our aim is to extend the result from [3] to weigh ted geo desic random w alks in Riemannian manifolds. Geo desic random w alks are piecewise geo desic paths, where the directions of the geo desics are c hosen at random. The w eights then determine the time for which w e follow eac h direction. W e refer to Section 2.1 for a detailed description. In [9] it w as sho wn ho w Cramér’s theorem for general vector spaces can b e extended to geodesic random w alks. Up on analysing the pro of in [3], one realizes that the result remains v alid in R d , and ultimately in a general vector space V , see Theorem 2.3 for a precise statemen t. This op ens up the p ossibilit y to ap ply tec hniques from [9] to stud y large deviations for weigh ted geo desic random w alks based on the results in [3]. A k ey step in this metho dology is the splitting of the geodesic random w alks in smaller pieces. Where in [9] these pieces are identically distributed, b ecause of the weigh ts this is no longer the case. W e o vercome this b y using the symmetries of the measure σ on the w eights (see Section 2.1.2) showing that each piece of the weigh ted geo desic random w alk still follows the same large deviations, which is ultimately what we need. F urthermore, our w ork demonstrates that the approach in [9] is rather robust, and emphasizes the relev an t prop erties of the sto c hastic processes for the metho dology to w ork. In particular, it motiv ates that the metho ds in [9] can b e extended to a general framework to study large deviations for discrete-time pro cesses in Riemannian manifolds from their Euclidean counterparts. This can for instance b e used to obtain a Riemannian analogue of the Gartner-Ellis theorem (see [2, Section 2.3]) and large deviations for Mark o v c hains with v alues in Riemannian manifolds. F urthermore, it can be used to extend recen t results on large deviations for random pro jections of l p -balls ([4, 1, 8]) and projections on finite-dimensional subspaces ([5]) to Riemannian manifolds. This will b e the topic of future work. The pap er is structured as follows. In Section 2 we define w eighted geo desic random walks and state our main theorem (Theorem 2.2). F urthermore, we formulate the extension of large deviation for w eighted random walks as in [3] to general v ector spaces. In Section 3 we pro v e Theorem 2.2. F ollowing the ideas from [9], the pro of is split in t wo parts, proving the upp er bound and lo wer b ound for the large deviation principle for the weigh ted geo desic random w alks separately . 2 Randomly w eigh ted geo desic random w alks Let ( M , g ) b e a complete Riemannian manifold. Denote by d the Riemannian distance function. In this section w e define w eigh ted geo desic random w alks in M , and state our main result, Theorem 2.2. F urthermore, w e pro vide the extension of the large deviation result from [3] to general vector spaces, which is essential for the pro of of Theorem 2.2. W e conclude with a d iscussion on ho w the result for v ector spaces relates to large deviations for k -dimensional projections as in [5]. 2 2.1 Geo desic random w alks with w eighted incremen ts In a manifold, w e cannot define random walks as sums of random v ariables. Instead, ge o desic r andom walks are defined recursively by following pieces of geo desics (see e.g. [7, 9]). W e then in tro duce the w eigh ts as the time for which w e follo w each piece of geodesic. The pro cedure of follo wing geo desics is enco ded b y the Riemannian exp onential map . The map Exp : T M → M is defined by Exp x v = Exp( x, v ) = γ (1) , where γ : [0 , 1] → M is the geo desic with γ (0) = x and ˙ γ (0) = v . Since we assume M is complete, Exp is defined on all of T M . With this notation at hand, w e define weigh ted geodesic random walks. Definition 2.1 (W eighted geo desic random w alks) . Fix x 0 ∈ M , n ∈ N and let α n ∈ R n . A p air ( {S α k } 0 ≤ k ≤ n , { X k } 1 ≤ k ≤ n ) is c al le d a weigh ted geodesic random walk with incr ements { X k } 1 ≤ k ≤ n and weights α n , and starte d at x 0 , if the fol lowing hold: 1. S α n 0 = x 0 , 2. X k ∈ T S α n k − 1 M for al l 1 ≤ k ≤ n , 3. S α n k = Exp S α n k − 1 ( α n k X k ) for al l 1 ≤ k ≤ n . W e will consider the increments { X n } n ≥ 1 to be random v ariables. Note that the tangen t space from which the next increment is drawn dep ends on the current p osition of the geo desic random w alk. Therefore, we will put a collection { µ x } x ∈ M of probability measures on the tangent bundle, where µ x ∈ P ( T x M ) is a probabilit y measure on T x M for ev ery x ∈ M . 