# Cramér type moderate deviation expansions for entries of products of random invertible matrices

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## Classical results on precise large and moderate devia-tions

In this section we briefly recall some classical results on precise large and moderate deviations for sums of i.i.d. real-valued random variables. Let (Xi)i>1 be a sequence of i.i.d. real-valued random variables and Sn = Pni=1 Xi. Denote by IΛ the set of real numbers s > 0 such that Λ(s) := log E[esX1 ] < +∞ (1.2.9) and by IΛ◦ the interior of the interval IΛ. Let Λ∗ be the Frenchel-Legendre transform of Λ. Assume that s ∈ IΛ◦ and q are related by q = Λ0(s). Set σs2 = Λ00(s). The following theorem is due to Bahadur and Rao [4]: Theorem 1.2.1 ([4]). Let s ∈ IΛ◦ and q = Λ0(s). Assume that the law of X1 is non-lattice. Then, as n → ∞, exp(−nΛ∗(q)) P(Sn > nq) = √ (1 + o(1)). (1.2.10) sσs 2πn When the law of X1 is lattice, an analogous precise large deviation result has also been established in [4]. Later on, Petrov [73] improved Theorem 1.2.1 by considering a vanishing pertur-bation l on q: Theorem 1.2.2 ([73]). Let s ∈ IΛ◦ and q = Λ0(s). Assume that the law of X1 is non-lattice. Then, for any positive sequence (ln)n>1 satisfying limn→∞ ln = 0, we have, as n → ∞, uniformly in |l| 6 ln, P (S n > n(q + l)) = exp(−nΛ∗(q + l)) (1 + o(1)). (1.2.11) sσs√ 2πn.
Note that the exact asymptotics for the lower tail large deviation probabilities can be deduced easily from upper tail large deviation asymptotics (1.2.10) and (1.2.11) by considering −X1 instead of X1. Denote γk = Λ(k)(0), k > 1, where Λ is the cumulant generating function of X1 defined in (1.2.9). Let λ := γ1 = EX1 and σ2 := γ2 = E(X1 − λ)2 be the mean and variance of X1. Denote by ζ the Cramér series of Λ (see [26] and [74]): ζ(t) = γ3 + γ4γ2 − 3γ32 t + γ5γ22 − 10γ4γ3γ2 + 15γ33 t2 + · · · 6γ3/2 24γ23 120γ9/2 2 2.
which converges for |t| small enough. Let Φ be the standard normal distribution function on R. We recall the following Cramér type moderate deviation expansion for Sn.

### Presentation of main results of the thesis

As already mentioned, the main goal of the present thesis is to study precise large and moderate deviation asymptotics for products of random matrices. The remaining part of the thesis consists of six chapters.
Chapter 2 is mainly devoted to the study of Bahadur-Rao type and Petrov type large deviations for the norm cocycle log |Gnx|, for both invertible matrices and pos-itive matrices; see Section 1.3.2 for the presentation of the main results. Using the spectral gap theory for products of random matrices and the saddle point method, we establish Bahadur-Rao-Petrov type exact asymptotics of the upper and lower tail large deviation probabilities for the norm cocycle. More generally, we also prove analogous Bahadur-Rao-Petrov type large deviation results for the couple (Xnx, log |Gnx|) with target functions. As applications, we deduce new results on large deviation principles for the operator norm kGnk for invertible matrices, and reinforced large deviation prin-ciples for kGnk for positive matrices. Moreover, we derive precise local limit theorems with large deviations for the norm cocycle.
In Chapter 3, our main goal is to establish Bahadur-Rao type and Petrov type large deviations for the entries Gi,jn, for both invertible matrices and positive matrices; see Section 1.3.3 for the presentation of the main results. As in Chapter 2, we also prove Bahadur-Rao-Petrov type upper and lower tail large deviation results for the couple (Xnei , log |Gi,jn|) with target functions. In particular, we obtain the large deviation principle with an explicit rate function, thus improving significantly the large deviation bounds established earlier. In our proof, a very important issue is to prove the Hölder regularity property for the stationary measure corresponding to the Markov chain Xnx under the changed measure on the projective space, which is one of the central points of the proof and is of independent interest. As applications, we obtain new results on precise local limit theorems with large deviations for the entries and on reinforced large deviation principles for the spectral radius of products of positive random matrices.
Chapter 4 is devoted to investigating the Berry-Esseen bound and Cramér type moderate deviation expansion for the norm cocycle of products of random matrices; see Section 1.3.4 for the presentation of the main results. We first establish the Berry-Esseen bound and the Edgeworth expansion for the couple (Xnx, log |Gnx|) with a target function ϕ on the Markov chain Xnx, for both invertible matrices and positive matrices. This is proved by elaborating a new approach based on a smoothing inequality in the complex plane and on the saddle point method. Using the Berry-Esseen bound under the changed measure, we then establish Cramér type moderate deviation expansion for the couple (Xnx, log |Gnx|).
In Chapter 5, we study Berry-Esseen bounds and Cramér type moderate deviation expansions for the operator norm kGnk, entries Gi,jn and spectral radius ρ(Gn) of products of positive random matrices; see Section 1.3.5 for the presentation of the main results. The results for the operator norm kGnk are proved under general conditions; the results for the entries Gi,jn and the spectral radius ρ(Gn) are established under a boundedness condition weaker than that of Furstenberg-Kesten.
Chapter 6 is devoted to studying the Berry-Esseen type bounds and moderate deviations for the operator norm kGnk and the spectral radius ρ(Gn) of products of random matrices in the general linear group GLd(R); see Section 1.3.6 for the presentation of the main results. Under the proximality condition, we first prove the moderate deviation principles for the couples (Xnx, log kGnk) and (Xnx, log ρ(Gn)) with target functions based on the moderate deviation results for the norm cocycle log |Gnx| established in Chapter 2. Then we prove the moderate deviation principles for kGnk and ρ(Gn) without assuming the proximality condition. We also prove the moderate deviation expansions in the range [0, o(n1/6)] for the couples (Xnx, log kGnk) and (Xnx, log ρ(Gn)) with target functions.
In Chapter 7, we establish the Cramér type moderate deviation expansions for the entries Gi,jn of products of invertible matrices in the special linear group SL2(R); see Section 1.3.7 for the presentation of the main results. Our result implies the moderate deviation principle for log |Gi,jn| and local limit theorems with moderate deviations, which are also new. In our proof, we use the saddle point method, the Hölder regularity of the stationary measure corresponding to the Markov chain Xnx, and the recent progress on the strong non-lattice result for the perturbed transfer operator.

