Berry-Esseen bound and precise moderate deviations for products of random matrices

Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed (i.i.d.) $d\times d$ real random matrices. Set $G_n = g_n g_{n-1} \ldots g_1$ and $X_n^x = G_n x/|G_n x|$, $n\geq 1$, where $|\cdot|$ is an arbitrary norm in $\mathbb R^d$ and $x\in\mathbb{R}^{d}$ is a starting point with $|x|=1$. For both invertible matrices and positive matrices, under suitable conditions we prove a Berry-Esseen type theorem and an Edgeworth expansion for the couple $(X_n^x, \log |G_n x|)$. These results are established using a brand new smoothing inequality on complex plane, the saddle point method and additional spectral gap properties of the transfer operator related to the Markov chain $X_n^x$. Cram\'{e}r type moderate deviation expansions are derived for the couple $(X_n^x, \log |G_n x|)$ with a target function $\varphi$ on the Markov chain $X_n^x$. A local limit theorem with moderate deviations is also obtained.

1. Introduction 1.1. Background and objectives. For any integer d 2, denote by GL(d, R) the general linear group of d × d invertible matrices. Equip R d with any norm | · |, denote by P d−1 = {x ∈ R d , |x| = 1}/± the projective space in R d , and let g = sup x∈P d−1 |gx| be the operator norm for g ∈ GL(d, R). Let (g n ) n 1 be a sequence of i.i.d. d × d real random matrices of the same law µ on GL(d, R), and consider the product G n = g n g n−1 . . . g 1 and the process X x n = G n x/|G n x|, n 1, with starting point x ∈ P d−1 . The study of the asymptotic properties of the Markov chain (X x n ) n 1 and of the product (G n ) n 1 has attracted a good deal of attention since the groundwork of Furstenberg and Kesten [14], where the strong law of large numbers for log G n has been established, which is a fundamental result for the products of random matrices. Furstenberg [15] proved the following version of the law of large numbers: for any x ∈ R d , lim n→∞ 1 n log |G n x| = lim n→∞ 1 n E log |G n x| = λ P-a.s., where the real number λ is called upper Lyapunov exponent associated with the product G n . Another cornerstone result is the central limit theorem (CLT) for the couple (X x n , log |G n x|), established under contracting type assumptions by Le Page [29]: for any fixed y ∈ R and any Hölder continuous function ϕ : P d−1 → R, it holds uniformly in x ∈ P d−1 that where ν is the unique stationary probability measure of the Markov chain X x n on P d−1 , σ 2 = lim n→∞ 1 n E (log |G n x| − nλ) 2 is the asymptotic variance independent of x, and Φ is the standard normal distribution function. The optimal conditions for the CLT to hold true have been established recently by Benoist and Quint [2].
A very important topic is the study of large and moderate deviation probabilities, which describe the rate of convergence in the law of large numbers. For an account to the theory of large deviations for sums of independent random variables we refer to Cramér [8], Petrov [30], Strook [34], Varadhan [35] and Dembo and Zeitouni [11]. For products of random matrices, precise large deviations asymptotics have been considered e.g. by Le Page [29], Buraczewski and Mentemeier [6], Guivarc'h [16], Benoist and Quint [3], Sert [33], Xiao, Grama and Liu [37]. For moderate deviations, very little results are known. Benoist and Quint [3] have recently established the asymptotic for the logarithm of probabilities of moderate deviations for reductive groups, which in our setting reads as follows: for any interval B ⊆ R, and positive sequence (b n ) n 1 satisfying bn n → 0 and bn √ n → ∞, it holds, uniformly in x ∈ P d−1 , that lim n→∞ n b 2 n log P log |G n x| − nλ b n ∈ B = − inf y∈B y 2 2σ 2 . (1.1) A functional moderate deviation principle has been established by Cuny, Dedecker and Jan [7]. The first objective of our paper is to improve on the result (1.1) by establishing a Cramér type moderate deviation expansion for log |G n x|: we prove that uniformly in x ∈ P d−1 and y ∈ [0, o( √ n)], P log |G n x| − nλ √ nσy where t → ζ(t) is the Cramér series of the logarithm of the eigenfunction related to the transfer operator of the Markov walk associated to the product of random matrices (see Section 2.3). It will be seen that the expansion (1.2) implies the following local limit theorem with moderate deviations: for any bounded Borel set B ⊂ R with boundary ∂B satisfying ℓ(∂B) = 0, where ℓ is the Lebesgue measure, and for any positive sequence (y n ) n 1 satisfying y n → 0 and √ ny n → ∞, we have, uniformly in x ∈ P d−1 and y ∈ [y n , o( √ n)], It is clear that when y = o(n 1/6 ) the term y 3 √ n ζ( y √ n ) tends to 0 and can be removed in (1.3). Thus (1.3) enlarges the range y = O( √ log n) in the local limit theorem in [3,Theorem 17.10] established under different assumptions. Local limit theorems of type (1.3) are used for instance in [1] for studying dynamics of group actions on finite volume homogeneous spaces. It is useful to extend the moderate deviation expantion (1.2) for the couple (X x n , log |G n x|) which describes completely the random walk (G n x) n 1 . We prove that, for any Hölder continuous function ϕ on P d−1 , uniformly in x ∈ P d−1 and y ∈ [0, o( √ n)], (1.4) see Theorem 2.3. Our second objective, which is also the key point in proving (1.4), is a Berry-Esseen bound for the couple (X x n , log |G n x|): for any Hölder continuous function ϕ on P d−1 , (1.5) see Theorem 2.1. This extends the result of Le Page [29] established for the particular target function ϕ = 1 (see also Jan [26]). We further upgrade (1.5) to an Edgeworth expansion under a non-arithmeticity condition, see Theorem 2.2, which is new even for ϕ = 1.
All the results stated above concern invertible matrices, but we also establish analogous theorems for positive matrices. Some limit theorems for log |G n x| in case of positive matrices such as central limit theorem and Berry-Esseen theorem have been established earlier by Furstenberg and Kesten [14], Hennion [19], and Hennion and Hervé [21]. Here, we extend the Berry-Esseen theorem of [21] to the couple (X x n , log |G n x|) with a target function ϕ on the Markov chain X x n . We also complement the results in [14,19,21] by giving a Cramér type moderate deviation expansion and a local limit theorem with moderate deviations.