2.1.1 Iden tically distributed incremen ts and parallel transport T o compare probabilit y distributions on different tangent spaces, w e identify tangent spaces at differen t p oin ts using parallel transp ort. F or x, y ∈ M and a curve γ connecting x and y , we denote b y τ γ ; xy : T x M → T y M parallel transp ort along γ . If γ is a shortest geo desic betw een x and y , we simply write τ xy , omitting the reference to γ . Generally , we only use this notation when x and y are sufficiently close, so that the shortest geodesic is unique. W e sa y the measures µ x and µ y are identic al if for an y piecewise smo oth curve γ connecting x and y we ha ve µ x = µ y ◦ τ γ ; xy , i.e., the distributions are inv ariant under parallel transp ort along any piecewise smooth curv e. 2.1.2 W eigh t distribution Our goal is to show that for almost all w eights, the sequence of w eighted geodesic random w alks, when suitably scaled, satisfies a large deviation principle with a universal rate function. F or suc h statemen ts to mak e sense, we need to introduce a probabilit y distribution on the space of w eigh ts. F or this, w e follow [3]. Let S n − 1 b e the unit sphere in R n and denote by σ n − 1 the uniform measure on S n − 1 . Define the pro duct space S = Q ∞ n =1 S n − 1 with projections π n : S → S n − 1 . W e consider a probabilit y measure σ on S suc h that σ ◦ π n = σ n − 1 for all n ∈ N . 3 2.2 Main result Our main result extends Theorem 2 from [3] to weigh ted geo desic random walks. F or this, consider θ ∈ S a triangular array of w eights. W e define {S θ n k } 0 ≤ k ≤ n to b e the weigh ted geodesic random w alk with resp ect to the scaled weigh ts α n = 1 √ n θ n (see Definition 2.1). Our main result concerns the large deviations for the sequence {S θ n n } n ∈ N . Theorem 2.2. L et θ ∈ S and let {S θ n n } n ∈ N b e the weighte d ge o desic r andom walk as define d ab ove. A ssume the incr ements of the ge o desic r andom walks ar e b ounde d, indep endent and identic al ly distribute d. L et σ b e a me asur e on S as in Se ction 2.1. Then for σ -almost every θ , the se quenc e {S θ n n } n ∈ N satisfies a lar ge deviation principle with r ate function I ( x ) = inf v ∈ exp − 1 x 0 x sup λ ∈ T x 0 M ⟨ λ, v ⟩ − Ψ( λ ) , wher e Ψ( λ ) = E (Λ x 0 ( Z λ )) with Z ∼ N (0 , 1) and Λ x 0 ( λ ) = log R T x 0 M e ⟨ λ,w ⟩ µ x 0 (d w ) . The pro of is inspired by the proof of Cramér’s theorem for geodesic random walks in [9]. The k ey idea is to asso ciate the weigh ted geo desic random walk S θ n n to a weigh ted random w alk in T x 0 M . The large deviations for this asso ciated random w alk follow from Theorem 2.3. Unfortunately , the connection b et w een the tw o random walks does not immediately allow us to tranfer the large deviations to S θ n n . Instead, we carefully analyse the connection b et w een the tw o pro cesses and pro ve the upp er b ound and low er b ound of the large deviation principle for {S θ n n } n ∈ N separately . In particular, Theorem 2.2 follows immediately from Propositions 3.5 and 3.7. 2.2.1 Large deviations for w eighted random w alks in v ector spaces T o pro ve Theorem 2.2, we need an extension of [3, Theorem 2] to and arbitrary v ector space V , which for us will b e the tangen t space T x 0 M . Let X 1 , X 2 , . . . b e a sequence of indep enden t, iden tically distributed random v ariables in V . Let θ ∈ S b e a sequence of coefficients. W e define W θ n k := 1 √ n k X i =1 θ n i X i (2.1) for k = 1 , 2 , . . . , n . The pro of in [3] for the large deviations for w eigh ted random w alks in R extends to higher dimen- sional Euclidean spaces. Up on choosing a basis, we get the follo wing. Theorem 2.3. L et X 1 , X 2 , . . . b e a se quenc e of indep endent, identic al ly distribute d r andom vari- ables in a ve ctor sp ac e V . Denote by Λ( λ ) = log E ( e ⟨ λ,X 1 ⟩ ) the lo g-moment gener ating function of X 1 . Define the r andom variables W θ n k as in (2.1) and let σ b e as in Se ction 2.1.2. Then for σ -almost every θ , the se quenc e { W θ n n } n ∈ N satisfies the lar ge deviation principle with go o d r ate function I ( v ) = Ψ ∗ ( v ) = sup λ ∈ V {⟨ v , λ ⟩ − Ψ( λ ) } , (2.2) wher e Ψ( λ ) = E Z (Λ( Z λ )) with Z ∼ N (0 , 1) . By using V aradhan’s lemma (see e.g [2, Theorem 4.3.1]), and the symmetry prop erties of the measure σ , we can deduce the follo wing from Theorem 2.3. This is essen tial for our proof of Theorem 2.2, where w e split up the w eighted geo desic random w alks in smaller pieces. 