#### Precise large deviations for the norm cocycle of prod-ucts of random matrices

The large deviation theory, which is an important and active research area in proba-bility theory, allows us to describe the rate of convergence in the law of large numbers. For sums of i.i.d. real-valued random variables, the most remarkable large deviation results in this direction are due to Bahadur-Rao [4] and Petrov [73]. These milestone results have numerous applications in various domains of probability and statistics; see, for example, Buraczewski, Collamore, Damek and Zienkiewicz [18] for a recent application to the asymptotic of the ruin time in some models of financial mathemat-ics. Our main goal of this section is to present the analogous Bahadur-Rao type and Petrov type precise large deviation asymptotics for the norm cocycle log |Gnx|. Our results are valid for both invertible matrices and positive matrices. As applications we improve previous results on large deviation principles for the operator norm kGnk and we obtain precise local limit theorems with large deviations.
The standard approach to establish Bahadur-Rao [4] and Petrov [73] types large deviation asymptotics for sums of i.i.d. real-valued random variables consists of making a change of measure and then proving an Edgeworth expansion under the changed measure, see also Dembo and Zeitouni [30]. In the case of products of random matrices, this approach has been recently employed by Buraczewski and Mentemeier [17], where the main result is the following: Theorem 1.3.1 ([17]). Let s ∈ Iµ◦ and q = Λ0(s). Assume either conditions A1, A3 for invertible matrices, or conditions A1, A4, A8 for positive matrices. Then, n→∞ x∈S s 2 E h s( n ) {log |Gnx|>nq}i − s lim sup sσ √ enΛ∗(q) r Xx 1 r (x) = 0. πn In particular, there exist constants c, C > 0 such that for all x ∈ S, c < lim inf √n enΛ∗(q)P log |Gnx| > nq n→∞ lim sup √n enΛ∗(q)P log |Gnx| > nq < C. n→∞ (1.3.6) (1.3.7) It is worth mentioning that Theorem 1.3.1 turns out to be useful in [17] to inves-tigate the precise tail asymptotics for multidimensional aﬃne stochastic recursion.
The appearance of the eigenfunction rs inside the expectation in the statement (1.3.6) is somehow unpleasant, even though we know that rs is strictly positive and uniformly bounded on the projective space S. We would like to sharpen the inequality (1.3.7) by giving an exact limit instead of upper and lower bounds. To achieve this goal, our approach becomes diﬀerent from the standard one employed in [4, 73, 30, 17], as mentioned above. Our proof is carried out by making use of the spectral gap theory developed in [50] for invertible matrices and in [16, 17] for positive matrices, by employing the saddle point method (see for instance [34]), and by using smoothing and approximation techniques; we refer to Chapter 2 for details. The following result concerns the exact asymptotics of the upper tail large deviation probabilities for the norm cocycle.