Key ideas of the approach.
For the moderate deviation expansions (1.2) and (1.4), our proof is different from those in [3] and [7]: in [3] the moderate deviation principle (1.1) is obtained by following the strategy of Kolmogorov [28] suited to show the law of iterated logarithm (see also de Acosta [10] and Wittman [36]); in [7] the proof of the functional moderate deviation principle is based on the martingale approximation method developed in [2].
In order to prove (1.4) we have to rework the spectral gap theory for the transfer operators P z and R s,z , by considering the case when s can take values in the interval (−η, η) with η > 0 small, and z belongs to a small complex ball centered at the origin, see Section 3. This allows to define the change of measure Q x s and to extend the Berry-Esseen bound (1.5) for the changed measure Q x s , see Theorem 5.1. The moderate deviation expansion (1.4) is established by adapting the techniques from Petrov [30].
It is surprising that the proof of the Berry-Esseen bound and of the Edgeworth expansion with a non-trivial target function ϕ = 1 is way more difficult than the analogous results with ϕ = 1. This can be seen from the following sketch of the proof.
For simplicity, we assume that σ = 1. Introduce the transfer operator P z : for any Hölder continuous function ϕ on P d−1 and z ∈ C, Let F be the distribution function of log |Gnx|−nλ √ n and f be its Fourier transform: f (t) = e it √ nλ (P n −it/ √ n 1)(x), t ∈ R. The Berry-Esseen bound (1.5) with target function ϕ = 1 is usually proved using Esseen's smoothing inequality: for all T > 0, Inserting the spectral gap decomposition into (1.7) allows us to obtain the Berry-Esseen bound (1.5) with ϕ = 1: after some straightforward calculations, it reduces to showing that, with The bound (1.9) is proved using Taylor's expansion L n √ n, and the fact that L n 0 1 = 0. However, when we replace the unit function 1 by a target function ϕ for which in general L n 0 ϕ = 0, instead of (1.9), we have (1.10) even though |L n 0 ϕ(x)| decays exponentially fast to 0 as n → ∞. To overcome it, we have elaborated a new approach based on smoothing inequality on complex contours and on the saddle point method, see Daniels [9] and Fedoryuk [13].
For simplicity, we formulate our smoothing inequality only for y 0: where the integration is taken over the complex contour C − T = {z ∈ C : |z| = T, ℑz < 0} and ρ T is the analytic extension of the Fourier transform of a smoothing density ρ T on the real line (see Section 4). An important issue is to construct the density function ρ T such that ρ T has a compact support on the real line R and can be extended analytically on a domain containing C − T . This enables us to use Cauchy's integral theorem for establishing (1.11) and also for the estimation of the integrals therein. The smoothing inequality (1.11) together with the spectral gap property (1.8) leads to the estimation of the following integrals: The integral (1.12) is handled by using the saddle point method choosing a suitable path for the integration in Section 5.2, which is one of the challenging parts of the proof. For the integral (1.13) we use the facts that |L n z ϕ(x)| decays exponentially fast as n → ∞ and that | e izy z | 1 T on the contour C − T for y 0, where T = c √ n. In contrast to (1.10), this shows that (1.13) is bounded by Ce −cn uniformly in y. The case y > 0 is treated similarly, which allows us to establish (1.5). Note that the non-arithmeticity condition is not needed for the validity of (1.5). Under the non-arithmeticity condition, in Theorem 2.2 we obtain an Edgeworth expansion for (X x n , |G n x|) with the target function ϕ on X x n , which is of independent interest.

Main results
2.1. Notation and conditions. Let N = {0, 1, 2, . . .} and N * = N \ {0}. The real part, imaginary part and the conjugate of a complex number z are denoted by ℜz, ℑz and z respectively. For y ∈ R, we write φ(y) = 1 and Φ(y) = y −∞ φ(t)dt. For any η > 0, set B η (0) = {z ∈ C : |z| < η} for the ball with center 0 and radius η in the complex plane C. We denote by c, C, positive absolute constants whose values may change from line to line. By c α , C α we mean positive constants depending only on the index α. We write ½ A for the indicator function of an event A. For a measure ν and a function ϕ we denote ν(ϕ) = ϕdν.
For d 2, let M (d, R) := M be the set of d×d matrices with entries in R. We shall work with products of invertible or non-negative matrices. Denote by G = GL(d, R) the group of invertible matrices of M . A non-negative matrix g ∈ M is said to be allowable, if every row and every column of g has a strictly positive entry. Denote by G + the multiplicative semigroup of allowable non-negative matrices of M , which will be called simply positive. We write G • + for the subsemigroup of G + with strictly positive entries. The space R d is equipped with any given norm | · |. Denote by S d−1 = {x ∈ R d , |x| = 1} the unit sphere, and by S d−1 + = {x 0 : |x| = 1} the intersection of the unit sphere with the positive quadrant. It will be convenient to consider the projective space To unify the exposition, we use the symbol S to denote P d−1 in case of invertible matrices and S d−1 + in case of positive matrices. For x ∈ S and g ∈ G or g ∈ G + , we write g · x = gx |gx| for the projective action of g on S. The space S is endowed with the metric d: for invertible matrices, d is the angular distance, i.e., for any x, y ∈ P d−1 , d(x, y) = | sin θ(x, y)|, where θ(x, y) is the angle between x and y; for positive matrices, d is the Hilbert cross-ratio metric, i.e., for any x = (x 1 , . . . , x d ) and y = (y 1 , . . . , 1+m(x,y)m(y,x) , where m(x, y) = sup{λ > 0 : λy i x i , ∀i = 1, . . . , d}. In both cases, there exists a constant C > 0 such that |x − y| Cd(x, y), for any x, y ∈ S. (2.1) We refer to [17] and [19] for more details. Let C(S) be the space of continuous complex-valued functions on S and 1 be the constant function with value 1. Let γ > 0. For any ϕ ∈ C(S), set Introduce the Banach space B γ := {ϕ ∈ C(S) : ϕ γ < +∞}. Let (g n ) n 1 be a sequence of i.i.d. random matrices with the same law µ, defined on some probability space (Ω, F, P). Set G n = g n . . . g 1 , n 1, then for any starting point x ∈ S, the process X x 0 = x, X x n = G n ·x, n 1 forms a Markov chain on S. The goal of the present paper is to establish a Berry-Esseen bound and a Cramér type moderate deviation expansion for the couple (X x n , log |G n x|) with a target function ϕ on the Markov chain (X x n ), for both invertible matrices and positive matrices. For any g ∈ M , set g = sup x∈S |gx| and ι(g) = inf x∈S |gx| > 0, where ι(g) > 0 for both g ∈ G and g ∈ G + . In the following we use the notation N (g) = max{ g , ι(g) −1 }. From the Cartan decomposition it follows that the norm g coincides with the largest singular value of g, i.e. g is the square root of the largest eigenvalue of g T g, where g T denotes the transpose of g. For an invertible matrix g ∈ G , ι(g) = g −1 −1 , hence ι(g) is the smallest singular value of g and N (g) = max{ g , g −1 }. We need the two-sided exponential moment condition: We denote by Γ µ := [supp µ] the smallest closed semigroup of M generated by supp µ, the support of µ.
For invertible matrices, we will need the strong irreducibility and proximality conditions. Recall that a matrix g is said to be proximal if g has an eigenvalue λ g satisfying |λ g | > |λ ′ g | for all other eigenvalues λ ′ g of g. The normalized eigenvector v g (|v g | = 1) corresponding to the eigenvalue λ g is called the dominant eigenvector. It is easy to verify that λ g ∈ R.
For positive matrices, we will use the allowability and positivity conditions: A3. (i) (Allowability) Every g ∈ Γ µ is allowable.
(ii) (Positivity) Γ µ contains at least one matrix belonging to G • + . It follows from the Perron-Frobenius theorem that every g ∈ G • + has a dominant eigenvalue λ g > 0, with the corresponding eigenvector v g ∈ S d−1 + . Under conditions A1 and A2 for invertible matrices, or conditions A1 and A3 for positive matrices, there exists a unique µ-stationary probability measure ν on S ( [17,5]): for any ϕ ∈ C(S), Moreover, for invertible matrices, supp ν (the support of ν) is given by for positive matrices, supp ν is given by In addition, for both cases, V (Γ µ ) is the unique minimal Γ µ -invariant subset (see [17] and [5]). For positive matrices, it will be shown in Proposition 3.14 that under conditions A1 and A3, the asymptotic variance exists with value in [0, ∞). To establish the Berry-Esseen theorem and the moderate deviation expansion, we need the following condition: A4. The asymptotic variance σ 2 satisfies σ 2 > 0.