4 Corollary 2.4. L et the assumptions of The or em 2.3 b e satisfie d. Then for σ -almost every θ we have lim n →∞ 1 n log E  e n ⟨ λ,W θ ⌊ k − 1 n ⌋ ⟩  = 1 k Ψ( λ ) . Pr o of. By V aradhan’s lemma ([2, Theorem 4.3.1]) it follo ws from the large deviation principle in Theorem 2.3 that lim n →∞ 1 n log E  e n ⟨ λ,W θ n n ⟩  = Ψ( λ ) , By comparing sequences θ and ˜ θ for which θ n and ˜ θ n only differ in the first ⌊ nk − 1 ⌋ elemen ts, the ab o v e implies that Ψ k ( λ ) := lim n →∞ 1 n log E  e n ⟨ λ,W θ n ⌊ nk − 1 ⌋ ⟩  is indep enden t of θ for σ -almost ev ery θ . No w write n i = i ⌊ nk − 1 ⌋ for i = 1 , . . . , k − 1 and n k = n . Since the uniform distribution on S n − 1 is p erm utation inv ariant, it follows that for σ -almost every θ we hav e lim n →∞ 1 n log E  e n ⟨ λ,W θ n n i +1 − W θ n n i ⟩  = Ψ k ( λ ) . for all i = 1 , . . . , k − 1 . Since also lim n →∞ 1 n log E  e n ⟨ λ,W θ n n ⟩  = k − 1 X i =0 lim n →∞ 1 n log E  e n ⟨ λ,W θ n n i +1 − W θ n n i ⟩  , b y independence of the increments, it follows that Ψ k ( λ ) = 1 k Ψ( λ ) as desired. 2.2.2 Connection to large deviations for k -dimensional pro jections In [5], large deviations are studied for random multidimensional pro jections. In particular, for a sequence Y 1 , Y 2 , . . . of indep enden t, identically distributed random v ariables in R , the authors consider (among many other things) sequences of k -dimensional pro jections of Y ( n ) := ( Y 1 , . . . , Y n ) . More precisely , for a sequence a = { a n,k } n ≥ k with each a n,k an orthonormal k -frame for R n , large deviations are considered for the sequence P n := 1 √ n a T n,k Y ( n ) ∈ R k . It is shown that for almost all suc h sequences of pro jections (with respect to the Haar measure on the Stiefel manifold of orthogonal k -frames on R n ), the sequence { P n } n ≥ k satisfies the large deviation principle with rate function I proj ( v ) = sup λ ∈ R k {⟨ λ, v ⟩ − E Z 1 ,...,Z k (Λ( λ 1 Z 1 + · · · + λ k Z k )) } , where Z 1 , . . . , Z k are indep endent, standard normal random v ariables and Λ is the log-moment generating function of Y 1 . F or a sequence X 1 , X 2 , . . . of independent, identically distributed random v ariables in R k , the random v ariables W θ n n as defined in (2.1) can also b e interpreted as such k -dimensional pro jections. Indeed, one pro jects the sequence ( X 1 1 , . . . , X k 1 , . . . , X 1 n , . . . , X k n ) ∈ R nk on to R k . Therefore, one ma y wonder to what extent the results from [5] are connected to the statement in Theorem 2.3. There are t w o main cav eats: 5 1. F or the results of [5] to apply , the co ordinates of X i m ust indep enden t and identically dis- tributed, i.e., the distribution µ of X i has to be of the form µ ⊗ k 1 . 2. The pro jections we consider map the v ariables X i 1 , X i 2 , . . . , X i n on to the i -th co ordinate. With resp ect to the measure considered in [5], the collection of suc h pro jections has measure 0. A dditionally , for such pro jections, the co ordinates are indep enden t, iden tically distributed. Because the set of pro jections we consider has measure 0, it is inconclusive whether the asso ciated k -projections P n satisfy the large deviation principle with universal rate function I proj . T o inv estigate this, note that w e are comparing pro jections of the form P nk to randomly w eigh ted sums of the form W θ n n . The scaling of these random v ariables is different by a factor √ k . F urther- more, the large deviations of P nk are at rate nk , while those for W θ n n are at rate n . If we denote by I k the universal rate function in (2.2) for { W θ n n } n ∈ N , the question th us becomes whether we ha ve I k ( v ) = k I proj  v √ k  . W e first consider tw o examples. Example 2.5 . Supp ose µ 1 is a standard normal distribution, so that Λ µ 1 ( t ) = 1 2 t 2 . Since µ = µ ⊗ k 1 w e ha v e I k ( v ) = sup λ ∈ R k ( ⟨ λ, v ⟩ − k X i =1 E Z (Λ µ 1 ( λ i Z )) ) = sup λ ∈ R k ( ⟨ λ, v ⟩ − k X i =1 1 2 λ 2 i ) = k X i =1 1 2 v 2 i = 1 2 | v | 2 . Lik ewise, w e find that I proj ( v ) = sup λ ∈ R k  ⟨ λ, v ⟩ − 1 2 E Z 1 ,...,Z k (( λ 1 Z 1 + · · · + λ k Z k ) 2 )  = sup λ ∈ R k  ⟨ λ, v ⟩ − 1 2 | λ | 2  = 1 2 | v | 2 Here w e used that Z 1 , . . . , Z k are independent and E ( Z 2 i ) = 1 . This sho ws that I k ( v ) = k I proj  v √ k  . Note that this relies on the fact that I proj ( v ) is not affected by the sp ecific scaling with k . Example 2.5 remains true when µ 1 is a general normal distribution. Intuitiv ely , this can be explained as follows. T ak e an y orthonormal k -frame a of R n and let X ( n ) = ( X 1 , X 2 , . . . , X n ) b e a vector of indep enden t, identically distributed random v ariables with a normal distribution. Let us write a 1 , . . . , a n for the ro ws of a . Then a T X ( n ) = n X l =1 X l a l ∈ R k . In particular, this shows that eac h co ordinate of the pro jection has a normal distribution. F urther- more, we can compute Co v n X l =1 X l a i l , n X l =1 X l a j l ! = E ( X 2 1 ) n X l =1 a i l a j l = 0 since the columns of a are orthogonal. This sho ws the co ordinates of the pro jection are uncorrelated, and hence indep endent since they are normally distributed. This means we can treat the co ordinates separately , each one b eing a randomly weigh ted sum as in [3]. Since this also holds for our sp ecial pro jections, the result of Example 2.5 is indeed expected. 