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1 Introduction
1.1 Context
1.2 Background and main objectives
1.2.1 Background
1.2.2 Classical results on precise large and moderate deviations
1.2.3 Main objectives and previous results
1.3 Presentation of main results of the thesis
1.3.1 Notation and conditions
1.3.2 Precise large deviations for the norm cocycle of products of random matrices
1.3.3 Precise large deviations for entries of products of random matrices
1.3.4 Berry-Esseen bound and precise moderate deviations for the norm cocycle of products of random matrices
1.3.5 Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices
1.3.6 Berry-Esseen bounds and moderate deviation principles for the random walk on GLd(R)
1.3.7 Moderate deviation expansions for the entries of the random walk on SL2(R)
2 Precise large deviation asymptotics for products of random matrices
2.1 Introduction
2.1.1 Background and main objectives
2.1.2 Proof outline
2.2 Main results
2.2.1 Notation and conditions
2.2.2 Large deviations for the norm cocycle
2.2.3 Applications to large deviation principle for the matrix norm .
2.2.4 Local limit theorems with large deviations
2.3 Spectral gap theory for the norm
2.3.1 Properties of the transfer operator
2.3.2 Spectral gap of the perturbed operator
2.4 Proof of precise large deviations for the norm cocycle
2.4.1 Auxiliary results
2.4.2 Proof of Theorem 2.2.1
2.4.3 Proof of Theorem 2.2.3
2.5 Proof of precise large deviations with target functions
2.6 Proofs of LDP for log kGnk and local limit theorems with large deviations
3 Large deviation expansions for the entries of products of random matrices
3.1 Introduction
3.1.1 Background and objectives
3.1.2 Brief overview of the results
3.1.3 Proof strategy
3.2 Main results
3.2.1 Notation and conditions
3.2.2 Precise large deviations for the scalar product
3.2.3 Local limit theorems with large deviations
3.2.4 Large deviation principle for the spectral radius of positive matrices
3.3 Hölder regularity of the stationary measure
3.3.1 Spectral gap properties and a change of measure
3.3.2 Hölder regularity of the stationary measure
3.4 Auxiliary statements
3.5 Proof of the Hölder regularity of s for positive s
3.5.1 Regularity of the stationary measure for invertible matrices
3.5.2 The regularity of the stationary measure for positive matrices .
3.6 The Hölder regularity of s for negative s
3.7 Proof of precise large deviations for scalar products
3.7.1 Proof of Theorems 3.2.2 and 3.2.4
3.8 Proofs of large deviation principles for spectral radius
4 Berry-Esseen bound and precise moderate deviations for products of random matrices
4.1 Introduction
4.1.1 Background and objectives
4.1.2 Key ideas of the approach
4.2 Main results
4.2.1 Notation and conditions
4.2.2 Berry-Esseen bound and Edgeworth expansion
4.2.3 Moderate deviation expansions
4.3 Spectral gap theory
4.3.1 Properties of the transfer operator Pz
4.3.2 Definition of the change of measure Qxs
4.3.3 Properties of the Markov operator Qs
4.3.4 Quasi-compactness of the operator Qs+it
4.3.5 Spectral gap properties of the perturbed operator Rs,z
4.4 Smoothing inequality on the complex plane
4.5 Proofs of Berry-Esseen bound and Edgeworth expansion
4.5.1 Berry-Esseen bound and Edgeworth expansion under the changed measure
4.5.2 Proof of Theorem 4.5.2
4.5.3 Proof of Theorem 4.5.1
4.6 Proof of moderate deviation expansions
5 Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices
5.1 Introduction
5.2 Main results
5.2.1 Notation and conditions
5.2.2 Berry-Esseen bounds
5.2.3 Precise moderate deviation expansions
5.2.4 Formulas for the asymptotic variance
5.3 Proofs of Berry-Esseen bounds
5.4 Proofs of moderate deviation expansions
6 Berry-Esseen bounds and moderate deviations for the random walk on GLd(R)
6.1 Introduction
6.1.1 Background and previous results
6.1.2 Objectives
6.1.3 Proof outline
6.2 Main results
6.2.1 Berry-Esseen type bounds
6.2.2 Moderate deviation principles
6.2.3 Moderate deviation expansions
6.3 Berry-Esseen type bounds
6.4 moderate deviation principles
6.4.1 Proof of Theorem 6.2.3
6.4.2 Proof of Theorem 6.2.4
6.5 Moderate deviation expansions
7 Cramér type moderate deviation expansions for entries of products of random invertible matrices
7.1 Introduction
7.1.1 Background and main objective
7.1.2 Proof strategy
7.2 Main results
7.3 Spectral gap theory
7.3.1 A change of measure
7.3.2 Spectral gap and strong non-lattice
7.4 Regularity of the stationary measure