A5. (Non-arithmeticity) The measure µ is non-arithmetic.
A simple sufficient condition introduced in [27] for the measure µ to be non-arithmetic is that the additive subgroup of R generated by the set . We end this subsection by giving some implications among the above conditions. For invertible matrices, it was proved in [18] that condition A2 implies condition A5. For positive matrices, conditions A1, A3 and A5 imply condition A4, see Proposition 3.14.

Berry-Esseen bound and Edgeworth expansion.
In this subsection we formulate the Berry-Esseen theorem and the Edgeworth expansion for (X x n , log |G n x|). We first state the Berry-Esseen theorem with a target function on X x n . Through the rest of the paper we assume that γ > 0 is a fixed small enough constant so that the spectral properties stated in Proposition 3.1 hold true. Theorem 2.1. Assume either conditions A1 and A2 for invertible matrices, or conditions A1, A3 and A4 for positive matrices. Then, there exists a constant C > 0 such that for all n 1, x ∈ S, y ∈ R and ϕ ∈ B γ , The proof of this theorem follows the same line as the proof of the Edgeworth expansion in Theorem 2.2 formulated below, and will be sketched at the end of Section 5. The presence of the target function in Theorem 2.1 turns out to be crucial in the study of the asymptotic of moderate deviations of the scalar product log | f, G n x |, which will be done in a forthcoming paper.
Theorem 2.1 extends the Berry-Esseen bounds from [29,26] for invertible matrices, and [21] for positive matrices to versions with target functions on X x n . Note that the results in [26,21] have been established under some polynomial moment conditions. However, proving (2.5) with the target function ϕ = 1 under the polynomial moments is still an open problem.
The following result gives an Edgeworth expansion for log |G n x| with the target function ϕ on X x n . To formulate the result, we introduce the necessary notation. Consider the following transfer operator: for any s ∈ (−η, η) with η > 0 small, and ϕ ∈ C(S), It will be shown in Proposition 3.1 that there exist a measure ν s and a Hölder continuous function r s on S such that ν s P s = κ(s)ν s and P s r s = κ(s)r s , (2.6) where κ(s) is the unique dominant eigenvalue of P s . Set Λ(s) = log κ(s). It is shown in Lemma 3.10 that for any ϕ ∈ B γ , the function is well defined, belongs to B γ and has an equivalent expression (3.39) in terms of derivative of the projection operator Π 0,z , see Proposition 3.8.

Theorem 2.2.
Assume either conditions A1 and A2 for invertible matrices, or conditions A1, A3 and A5 for positive matrices. Then, as n → ∞, uniformly in x ∈ S, y ∈ R and ϕ ∈ B γ , The proof of this theorem is postponed to Section 5 and is based on a new smoothing inequality (Proposition 4.1) and the saddle point method. Even for ϕ = 1, Theorem 2.2 is new.
We start by formulating a Cramér type moderate deviation expansion for the couple (X x n , log |G n x|) with target function on X x n , for both invertible matrices and positive matrices. Theorem 2.3. Assume either conditions A1 and A2 for invertible matrices, or conditions A1, A3 and A4 for positive matrices. Then, uniformly in x ∈ S, y ∈ [0, o( √ n)] and ϕ ∈ B γ , as n → ∞, Note that the above asymptotic expansions remain valid even when ν(ϕ) = 0. In this case, for example, the first expansion becomes It is an open question to extend the results of Theorem 2.3 to higher order expansions under the additional condition of non-arithmeticity. We refer to Saulis [32] and Rozovsky [31] for relevant results in the i.i.d. real-valued case.
In the case of products of random matrices this problem seems to us challenging because of the presence of the derivatives in s of the eigenfunction r s and of the eigenmeasure ν s in the higher order terms. In particular, under conditions of Theorem 2.3, with ϕ = 1 we obtain: When ϕ ∈ B γ is a real-valued function satisfying ν(ϕ) > 0, Theorem 2.3 clearly implies the following moderate deviation principle for log |G n x| with target function on X x n : for any Borel set B ⊆ R, and positive sequence (b n ) n 1 satisfying bn n → 0 and bn √ n → ∞, uniformly in x ∈ P d−1 , 2.4. Local limit theorem with moderate deviations. In this subsection we state a local limit theorem with moderate deviations for log |G n x|, which is a consequence of Theorem 2.3. Recall that ℓ denotes the Lebesgue measure on R, and ∂B denotes the boundary of a set B on the real line.
In the case of invertible matrices, a similar local limit theorem has been established in [3] in a more general setting and plays an important role in studying dynamics of group actions on finite volume homogeneous spaces, see [1,Proposition 4.7]. Specifically, from [3, Theorem 17.10], by simple calculations we deduce that for any a 1 < a 2 , it holds uniformly in x ∈ P d−1 and y ∈ [0, O( √ log n)] that, as n → ∞, Theorem 2.4 extends the range of y in (2.11) beyond O( √ log n) and moreover, allows a target function ϕ on the Markov chain X x n . Note also that in [3] the group SL(d, R) is considered instead of GL(d, R), and the proximality condition A2(ii) is replaced by the condition that Γ µ is unbounded. For positive matrices, Theorem 2.4 and its consequence (2.11) are new.