6 Example 2.6 . T ake k = 2 and µ 1 to b e Poi (1) . A computation sho ws that I 2 ( v ) = sup λ ∈ R 2 n λ 1 v 1 + λ 2 v 2 − e 1 2 λ 2 1 − e 1 2 λ 2 2 + 2 o and I proj ( v ) = sup λ ∈ R 2 n λ 1 v 1 + λ 2 v 2 − e 1 2 | λ | 2 + 1 o Numerically , we obtain that I 2 ((1 , 2)) ≈ 1 . 7940 , 2 I proj  1 √ 2 (1 , 2)  ≈ 1 . 8662 . W e thus find v for which I k ( v )  = k I proj  v √ k  . As Example 2.6 demonstrates, large deviation principles asso ciated to the sp ecial sequences of k - pro jections w e consider in general ha ve a rate function different from the universal coming from [5]. The reason is that for our sp ecial k -pro jections, the co ordinates are indep enden t and ha v e the same distribu tion. This is in con trast to a t ypical sequence of k -pro jections, in whic h all elements are mapp ed to any of the co ordinates, making their joint distribution differ significantly from a pro duct distribution. Ho wev er, as mentioned ab ov e, for our special pro jections, eac h coordinate is indep enden t and has the same distribution. Moreov er, w e can in terpret co ordinate i as a 1-pro jection of the sequence ( X i 1 , X i 2 , . . . , X i n ) ∈ R n . This motiv ates the follo wing result. Prop osition 2.7. L et µ 1 b e a pr ob ability me asur e on R and set µ = µ ⊗ k 1 . Denote by Λ µ , Λ µ 1 the lo g-moment gener ating function of the me asur e µ , r esp e ctively µ 1 . Define I k ( v ) = sup λ ∈ R k {⟨ v , λ ⟩ − E Z (Λ µ ( Z λ )) } and I proj ( v ) = sup λ ∈ R k {⟨ v , λ ⟩ − E Z 1 ,...,Z k (Λ µ ( λ 1 Z 1 + · · · + λ k Z k ) } , wher e Z , Z 1 , . . . , Z k ar e indep endent, standar d normal r andom variables. Then for al l c ∈ R we have I k ( c 1 k ) = k I proj  c √ k 1 k  , wher e 1 k ∈ R k denotes the al l-ones ve ctor. Pr o of. By indep endence we hav e I k ( c 1 k ) = k X i =1 sup λ i ∈ R { cλ i − E Z (Λ µ 1 ( λ i Z )) } = k sup λ ∈ R { cλ − E Z (Λ µ 1 ( λZ )) } On the other hand, if Z 1 , . . . , Z k are indep enden t, standard normal random v ariables, then for λ ∈ R k w e ha v e ⟨ λ, ( Z 1 , . . . , Z k ) ⟩ d = | λ | Z with Z standard normal. This implies that I proj ( v ) = sup λ ∈ R k {⟨ λ, v ⟩ − E Z (Λ µ 1 ( | λ | Z )) } F or v = c 1 k , the optimal λ will be of the form t 1 k . As a consequence, we get I proj ( c 1 k ) = sup t ∈ R n k tc − E Z (Λ µ 1 ( √ k | t | Z )) o = sup t ∈ R n k tc − E Z (Λ µ 1 ( √ k tZ )) o , 7 where we used that | t | Z d = tZ . F rom this it follows that k I proj  c √ k 1 k  = k sup t ∈ R n √ k tc − E Z (Λ µ 1 ( √ k tZ )) o = k sup t ∈ R { tc − E Z (Λ µ 1 ( tZ )) } = I k ( c 1 k ) as desired. 3 Pro of of Theorem 2.2 As explained b elo w Theorem 2.2, w e pro ve the lo w er and upper bound of the large deviation principle separately . In particular, Theorem 2.2 follows immediately from Prop ositions 3.7 and 3.5. Before we get to those, w e first introduce some notation and preliminary results. 3.1 Notation and preliminary results Our proof is inspired b y the metho ds in [9]. Therefore, we rely on connecting the random w alks {S θ n l } 0 ≤ l ≤ n to weigh ted random w alks in the tangent space T x 0 M . Naively , w e could consider v n l := exp − 1 x 0  S θ n l  ∈ T x 0 M . Ho wev er, t wo problems arise. Most imp ortan tly , the Riemannian exp onen tial map need not b e in vertible, so that v n l cannot b e uniquely defined. But even if it is inv ertible, it turns out that cur- v ature forms an obstruction to compare v n n to a random w alk with increments distributed according to µ x 0 . Indeed, one exp ects to compare v n n to the w eigh ted random walk 1 √ n n X l =1 θ n l τ − 1 x 0 S θ n l − 1 X n l ∈ T x 0 M . Because the distributions µ x of the increments are inv ariant under parallel transp ort, w e indeed ha ve τ − 1 x 0 S θ n l − 1 X n l ∼ µ x 0 . Therefore, by Theorem 2.3, these random walks in T x 0 M satisfy for σ -almost ev ery θ a large d eviation principle with rate function Ψ ∗ x 0 . One then hop es to use a con traction principle to also obtain a large deviations for {S θ n n } n . Unfortunately , from Prop osition 3.2 (by taking l = n ) w e see that the difference b et w een v n n and the prop osed random w alk in T x 0 M is O (1) . The connection to a w eighted random walks in the tangent space therefore needs to b e refined. W e solve b oth issues by introducing a parameter m ∈ N and partitioning the random walk in m parts, each consisting of (roughly) ⌊ nm − 1 ⌋ steps. More precisely , define n i = i ⌊ nm − 1 ⌋ for i = 0 , 1 . . . , m − 1 and set n m = n . W e hav e the following b ound on how far our random w alks can w ander in ⌊ nm − 1 ⌋ steps. Lemma 3.1. Fix θ n ∈ S n − 1 and let r b e the uniform upp er b ound on the incr ements of S θ n k . Then d  S θ n k , x 0  ≤ r √ k √ n ≤ r for k = 1 , . . . , n . Similarly, if n i = i ⌊ nm − 1 ⌋ , we have d  S θ n n i , S θ n n i + k  ≤ r √ m for 1 ≤ k ≤ ⌊ nm − 1 ⌋ . 