Spectral gap theory
This section is devoted to investigating the spectral gap properties of some operators to be introduced below: the transfer operator P z , its normalization Q s which is a Markov operator, and the perturbed operator R s,z , for realvalued s and complex-valued z. The properties for these operators have been intensively studied in recent years, for instance in [29,5,17,6,3], where various results have been established under different restrictions on s and z, which are not enough for obtaining the results of the paper. We shall complete these results by investigating the case when s ∈ (−η, η) with η > 0 small, and z belongs to a small ball of the complex plane centered at the origin. The case of s < 0 turns out to be more difficult than the case s 0 and requires a deeper analysis. We also complement the previous results with some new properties to be used in the proofs of the main results of the paper.
3.1. Properties of the transfer operator P z . Recall that the Banach space B γ consists of all the γ-Hölder continuous complex-valued functions on S. We write B ′ γ for the topological dual of B γ endowed with the norm be the set of all bounded linear operators from B γ to B γ equipped with the operator norm · Bγ →Bγ . Denote by ̺(Q) the spectral radius of an operator Q ∈ L(B γ , B γ ), and by Q| E its restriction to the subspace E ⊆ B γ .
For any z ∈ C with |z| < η 0 , where η 0 is given in condition A1, define the transfer operator P z as follows: for any ϕ ∈ C(S), The transfer operator P z acts from C(S) to the space of bounded functions on S. The following proposition gives the spectral gap properties of the operator P z for z in a small enough neighborhood of 0 in the complex plane. (0) → L(B γ , B γ ) is analytic for γ > 0 small enough, where η 0 is given in condition A1. Moreover, there exists a small η > 0 such that for any z ∈ B η (0) and n 1, we have the decomposition where the operator M z := ν z ⊗ r z is a rank one projection on B γ defined by 2); all these mappings are analytic in B η (0), and possess the following properties: (c) κ(0) = 1, r 0 = 1, ν 0 = ν, and κ(s) and r s are real-valued and satisfy κ(s) > 0 and r s (x) > 0 for any s ∈ (−η, η) and x ∈ S; Let us point out the differences between Proposition 3.1 and the previous results in [29,5,3]. Firstly, we complement the results in [29,3] by giving the explicit formula M z ϕ = νz(ϕ) νz(rz) r z in (3.2), for z ∈ B η (0), which is one of the crucial points in the proofs of the results of the paper. Basically, it permits us to deduce the spectral gap properties of the operators Q s and R s,z from those of P z . In particular this will enable us to obtain an explicit formula for the operators N s and N s,z in Propositions 3.4 and 3.8, and the uniformity of the bounds (3.36) and (3.37). Secondly, for positive matrices, some points of Proposition 3.1 have been obtained in [5] only for real z 0. The difficulty here is the case when z ∈ R is negative and when z is not real, so Proposition 3.1 is new for positive matrices when |z| η. Thirdly, we show that κ(z) and r z take real positive values when z is real, which allows to define the change of measure Q x s for real s, for both invertible matrices and positive matrices. Previously it was shown in [3] that κ(z) is real-valued for real z ∈ (−η, η) for invertible matrices.
In the sequel, without explicitly stated, we always assume that γ > 0 is a sufficiently small constant.

Remark 3.2.
Define the conjugate transfer operator P * z by where z ∈ C with ℜz ∈ (−η 0 , η 0 ), and g T 1 denotes the transpose of the matrix g 1 . One can verify that P * z satisfies all the properties of Proposition 3.1: under conditions of Proposition 3.1, we have the decomposition and all the assertions in Proposition 3.1 hold for Proof of Proposition 3.1. We split the proof into three steps. In steps 1 and 2 we concentrate on the case of positive matrices, since for invertible matrices the results of these steps have been proved in [29,3]. In step 1 we follow the same lines as in [29,3]. In step 2 we follow [22] to prove the spectral gap property of the operator P 0 and we use the perturbation theory to extend it to P z . In step 3 the proof is new and is provided for both invertible and positive matrices by complementing the results in [29,5,3].
Step 1. We only need to consider the case of positive matrices. We will show that there exists γ ∈ (0, η 0 6 ) such that P z ∈ L(B γ , B γ ), and that the It suffices to show that for z ∈ B η 0 2 (0) and θ ∈ B η 0 6 (0), and that there exists a constant C > 0 not depending on θ and z such that . Moreover, the bound (3.5) ensures the validity of (3.4) which implies the analyticity of the mapping z → P z on B η 0 2 (0).
It remains to prove (3.5). We first give a control of P We then control each of the three terms I 1,m , I 2,m , I 3,m . Control of I 1,m . Since for any a, b ∈ C, m ∈ N and 0 < γ < 1, we get Using (2.1), we deduce that for any g ∈ Γ µ , (3.10) Control of I 2,m . Using (3.8), we deduce that for any z 1 , z 2 ∈ C, By this inequality, we find that for any g ∈ Γ µ , Combining this with (3.6), (3.7), (3.10) and (3.12), we obtain (3.5).
Step 2. Again we need only to consider the case of positive matrices. We will prove the decomposition formula (3.2) together with parts (a), (b) and (d). Our proof follows closely [22]. Define the operator M on B γ by We next show that ̺(P | E ) < 1, where P = P 0 (see (3.1)). For any x, y ∈ S d−1 + , x = y, and ϕ ∈ B γ , there exists a ∈ (0, 1) such that for large n 1, where for the last inequality we use [19,Lemma 3.2]. Observe that for any ϕ ∈ B γ , we have ϕ − M ϕ ∈ E, thus P n (ϕ − M ϕ) ∈ E for any n 1 since νP = ν. Combining this with (3.13) and the above inequality, we get which implies ̺(P | E ) < 1. This, together with the definition of E and the fact that P 1 = 1, shows that 1 is the isolated dominant eigenvalue of the operator P . Using this and the analyticity of P z ∈ L(B γ , B γ ), and applying the perturbation theorem (see [20,Theorem III.8]), we obtain the decomposition formula (3.2) with M z (ϕ) = c 1 ν z (ϕ)r z for some constant c 1 = 0, as well as parts (a), (b) and (d). Using P z r z = κ(z)r z , we get c 1 = 1/ν z (r z ) and thus M z ϕ = νz(ϕ) νz(rz) r z for any ϕ ∈ B γ . Step 3. We prove part (c) for invertible matrices and positive matrices. From P 1 = 1, we see that κ(0) = 1 and r 0 = 1. Letting z = 0 in ν z P z = κ(z)ν z , we get ν 0 P = ν 0 and thus ν 0 = ν since ν is the unique µ-stationary probability measure. Now we fix z ∈ (−η, η) and we show that κ(z) and r z are real-valued. Taking the conjugate in the equality P z r z = κ(z)r z , we get P z r z = κ(z)r z , so that κ(z) is an eigenvalue of the operator P z . By the uniqueness of the dominant eigenvalue of P z , it follows that By part (a), the space of eigenvectors corresponding to the eigenvalue κ(z) is one dimensional. Therefore, we have either u z = cv z for some constant c ∈ R, or v z = 0. However, the equality u z = cv z is impossible because we have seen that ν(u z ) = 1 and ν(v z ) = 0. Hence v z = 0 and r z is real-valued for z ∈ (−η, η). The positivity of κ(z) and r z then follows from κ(0) = 1, r 0 = 1 and the analyticity of the mappings z → κ(z) and z → r z . This ends the proof of part (c), as well as the proof of Proposition 3.1.