8 Pr o of. By the Cauch y-Sch warz inequalit y and the fact that θ n ∈ S n − 1 , we ha ve k X i =1 | θ n i | ≤ √ k k X i =1 | θ n i | 2 ≤ √ k . (3.1) The triangle inequalit y then giv es us d  S θ n k , x 0  ≤ 1 √ n k X i =1 | θ n i || X n i | ≤ r √ k √ n . The second claim follo ws similarly by using that k ≤ n m . Lemma 3.1 shows that the distance b et w een p oin ts in the same piece of length ⌊ nm − 1 ⌋ of the random w alk deca ys with m . This allo ws us to refine the estimate in Proposition 3.2, and let m tend to infinit y in the end. F urthermore, Lemma 3.1 als o resolves the issue of the non-inv ertibility of the Riemannian exp onen tial map. T o see this, for x ∈ M w e first define the inje ctivity r adius ι ( x ) = sup { t > 0 | exp x is injective on B (0 , t ) } . It turns out that ι is con tinuous on M (see e.g. [6]). By Lemma 3.1, the random w alks {S θ n l } 0 ≤ l ≤ n all remain in the set K = B ( x 0 , r ) , whic h is compact because M is complete. As a consequence, ι attains a minim um ι K > 0 on K , meaning that exp x is injectiv e on B (0 , ι K ) ⊂ T x M for all x ∈ K . In particular, if w e take m large enough so that r √ m < ι K , by Lemma 3.1 w e can uniquely define the random v ectors ˜ v n,m,i k ∈ Exp − 1 S θ n n i − 1  S θ n n i − 1 + k  ⊂ T S θ n n i − 1 M (3.2) of minimal length. The follo wing result sho ws ho w w e can compare the random v ariables v n l := ˜ v n,m, 1 l (whic h is indep enden t of m , since S n 0 = x 0 ) to random v ariables of the form 1 √ n l X k =1 θ n k τ − 1 x 0 S θ n k − 1 X n k . As discussed abov e, the latter is a random walk in T x 0 M with incremen ts distributed as µ x 0 . Prop osition 3.2. Fix n ∈ N , θ n ∈ S n − 1 and let r > 0 b e such that | X n l | ≤ r for al l 1 ≤ l ≤ n . Then ther e is a c onstant C > 0 such that      v n l − 1 √ n l X k =1 θ n k τ − 1 x 0 S θ n k − 1 X n k      ≤ C 1 n l X k =1 ( θ n k ) 2 + C r 3 l 3 / 2 n 3 / 2 . for al l 1 ≤ l ≤ n for which d( S θ n k , x 0 ) < ι ( x 0 ) for al l 1 ≤ k ≤ l − 1 . Pr o of. F rom Lemma 3.1 w e know that S θ n k remains inside the compact set B ( x 0 , r ) . Therefore, by [9, Prop osition 5.4], there exists a constan t C > 0 suc h that     v n l +1 −  v n l + θ n l √ n d(Exp x 0 ) − 1 v n l X n k +1      ≤ C ( θ n l ) 2 n . (3.3) T o use this to prov e the desired statement, w e first apply the triangle inequalit y to obtain 9      v n l − 1 √ n l X k =1 θ n k τ − 1 x 0 S θ n k − 1 X n k      ≤      v n l − 1 √ n l X k =1 θ n k d(Exp x 0 ) − 1 v n k − 1 X n k      + 1 √ n l X k =1 | θ n k |     d(Exp x 0 ) − 1 v n k − 1 X n k − τ − 1 x 0 S θ n k − 1 X n k     . W e estimate b oth terms separately . By telescoping, w e can use (3.3) to estimate the first term:      v n l − 1 √ n l X k =1 θ n k d(Exp x 0 ) − 1 v n k − 1 X n k      ≤ l X k =1 | v n k − v n k − 1 − θ n k d(Exp x 0 ) − 1 v n k − 1 X n k | ≤ C 1 n l X k =1 ( θ n k ) 2 . F or the second term, we apply [9, Corollary 5.8] to obtain 1 √ n l X k =1 | θ n k |     d(Exp x 0 ) − 1 v n k − 1 X n k − τ − 1 x 0 S θ,n k − 1 X n k     ≤ C r √ n l X k =1 | θ n k || v n k − 1 | 2 ≤ C r 3 n 3 / 2 l l X k =1 | θ n k | ≤ C r 3 l 3 / 2 n 3 / 2 . Here we used that | v n k − 1 | = d  S θ n k − 1 , x 0  together with Lemma 3.1 and (3.1). 3.2 The upp er b ound of the large deviation principle T o pro ve large deviation b ounds for {S θ n n } n ∈ N , w e first consider the random v ariables ˜ v n,m,k ⌊ m − 1 n ⌋ as defined in (3.2). Ho w ever, these random v ariables still live in different tangent spaces, which de- p end on the tra jectory of the random walk. Therefore, we first transp ort all these random tangen t v ectors bac k to T x 0 M . F or the upp er b ound of the large deviation principle, the only relev an t prop ert y is that the transp ort of ˜ v n,m,i ⌊ m − 1 n ⌋ is measurable with respect to σ ( S θ n l , 0 ≤ l ≤ n i − 1 ) . Therefore, w e choose to carry out the parallel transp ort via the intermediate p oints S θ n n 1 , . . . , S θ n n i − 1 of the random walk. F or m large enough, consecutiv e points of this form can be connected with a unique shortest geo desic, and w e denote b y τ S θ n n j − 1 S θ n n j parallel transp ort along this geo desic. Using this notation, we define parallel transp ort τ RW,i : T x 0 M → T S θ n n i M by τ RW,i = τ S θ n n i − 1 S θ n n i ◦ τ S θ n n i − 2 S θ n n i − 1 ◦ · · · ◦ τ x 0 S θ n n 1 W e then define v n,m,i ⌊ m − 1 n ⌋ = τ − 1 RW,i − 1 ˜ v n,m,i ⌊ m − 1 n ⌋ . (3.4) The aim is now to derive the upp er b ound of the large deviation principle for {S θ n n } n ∈ N b y studying the large deviations upp er b ound for the random v ariables  v n,m, 1 ⌊ m − 1 n ⌋ , . . . , v n,m,m ⌊ m − 1 n ⌋  ∈ ( T x 0 M ) m . With [2, Theorem 4.5.3] in mind, we first need to un derstand lim sup n →∞ 1 n log E  e n P m i =1 ⟨ λ i ,v n,m,i ⌊ m − 1 n ⌋ ⟩  . 