3.2.
Definition of the change of measure Q x s . Proposition 3.1 allows us to perform a change of measure. Note that this change of measure for positive s has been extensively studied in [5,6,17]; however, for negative s it is new. For any s ∈ (−η, η), x ∈ S and g ∈ Γ µ , denote Note that (q s n ) verifies the cocycle property: for any n, m 1 and g 1 , Since κ(s) and r s are strictly positive, q s n (x, G n )µ(dg 1 )...µ(dg n ), n 1, is a sequence of probability measures, and forms a projective system on M N * . By the Kolmogorov extension theorem, there is a unique probability measure Q x s on M N * with marginals q s n (x, G n )µ(dg 1 )...µ(dg n ). Denote by E Q x s the corresponding expectation. For any n ∈ N and any bounded measurable 3.3. Properties of the Markov operator Q s . For any s ∈ (−η, η) and ϕ ∈ B γ , define the Markov operator Q s by Under the changed measure Q x s , the process (X x n ) n∈N is a Markov chain with the transition operator given by Q s .
The following assertion will be useful to prove that the function κ is strictly convex (see Lemma 3.15). Recall that V (Γ µ ) is the support of the measure ν (cf. (2.3), (2.4)).
We state the spectral gap property of the Markov operator Q s , whose proof is postponed to Section 3.5.
where ν s , r s , L s are given in Proposition 3.1; (b) Π s N s = N s Π s = 0, and for each k ∈ N, there exist constants C k > 0 and a ∈ (0, 1) such that 3.4. Quasi-compactness of the operator Q s+it . For s ∈ (−η, η) and t ∈ R, define the operator Q s+it as follows: for any ϕ ∈ B γ , The spectral gap properties of the operator Q s+it for |t| small enough can be deduced from Proposition 3.1. However, this approach does not work for large |t|. In order to investigate the spectral gap properties of the operator Q s+it for t ∈ R, we first prove the Doeblin-Fortet inequality and then we apply the theorem of Ionescu-Tulcea and Marinescu [25] to establish the quasi-compactness of the operator Q s+it . Based on this property, we shall use the non-arithmeticicty condition A5 to prove that the spectral radius of Q s+it is strictly less than 1 when t is different from 0.
The following is the Doeblin-Fortet inequality for the operator Q s+it .

Lemma 3.5.
Assume the conditions of Proposition 3.1. Then, there exist constants 0 < a < 1, and η > 0 small enough, such that for any s ∈ (−η, η), t ∈ R, n 1 and ϕ ∈ B γ , we have For positive-valued s, analogous results can be found in [17] for invertible matrices and in [6] for positive matrices. The proofs in [17,6] rely essentially on the Hölder continuity of the mapping x → q s n (x, g) defined in (3.14). However, this property doesn't hold any more in the case when s is negative. Our proof of Lemma 3.5 is carried out using the Hölder inequality and the spectral gap properties of the operator P s established in Proposition 3.1.
Proof of Lemma 3.5. Using the definition of Q s+it and (3.15), we get that for any n 1, It follows that where Note that by Proposition 3.1, for any s ∈ (−η, η), we have min x∈S r s (x) > 0, max x∈S r s (x) < ∞ and κ(s) > 0. Control of J 1 (n). Observe that uniformly in x ∈ S, Since r s ∈ B γ , this implies that for any s ∈ (−η, η) and t ∈ R, Control of J 2 (n). Using the definition of P s+it and taking into account that r s is strictly positive and bounded on S, we have where Control of J 21 (n). Using (3.11) and the inequality log u u ε , u > 1, for ε > 0 small enough, we obtain |G n x| s+it − |G n y| s+it 2(N (G n )) |s|+ε log |G n x| − log |G n y| γ . (3.22) From the inequality (2.1), by arguing as in the estimate of (3.9), we get Using first (3.22) and then the last bound, we deduce that where the last inequality holds by condition A1.

27)
Control of J 23 (n). Using (3.26) and the fact that r s ∈ B γ , and applying similar techniques as in the control of J 22 (n), one can verify that there exists a constant 0 < a < 1 such that uniformly in n 1, (3.28) Inserting (3.23), (3.27) and (3.28) into (3.21), we conclude that Combining this with (3.20) and (3.19), we obtain the inequality (3.18).
From Lemma 3.5 and the fact that Q s+it ϕ ∞ C s ϕ ∞ for any s ∈ (−η, η) and t ∈ R, we can deduce that Q s+it ∈ L(B γ , B γ ). We next prove that the operator Q s+it is quasi-compact. Recall that an operator Q ∈ L(B γ , B γ ) is quasi-compact if B γ can be decomposed into two Q invariant closed subspaces B γ = E ⊕ F , such that dim E < ∞, each eigenvalue of Q| E has modulus ̺(Q), and ̺(Q| F ) < ̺(Q) (see [20] for more details). Proposition 3.6. Assume the conditions of Proposition 3.1. Then, there exists a small η > 0 such that for any s ∈ (−η, η) and t ∈ R, the operator Q s+it is quasi-compact.
Proof. The proof consists of verifying the conditions of the theorem of Ionescu-Tulcea and Marinescu [25]. We follow the formulation in [20,Theorem II.5].
Firstly, by the definition of Q s+it , there exists a constant C s > 0 such that Q s+it ϕ ∞ C s ϕ ∞ for any s ∈ (−η, η), t ∈ R and ϕ ∈ B γ . Secondly, by Lemma 3.5, the Doeblin-Fortet inequality (3.18) holds for the operator Q s+it .
The assertion of the proposition now follows from the theorem of Ionescu-Tulcea and Marinescu.
The following proposition shows that the spectral radius of the operator Q s+it is strictly less than 1 when t is different from 0. The proof which relies on the non-arithmeticity condition A5, follows the standard pattern in [17,6]; it is included for the commodity of the reader. Proposition 3.7. Assume either conditions A1 and A2 for invertible matrices, or conditions A1, A3 and A5 for positive matrices. Then, for any s ∈ (−η, η) with small η > 0, and any t ∈ R\{0}, we have ̺(Q s+it ) < 1.