10 Unfortunately , it cannot be computed exactly . Instead, we compare v n,m,i ⌊ m − 1 n ⌋ to sums of random v ariables giv en by Y n i = τ − 1 RW,i − 1 n i X k = n i − 1 +1 θ n k τ − 1 S θ n n i − 1 S θ n k − 1 X n k ∈ T x 0 M . (3.5) Computing the momen t generating function of sums of indep enden t, identically distributed ran- dom v ariables is straigh tforward. How ever, the w eights θ n mak e the sums in Y n i inhomogeneous. Therefore, contrary to the work on Cramér’s theorem for geo desic random w alks in [9], we now hav e to use of V aradhan’s Lemma, more precisely Corollary 2.4, to obtain asymptotics of the moment generating functions of the Y n i . Prop osition 3.3. L et the assumptions of The or em 2.2 b e satisfie d. Denote by r the uniform b ound on the incr ements of the ge o desic r andom walk. Consider the r andom variables  v n,m, 1 ⌊ m − 1 n ⌋ , . . . , v n,m,m ⌊ m − 1 n ⌋  define d in (3.4) . Then ther e exists a c onstant C > 0 such that for m lar ge enough and al l ( λ 1 , . . . , λ k ) ∈ ( T x 0 M ) m we have lim sup n →∞ 1 n log E  e n P m i =1 ⟨ λ i ,v n,m,i ⌊ m − 1 n ⌋ ⟩  ≤ 1 m m X i =1 Ψ x 0 ( λ i ) + C r 3 m 3 / 2 m X i =1 | λ i | for σ -almost every θ . Pr o of. Consider the random v ariables Y n i as defined in (3.5). By the Cauch y-Sch warz and triangle inequalit y , and the fact that parallel transport is an isometry , we ha ve      m X i =1 ⟨ λ i , v n,m,i ⌊ m − 1 n ⌋ ⟩ − 1 √ n m X i =1 ⟨ λ i , Y n i ⟩      ≤ m X i =1 | λ i |       ˜ v n,m,i ⌊ m − 1 n ⌋ − 1 √ n n i X k = n i − 1 +1 θ n k τ − 1 S θ n n i − 1 S θ n k X n k       . W e further estimate this using Prop osition 3.2 to obtain      m X i =1 ⟨ λ i , v n,m,i ⌊ m − 1 n ⌋ ⟩ − 1 √ n m X i =1 ⟨ λ i , Y n i ⟩      ≤ C m X i =1   1 n | λ i | n i X k = n i − 1 +1 ( θ n k ) 2 + | λ i | r 3 ( n i − n i − 1 ) 3 / 2 n 3 / 2   ≤ C max i | λ i | 1 n m X i =1 n i X k = n i − 1 +1 ( θ n k ) 2 + C r 3 m 3 / 2 m X i =1 | λ i | = C max i | λ i | 1 n + C r 3 m 3 / 2 m X i =1 | λ i | . Here we used that n i − n i − 1 = ⌊ nm − 1 ⌋ ≤ nm − 1 and m X i =1 n i X k = n i − 1 +1 ( θ n k ) 2 = n X k =1 ( θ n k ) 2 = 1 , b ecause θ n ∈ S n − 1 . Applying this estimate to the momen t generating function, we obtain E  e n P m i =1 ⟨ λ i ,v n,m,i ⌊ m − 1 n ⌋ ⟩  ≤ e C max i | λ i | e C r 3 nm − 3 / 2 P m i =1 | λ i | E  e √ n P m i =1 ⟨ λ i ,Y n i ⟩  . 11 It follows that lim sup n →∞ 1 n log E  e n P m i =1 ⟨ λ,v n,m,i ⌊ m − 1 n ⌋ ⟩  ≤ lim sup n →∞ C max i | λ i | n + C r 3 m 3 / 2 m X i =1 | λ i | + 1 n log E  e √ n P m i =1 ⟨ λ i ,Y n i ⟩  = C r 3 m 3 / 2 m X i =1 | λ i | + lim s up n →∞ 1 n log E  e √ n P m i =1 ⟨ λ i ,Y n i ⟩  W e no w compute the asymptotics for the remaining moment generating function. Because the incremen ts of the geodesic random w alk are parallel transport inv ariant and indep endent, the random v ariables τ − 1 RW,i − 1 τ − 1 S θ n n i − 1 S θ n k − 1 X n k are indep endent with distribution µ x 0 . This implies that Y n 1 , . . . , Y n m are indep enden t, and hence E  e √ n P m i =1 ⟨ λ i ,Y n i ⟩  = m Y i =1 E  e √ n ⟨ λ i ,Y n i ⟩  . Moreo ver, by applying Corollary 2.4 to the random v ariables 1 √ n Y n i , we find that for σ -almost every θ and all i = 1 , . . . , m w e hav e lim n →∞ 1 n log E  e √ n ⟨ λ i ,Y n i ⟩  = 1 m Ψ x 0 ( λ i ) , whic h completes the pro of. W e can no w prov e a large deviations upper bound for  v n,m, 1 ⌊ m − 1 n ⌋ , . . . , v n,m,m ⌊ m − 1 n ⌋  . Prop osition 3.4. L et the assumptions of The or em 2.2 b e satisfie d. Denote by r the uniform b ound on the incr ements of the ge o desic r andom walk. Then for m lar ge enough and any close d F ⊂ ( T x 0 M ) m we have lim sup n →∞ 1 n log P  v n,m, 1 ⌊ m − 1 n ⌋ , . . . , v n,m,m ⌊ m − 1 n ⌋  ∈ F  ≤ − inf ( v 1 ,...,v m ) ∈ F sup ( λ 1 ,...,λ m ) ∈ ( T x 0 M ) m 1 m m X i =1 n ⟨ λ i , mv i ⟩ − Ψ x 0 ( λ i ) − m − 1 2 C | λ i | r 3 o . Her e, C is a c onstant dep ending on the curvatur e of the c omp act set B (0 , r ) and the b ound r . Pr o of. F rom Lemma 3.1 it follows that the random w alks sta y in the compact set B (0 , r ) , so that it suffices to pro ve the statement for compact sets Γ . By [2, Theorem 4.5.3] we ha ve lim sup n →∞ 1 n log P  v n,m, 1 ⌊ m − 1 n ⌋ , . . . , v n,m,m ⌊ m − 1 