Spectral gap properties of the perturbed operator R s,z .
For any s ∈ (−η, η) and z ∈ C such that s + ℜz ∈ (−η 0 , η 0 ), define the perturbed operator R s,z as follows: for any ϕ ∈ B γ , With some calculations using (3.15), it follows that for any n 1, The following formula relates the operator R n s,z to the operator P n s+z and is of independent interest: for any ϕ ∈ B γ , n 1, s ∈ (−η, η) and z ∈ B η (0), The identity (3.31) is obtained by the definitions of R s,z and P z using the change of measure (3.16). There are two ways to establish spectral gap properties of the operator R s,z : one is to use the perturbation theory of operators [20, Theorem III.8], another is based on the Ionescu-Tulcea and Marinescu theorem [25] about the quasi-compactness of operators. The representation (3.31) allows us to deduce the spectral gap properties of R s,z directly from the properties of the operator P z . This has some advantages: it ensures the uniformity in s ∈ (−η, η), allows to deal with negative-vaued s and provides an explicit formula for the projection operator Π s,z and the remainder operator N n s,z defined below.
Recall that Λ = log κ, where κ is defined in (2.6). where r z , ν z and L z are given in Proposition 3.1. In addition, we have: (a) for fixed s, the mappings z → Π s,z : B δ (0) → L(B γ , B γ ), z → N s,z : B δ (0) → L(B γ , B γ ) and z → λ s,z : B δ (0) → C are analytic, (b) for fixed s and z, Π s,z is a rank-one projection with Π s,0 (ϕ)(x) = π s (ϕ) for any ϕ ∈ B γ and x ∈ S, and Π s,z N s,z = N s,z Π s,z = 0, (c) for k ∈ N, there exist 0 < a < 1 and C k > 0 such that Note that, for s > 0, similar results have been obtained in [6]. The novelty here is that s can account for negative values s ∈ (−η, 0] and that the bounds (3.36) and (3.37) hold uniformity in s ∈ (−η, η). This plays a crucial role in establishing Theorem 2.3.
Proof of Proposition 3.8. The proof is divided into three steps.
Step 3. We prove part (c). By Proposition 3.1, there exists η > 0 such that the mappings z → κ(z), z → r z , z → ν z are analytic and uniformly bounded on B 2η (0). Combining this with (3.34), we obtain (3.36). We now prove (3.37). Since the function r s is strictly positive on the compact set S, by Proposition 3.1(d), we deduce that there exists 0 < a 0 < 1 such that uniformly in ϕ ∈ B γ , Proof. Set ρ n (s, t) = R n s,it ϕ 1/n ∞ . Using the inequality |ρ n (s 1 , t 1 )−ρ n (s 2 , t 2 )| R n s 1 ,it 1 ϕ − R n s 2 ,it 2 ϕ 1/n ∞ , Proposition 3.1 and the definition of the operator R s,it , we get that for any fixed n ∈ N and ϕ ∈ B γ , the function ρ n is continuous on the compact set I η × K, where I η = [−η, η]. This implies that the function ρ := lim sup n→∞ ρ n is upper semicontinuous on I η × K, so that ρ attains the maximum at the point (s 0 , t 0 ) ∈ I η × K. By Proposition 3.7, we have ρ(s 0 , t 0 ) ̺(R s 0 +it 0 ) = ̺(Q s 0 +it 0 ) < 1 and thus the assertion follows.
We now give some properties of the function b s,ϕ defined as follows: for any s ∈ (−η, η) and ϕ ∈ B γ , In particular, with s = 0, b 0,ϕ = b ϕ , which is defined in (2.7).

Lemma 3.10. Assume the conditions of Proposition 3.1. Then the function
Proof. In view of Proposition 3.8, we have that for any ϕ ∈ B γ , From (3.33), we have λ s,0 = 1 and dλs,z dz | z=0 = 0. Differentiating both sides of the above equation w.r.t. z at the point 0 gives that for any x ∈ S, Using the bounds (3.36) and (3.37), we find that the first term on the righthand side of (3.40) belongs to B γ , and the second term converges to 0 exponentially fast as n → ∞. Hence, letting n → ∞ in (3.40), we obtain (3.39). This shows that the function b s,ϕ is well-defined and b s,ϕ ∈ B γ .
For any s ∈ (−η, η) with η > 0 small, define Q s = S Q x s π s (dx). The following result will be used to prove the strong law of large numbers for log |G n x| under the changed measure Q s . Lemma 3.11. Assume the conditions of Proposition 3.1. There exist η > 0 and c, C > 0 such that uniformly in s ∈ (−η, η), ϕ ∈ B γ and n 1, Proof. We follow the proof of the previous lemma. Integrating both sides of the identity (3.40) w.r.t. π s , we get, for any ϕ ∈ B γ , Integrating both sides of this equation w.r.t. π s and using the fact that Π s,0 = π s , we find that It follows from (3.37) that uniformly in ϕ ∈ B γ and s ∈ (−η, η), the second term on the right-hand side of (3.42) is bounded by C ϕ γ e −cn . Therefore, from (3.42) and (3.43) we obtain (3.41).
We now establish the strong laws of large numbers for log |G n x| under the measures Q x s and Q s , which are of independent interest. From (3.30) and Proposition 3.8, we deduce that there exist positive constants c, C independent of s, x, δ such that Using Taylor's formula and taking δ > 0 small enough, we conclude that which implies the assertion (3.44).  , θω), where ω ∈ Ω and θ is the shift operator on Ω. For any x ∈ S and ω ∈ Ω, set h(x, ω) = log |g 1 (ω)x|. Then h is Q s -integrable. Since log |G n x| = n−1 k=0 (h • θ k )(x, ω) and Q s is θ-ergodic, it follows from Birkhoff's ergodic theorem that 1 n log |G n x| converges Q s -a.s. to some constant c s as n → ∞. If we suppose that c s is different from Λ ′ (s), then this contradicts to (3.45). Thus c s = Λ ′ (s) and the assertion of the lemma follows. Now we give the third-order Taylor expansion of λ s,z defined by (3.33), w.r.t. z at the origin in the complex plane.
To show that the function Λ is strictly convex, we suppose, by absurd, that there exist s 1 = s 2 and some t ∈ (0, 1) such that κ(s ′ ) = κ t (s 1 )κ 1−t (s 2 ). Using this equality, the definition of Q s and (3.47), we get Q s ′ (r t for some constant c > 0. Substituting this equality and the identity κ(s ′ ) = κ t (s 1 )κ 1−t (s 2 ) into (3.47), we see that the Hölder inequality in (3.47) is actually an equality. This yields that there exists a function c(x) > 0 such that for any g ∈ Γ µ and x ∈ V (Γ µ ), we have κ(s 2 )rs 2 (x) . Substituting this into (3.48) and noting that s 1 = s 2 , we find that there exist a constant c 1 > 0 and a real-valued function ϕ on S such that |gx| = c 1 for any g ∈ Γ µ and x ∈ V (Γ µ ). This contradicts to condition A5. Recall that condition A2 implies condition A5 for invertible matrices. Hence Λ is strictly convex for invertible matrices under stated conditions. Proof of Proposition 3.14. The expansion (3.46) follows from the identity (3.33) and Taylor's formula.
For part (b), recall that it was shown in [6] that σ 0 > 0 for invertible matrices under the stated conditions, and for positive matrices under the additional condition of non-arithmeticity. Hence, using the continuity of the function Λ ′′ , we obtain that σ s > 0.
For part (c), by Proposition 3.8, we get that for |z| small, Taking the second derivative on both sides of the equation (3.49) with respect to z at 0, and using the expansions (3.50)-(3.52), we deduce that This, together with the definition of Q s and the fact that the constants c s,x,2 , C s,x,n,2 are bounded as functions of s, x, n, concludes the proof of part (c). For part (d), integrating both sides of the equations (3.49), (3.51) and (3.52) with respect to the invariant measure π s , and using the property (3.43) with ϕ = 1 (thus the second term on the right-hand side of (3.51) vanishes), in the same way as in the proof of (3.53), we get This implies the assertion in part (d).

Remark 3.16.
Inspecting the proof of Proposition 3.14, it is easy to see that the results in parts (c) and (d) can be reinforced to the following bounds: The first bound above also holds with the measure Q x s replaced by Q s .

Smoothing inequality on the complex plane
In this section we aim to establish a new smoothing inequality, which plays a crucial role in proving the Berry-Esseen theorem and Edgeworth expansion with a target function ϕ on X x n ; see Theorems 2.1, 2.2, 5.1 and 5.3.
From now on, for any integrable function h : R → C, denote its Fourier transform by h(t) = R e −ity h(y)dy, t ∈ R. If h is integrable on R, then using the inverse Fourier transform gives h(y) = 1 2π R e ity h(t)dt, for almost all y ∈ R with respect to the Lebesgue measure on R. Denote by h 1 * h 2 the convolution of the functions h 1 , h 2 on the real line. For any r > 0, set D r = {z ∈ C : |z| < r, ℑz = 0} and D r = {z ∈ C : |z| r}.
Following [37], we construct a density function ρ T which plays an important role in establishing a new smoothing inequality. Specifically, on the real line define ς(t) = e Then ρ is a non-negative Schwartz function with R ρ(y)dy = 1. Its Fourier transform ρ is given by We see that ρ is compactly supported on [−2, 2]. Moreover, it is proved in [37] that ρ has an analytic extension on the domain D 1 := {z ∈ C : |z| < 1, ℑz = 0} and has a continuous extension on the domain D 1 = {z ∈ C : |z| 1}. The Hölder inequality implies that ρ is bounded by 1 π . Since ρ is a density function on R and ρ is non-negative, we have 0 ρ 1 on R.
For any T > 0 and the fixed constant b > 0 satisfying b −b ρ(y)dy = 3/4, define the density function whose Fourier transform ρ T is given by ρ T (t) = e −2ibt/T ρ(2t/T ), t ∈ R. We see that on the real line ρ T is compactly supported on [−T, T ] since ρ is compactly supported on [−2, 2]. Since ρ has an analytic extension on the domain D 1 , we can extend the function ρ T analytically to the domain D T as follows: Note that ρ T has a continuous extension on the domain D T since ρ has a continuous extension on D 1 . Now we are ready to establish a new smoothing inequality. Its proof is based on the properties of the density function ρ T , Cauchy's integral theorem and some techniques from [12,30].

Proposition 4.1. Assume that F is non-decreasing on R, and that H is differentiable of bounded variation on
Suppose that r > 0 and that f and h have analytic extensions on D r , and have continuous extensions on D r . Then, for any T r, where b > 0 is a fixed constant satisfying b −b ρ(y)dy = 3/4, and the semicircles C − r and C + r are given by Upper bound. Since the function F is non-decreasing on R and ρ T is a density function on R, we find that for any y ∈ R, Restricted on the real line, the function ρ T is supported on [−T, T ]. By the inversion formula we get We shall use Cauchy's integral theorem to change the integration path [−T, T ] to a contour in the complex plane. In order to estimate the difference |F 1 (y) − H 1 (y)|, we are led to consider two cases: y 0 and y > 0.
T ] and the lower semicircle C − r is given in (4.2). Since the functions f , h and ρ T are analytic on the domain D r , and have continuous extensions on its closure D r , applying Cauchy's integral theorem gives where the integration is over the complex curve C − oriented from −T to T . The second integral in (4.4) converges to 0 as v → −∞, by using the Riemann-Lebesgue lemma on the real segment C r,T and by applying the Lebesgue convergence theorem on the semicircle C − r . Note that and hence sup y 0 T ] and the upper semicircle C + r is given in (4.2). In an analogous way as in (4.4), we have where the integration is over the complex curve C + also oriented from −T to T . The second integral in (4.6) converges to 0 as v → +∞, by using again the Riemann-Lebesgue lemma on the real segment C r,T and by applying the Lebesgue convergence theorem on the upper semicircle C + r . Note that F 1 (∞) = H 1 (∞) since F (∞) = H(∞). Hence, letting v → +∞ in (4.6), similarly to (4.5), we obtain As a result, putting together (4.5) and (4.7) leads to Then, taking into account that ρ T is a density function on R, using (4.8) and the fact that Substituting this inequality into (4.3), we obtain the following upper bound: Lower bound. Similarly to (4.3), we have for any y ∈ R, Proceeding in the same way as in the proof of (4.8), we get Following the proof of (4.9), we obtain the lower bound: Combining (4.9) and (4.10), we conclude the proof of Proposition 4.1.

Berry-Esseen bound and Edgeworth expansion under the changed measure.
We first formulate a Berry-Esseen bound under the changed measure Q x s . Theorem 5.1. Assume either conditions A1 and A2 for invertible matrices, or conditions A1, A3 and A4 for positive matrices. Then there exist constants η > 0 and C > 0 such that for all n 1, s ∈ (−η, η), x ∈ S, y ∈ R and ϕ ∈ B γ , The following result gives an Edgeworth expansion for log |G n x| with the target function ϕ on X x n under Q x s . The function b s,ϕ (x), x ∈ S, which will be used in the formulation of this result, is defined in Lemma 3.10 and has an equivalent expression (3.39) in terms of derivative of the projection operator Π s,z , see Proposition 3.8.
The assertion of Theorem 5.3 follows from Theorem 5.2, since the bound (3.17) implies that there exist constants c, C > 0 such that uniformly in Theorems 2.1 and 2.2 follow from the above theorems taking s = 0 and recalling the fact that Λ ′ (0) = λ, σ 0 = σ and b 0,ϕ = b ϕ .