n ⌋  ∈ Γ  ≤ − inf ( v 1 ,...,v m ) ∈ Γ sup ( λ 1 ,...,λ m ) ∈ ( T x 0 M ) m ( m X i =1 ⟨ λ i , v i ⟩ − lim s up n →∞ 1 n log E  e n P m i =1 ⟨ λ i ,v n,m,i ⌊ m − 1 n ⌋ ⟩  ) . T ogether with Prop osition 3.3 this giv es us that for σ -almost all θ we hav e 12 lim sup n →∞ 1 n log P  v n,m, 1 ⌊ m − 1 n ⌋ , . . . , v n,m,m ⌊ m − 1 n ⌋  ∈ Γ  ≤ − inf ( v 1 ,...,v m ) ∈ Γ sup ( λ 1 ,...,λ m ) ∈ ( T x 0 M ) m ( m X i =1 ⟨ λ i , v i ⟩ − 1 m m X i =1 Ψ x 0 ( λ i ) − C r 3 m 3 / 2 m X i =1 | λ i | ) = − inf ( v 1 ,...,v m ) ∈ Γ sup ( λ 1 ,...,λ m ) ∈ ( T x 0 M ) m 1 m m X i =1 ( ⟨ λ i , mv i ⟩ − m X i =1 Ψ x 0 ( λ i ) − C r 3 m 1 / 2 m X i =1 | λ i | ) as desired. W e are now ready to deriv e the upp er b ound of the large deviation principle for {S θ n n } n ∈ N . F or this, w e need a suitable map that maps  v n,m, 1 ⌊ m − 1 n ⌋ , . . . , v n,m,m ⌊ m − 1 n ⌋  to S θ n n . Intuitiv ely , we construct a piece- wise geo desic path with the given tangent v ectors as directions, which w e parallel transp ort along the constructed path. More precisely , w e introduce the map T m : ( T x 0 M ) m → M that constructs this piecewise geo desic path γ : [0 , 1] → M recursively as follo ws. Set γ (0) = x 0 and supp ose γ has b een defined on h 0 , i m i . Then for t ∈ h i m , ( i +1) m i w e define γ ( t ) = Exp γ ( i m )  t − i m  τ γ ; x 0 γ ( i m ) v i +1  . Finally , we set T m ( v 1 , . . . , v m ) = γ (1) . (3.6) Ha ving Proposition 3.4, the pro of of the large deviations upp er b ound for {S θ n n } n ∈ N is analogous to the proof of [9, Prop osition 6.9]. W e provide a condensed v ersion of the proof, emphasizing the adaptations that need to b e made for the proof to b e v alid in our current setting. Prop osition 3.5. L et the assumptions of The or em 2.2 b e satisfie d. Then for σ -almost every θ and every F ⊂ M close d we have lim sup n →∞ 1 n log P  S θ n n ∈ F  ≤ − inf x ∈ F I M ( x ) , wher e I M ( x ) = inf { Ψ ∗ x 0 ( v ) | v ∈ Exp − 1 x 0 x } . Pr o of. F rom Prop osition 3.4 and the inequality | λ | ≤ | λ | 2 + 1 , we obtain lim sup n →∞ 1 n log P  S θ n n ∈ F  ≤ C r 3 √ m − inf ( v 1 ,...,v m ) ∈T − 1 m F sup ( λ 1 ,...,λ m ) ∈ ( T x 0 M ) m 1 m m X i =1 n ⟨ λ i , mv i ⟩ − Ψ x 0 ( λ i ) − m − 1 / 2 C r 3 | λ i | 2 o , where T m is as in (3.6). Because Λ x 0 is differen tiable, con vex and non-negative, so is Ψ x 0 . Hence, w e can follo w the pro of of [9, Prop osition 6.9] to get inf ( v 1 ,...,v m ) ∈T − 1 m F sup ( λ 1 ,...,λ m ) ∈ ( T x 0 M ) m 1 m m X i =1 n ⟨ λ i , mv i ⟩ − Ψ x 0 ( λ i ) − m − 1 / 2 C r 3 | λ i | 2 o = inf v ∈ Exp − 1 x 0 F sup λ ∈ T x 0 M n ⟨ λ, v ⟩ − Ψ x 0 ( λ ) − m − 1 / 2 C r 2 | λ | 2 o . Since inf x ∈ F I M ( x ) = inf v ∈ Exp − 1 x 0 F sup λ ∈ T x 0 M {⟨ λ, v ⟩ − Ψ x 0 ( λ ) } 13 it suffices to sho w that lim m →∞ inf v ∈ Exp − 1 x 0 F sup λ ∈ T x 0 M n ⟨ λ, v ⟩ − Ψ x 0 ( λ ) − m − 1 / 2 C r 3 | λ | 2 o = inf v ∈ Exp − 1 x 0 F sup λ ∈ T x 0 M {⟨ λ, v ⟩ − Ψ x 0 ( λ ) } . (3.7) When we restrict the infimum to v ∈ B (0 , 2 r E ( | Z | ) ∩ Exp − 1 x 0 F , this follo ws from compactness, together with the fact that lim m →∞ sup λ ∈ T x 0 M n ⟨ λ, v ⟩ − Ψ x 0 ( λ ) − m − 1 / 2 C r 3 | λ | 2 o = sup λ ∈ T x 0 M {⟨ λ, v ⟩ − Ψ x 0 ( λ ) } b ecause the sequence is increasing in m . F or v / ∈ B (0 , 2 r E ( | Z | ) , it follows by showing that b oth sides of (3.7) are infinite. F or this, first observ e that since the support of of µ x 0 is contained in B (0 , r ) , we ha ve Ψ x 0 ( λ ) ≤ E ( r | Z || λ | ) = r | λ | E ( | Z | ) . This gives us that sup λ ∈ T x 0 M {⟨ λ, v ⟩ − Ψ x 0 ( λ ) } ≥ sup λ ∈ T x 0 M {⟨ λ, v ⟩ − r E ( | Z | ) | λ |} = ∞ for | v | ≥ r E ( | Z | ) . On the other hand, sup λ ∈ T x 0 M n ⟨ λ, v ⟩ − Ψ x 0 ( λ ) − m − 1 / 2 C r 2 | λ | 2 o ≥ sup λ ∈ T x 0 M n ⟨ λ, v ⟩ − r | λ | E ( | Z | ) − m − 1 / 2 C r 3 | λ | 2 o = ( | v | − r E ( | Z | )) 2 √ m 4 C r 3 , whic h tends to infinity as m → ∞ , completing the pro of. 3.3 Lo w er b ound of the large deviation principle Similar to the large deviations upp er b ound, w e deduce the low er b ound for the large deviation principle for {S θ n n } n ∈ N from from a suitable large deviations lo wer b ound in ( T x 0 M ) m . F or the upp er b ound, we had some freedom in c ho osing how to parallelly transport v ectors to ev entually map ( T x 0 M ) m to M . Ultimately , w e defined the map T m as in (3.6) for this. Unfortunately , this map do es not hav e strong enough con tinuit y prop erties to transfer a large deviations low er b ound from ( T x 0 M ) m to one for {S θ n n } n ∈ N . In particular, we require the con tinuit y of the maps in a sense to b e uniform in m . Therefore, we in tro duce an adapted v ersion of this map in the next section. 3.3.1 F rom T x 0 M to M T o pro v e the low er b ound of the large deviation principle, for x ∈ M and v ∈ Exp − 1 x 0 x ⊂ T x 0 M we can fo cus on realizations of the random w alk whic h sta y close to the geo desic γ v ( t ) = Exp x 0 ( tv ) . W e parallelly transp ort the random incremen ts as m uch as p ossible along this geo desic, to limit deviations arising from curv ature. More sp ecifically , given m ∈ N , v ∈ T x 0 M , w e define a map T v ,m : ( T x 0 M ) m → M as follows (see also [9, Section 6.2]). W e first discretize the geo desic γ v ( t ) = Exp x 0 ( tv ) by setting x i := γ v  i m  , i = 1 , . . . , m . Next, w e define p oin ts y i , i = 0 , . . . , m recursively . First set y 0 = x 0 and next supp ose y k is defined. Define ˜ v k ∈ T y k M as ˜ v k = τ x k y k τ γ v ; x 0 x k v k . Contrary to the setting of the 14 upp er bound, here τ γ v ; x 0 x k denotes parallel transp ort along the geo desic γ v , and τ x k y k is parallel transp ort along any shortest length geodesic connecting x k and y k . Finally , we set T v ,m ( v 1 , . . . , v m ) := y m . (3.8) F rom [9, Section 6.2] w e hav e the follo wing, which shows that the contin uit y of the maps T v ,m is in a sense uniform in m . Lemma 3.6. Given v ∈ T x 0 M and ε > 0 , ther e exists a δ > 0 such that for al l m lar ge enough we have T v ,m ( v 1 , . . . , v m ) ∈ B (Exp x 0 v , ε ) whenever ( v 1 , . . . , v m ) ∈ B ( v , δ ) m . 3.3.2 Pro of of the large deviations low er b ound Since w e hav e adapted the map T m to T v ,m , we ha ve to adapt the sums Y n i as defined in (3.5) accordingly . Let v ∈ T x 0 M and set γ v ( t ) = Exp x 0 ( tv ) as in the previous section. No w define for i = 1 , . . . , m the random v ariables ¯ Y n i = τ − 1 γ v ; x 0 γ v ( i − 1 m ) τ − 1 γ v ( i − 1 m ) S θ n n i − 1 n i X k = n i − 1 +1 θ n k τ − 1 S θ n n i − 1 S θ n k − 1 X n k ∈ T x 0 M F or the same reasons as for Y n i , ¯ Y n i is a w eighted sum of indep enden t random v ariables with distribution µ x 0 . As a consequence, Corollary 2.4 implies that each of the ¯ Y n i satisfies a large deviation principle. The con tinuit y prop ert y of T v ,m as in Lemma 3.6, esp ecially the uniformit y in m , then allo ws us to transfer the corresp onding lo w er b ound to the low er b ound for the large deviation principle for {S θ n n } n . Prop osition 3.7. L et the assumptions of The or em 2.2 b e satisfie d. Then for σ -almost every θ and every G ⊂ M op en we have lim sup n →∞ 1 n log P  S θ n n ∈ G  ≥ − inf x ∈ G I M ( x ) , wher e I M ( x ) = inf { Ψ ∗ x 0 ( v ) | v ∈ Exp − 1 x 0 x } . Pr o of. It suffices to sho w that for every x ∈ M and ε > 0 w e hav e lim sup n →∞ 1 n log P  S θ n n ∈ B ( x, ε )  ≥ − I M ( x ) . By definition of I M ( x ) , it is actually enough to show that lim sup n →∞ 1 n log P  S θ n n ∈ B ( x, ε )  ≥ − Ψ ∗ ( v ) for all v ∈ Exp − 1 x 0 x . Fix such v . W rite γ v ( t ) = Exp x 0 ( tv ) and let m b e large enough such that w e can uniquely define ˜ v n,m,i k ∈ Exp − 1 S θ n n i − 1  S θ n n i − 1 + k  ⊂ T S θ n n i − 1 M 15 of minimal length (as in (3.2)) and set ¯ v n,m,i k = τ − 1 γ v ; x 0 γ v ( i − 1 m ) τ − 1 γ v ( i − 1 m ) S θ n n i − 1 ˜ v n,m,i k ∈ T x 0 M . By construction T v ,m ( ¯ v n,m, 1 ⌊ m − 1 n ⌋ , . . . , ¯ v n,m,m ⌊ m − 1 n ⌋ ) = S θ n n , where T v ,m is as in (3.8). By Lemma 3.6 there exists a δ > 0 suc h that for all m large enough we can estimate P  S θ n n ∈ B ( x, ε )  ≥ P  ¯ v n,m, 1 ⌊ m − 1 n ⌋ , . . . , ¯ v n,m,m ⌊ m − 1 n ⌋  ∈ B ( v , δ ) m  . W e compare the latter to ( ¯ Y n 1 , . . . , ¯ Y n n ) . Prop osition 3.2 and (3.1) imply that | ¯ v n,m,i ⌊ m − 1 n ⌋ − ¯ Y n i | =       v n,m,i ⌊ m − 1 n ⌋ − n i X k = n i − 1 +1 θ n k τ − 1 S θ n n i − 1 S θ n k − 1 X n k       ≤ C 1 n n i X k = n i − 1 +1 ( θ n k ) 2 + C r 3 1 m 3 / 2 ≤ C 1 n r m 1 / 2 + C r 3 1 m 3 / 2 . As a consequence, w e can take m large enough so that | ¯ v n,m,i ⌊ m − 1 n ⌋ − ¯ Y n i | < δ 2 . F rom this it follows we can further estimate P  ( ¯ v n,m, 1 ⌊ m − 1 n ⌋ , . . . , ¯ v n,m,m ⌊ m − 1 n ⌋ ) ∈ B ( v , δ ) m  ≥ P  ( ¯ Y n 1 , . . . , ¯ Y n m ) ∈ B ( v , δ / 2) m  . Since the incremen ts of the random walk S θ n n are indep enden t, so are the ¯ Y n i . Therefore, 1 n log P  ( ¯ Y n 1 , . . . , ¯ Y n m ) ∈ B ( v , δ / 2) m  = 1 n m X i =1 log P  ¯ Y n i ∈ B ( v , δ / 2)  Since moreo ver the distributions of the increments of S θ n n are in v arian t under parallel transp ort, it follo ws that ¯ Y n i is a sum of indep enden t random v ariables with distribution µ x 0 . As a consequence, it follo ws from Corollary 2.4 and the Gartner-Ellis theorem ([2, Theorem 2.3.6]) that Y n i satisfies a large deviation principle with rate function 1 m Ψ ∗ x 0 . 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