Proof of Theorem 5.2.
Without loss of generality, we assume that ϕ is non-negative. Denote By straightforward calculations we have (5.5) By the formula (3.39) and the bound (3.36), we get that uniformly in ϕ ∈ B γ , Using the bounds (5.9) and (5.10), and taking into account that σ 2 s > 0 and Λ ′′′ (s) ∈ R are bounded by a constant independent of s ∈ (−η, η), we obtain that |H ′ (y)| is bounded by c 1 ϕ γ , uniformly in s ∈ (−η, η), x ∈ S, y ∈ R and ϕ ∈ B γ . Hence, for any ε > 0, we can choose a > 0 large enough, such that, for T = a √ n, uniformly in ϕ ∈ B γ , we have Control of I 2 . Since σ m := inf s∈(−η,η) σ s > 0, we can pick δ 1 small enough, such that 0 < δ 1 < min{a, δσ m /2}, where the constant δ > 0 is given in Proposition 3.8. Then with r = δ 1 √ n we bound I 2 as follows: Let σ M := sup s∈(−η,η) σ s < ∞. On the right-hand side of (5.12), using Proposition 3.9 with K = {t ∈ R : δ 1 /σ M |t| a/σ m }, the first integral is bounded by Ce −cn ϕ γ , uniformly in s ∈ (−η, η), x ∈ S and ϕ ∈ B γ ; the second integral, by the bounds (5.9) and (5.10) and direct calculations, is bounded by Ce −c √ n ϕ γ , also uniformly in s ∈ (−η, η), x ∈ S and ϕ ∈ B γ . Consequently, we conclude that uniformly in ϕ ∈ B γ , Recall that the term I 3 is decomposed into four terms in (5.7). We will only deal with I 31 , since I 32 , I 33 , I 34 can be treated in a similar way. In view of (5.4) and (5.5), by the spectral gap decomposition (3.32), we get where (5.18) With the above notation, we use the decomposition (5.14) to bound I 31 in (5.7) as follows: We now give bounds of A k , 1 k 4, in a series of lemmata. Let us start by giving an elementary inequality, which will be used repeatedly in the sequel. Let [z 1 , z 2 ] = {z 1 + θ(z 2 − z 1 )) : 0 θ 1} be the complex segment with the endpoints z 1 and z 2 .
Proof. The proof of this inequality can be carried out by induction. The inequality clearly holds for n = 1 since for any z 1 , z 2 ∈ D, For n 2, applying (5.20) to leads to the desired assertion. Now we are ready to establish a bound of each term A k . The proof is based on the saddle point method. To be more precise, we deform the integration path, which passes through a suitable point related to the saddle point, to minimise the integral in A k (see (5.19)).
Lemma 5.5. Let C − r be defined by (4.2) with r = δ 1 √ n and δ 1 > 0 small enough. Then, for T = a √ n with a > 0 large enough, uniformly in x ∈ S, s ∈ (−η, η) and ϕ ∈ B γ , Proof. In view of (3.33), using Λ = log κ and Taylor's formula, we have For brevity, for any z ∈ C − r , denote Then, in view of (5.15), the term A 1 can be rewritten as When −δ 1 √ n y 0, the saddle point iy belongs to D − 2r . By Cauchy's integral theorem, we change the integration in (5.23) to a rectangular path inside the domain on analyticity D − 2r which passes through the saddle point. When y < −δ 1 √ n is large, the saddle point iy is outside the domain D − 2r . In this case we choose a rectangular path inside D − 2r which passes through the point −iy n = −iδ 1 √ n. Note that π s (ϕ) is bounded by c 1 ϕ ∞ uniformly in s ∈ (−η, η) and ϕ ∈ B γ . Since the function h 1 has an analytic extension on the domain D − 2r with r = δ 1 √ n, applying Cauchy's integral theorem, we deduce that Control of A 11 . Using a change of variable, we get We first bound |h 1 (±δ 1 √ n − it)|. Since t ∈ [0, y n ] and y n δ 1 √ n, direct calculations give Observe that there exists a constant c > 0 such that uniformly in t ∈ [0, y n ] and s ∈ (−η, η), Since the function ρ T has a continuous extension on the domain D T , we infer that | ρ T (±δ 1 √ n + it)| is bounded uniformly in t ∈ [0, y n ] and n 1.
Combining this with the bounds (5.27) and (5.28), uniformly in s ∈ (−η, η), In view of (5.24), we have t y n −y and thus e t 2 2 +ty 1 for any t ∈ [0, y n ]. Note that y n δ 1 √ n by (5.24). Consequently, we obtain the bound: sup s∈(−η,η) Control of A 12 . Using a change of variable z = t − iy n leads to where the function h 1 is defined by (5.22). To estimate the term A 12 , the main task is to give a control of |h 1 (t − iy n )|. It follows from Lemma 5.4 that |e z 1 − e z 2 | e max{ℜz 1 ,ℜz 2 } |z 1 − z 2 | and |e z 2 − 1 − z 2 | 1 2 |z 2 | 2 e |z 2 | for any z 1 , z 2 ∈ C, and hence We shall make use of the inequality (5.31) to derive a bound of |h 1 (t − iy n )|. Since yn √ n δ 1 where δ 1 > 0 can be sufficiently small, for any |t| δ 1 √ n, we get that uniformly in s ∈ (−η, η), Moreover, elementary calculations yield that there exists a constant c > 0 such that uniformly in s ∈ (−η, η), End of the proof of Theorem 5.2. Combining Lemmata 5.5-5.8, we obtain that I 31 c n ϕ γ , uniformly in s ∈ (−η, η), x ∈ S and ϕ ∈ B γ . Now we give a control of the term I 32 defined in (5.7). Note that y > 0 in I 32 and the integral in I 32 is taken over the semicircle C + r , which lies in the upper part of the complex plane. In this case we have the saddle point equation d dz (− z 2 2 + izy) = 0 whose solution z = iy also lies in the upper part of the complex plane. Similarly to (5.24), we choose a suitable point y n = min{y, δ 1 √ n}. Proceeding in the same way as for bounding I 31 we obtain that I 32 c n ϕ γ , uniformly in s ∈ (−η, η), x ∈ S and ϕ ∈ B γ . Let us now bound the terms I 33 and I 34 defined in (5.7). Since the mapping z → ρ T (z) is analytic on C − r and C + r , the estimates of I 33 and I 34 are similar to those of I 31 and I 32 , respectively. From these bounds, one concludes that I 3 c ϕ γ /n, uniformly in s ∈ (−η, η), x ∈ S and ϕ ∈ B γ . Combining this with the bounds for I 1 and I 2 in (5.11) and (5.13), and using the fact that ε can be arbitrary small, we obtain (5.8), which finishes the proof. Then we obtain a similar inequality as (5.6) but with the term I 2 = 0. Since the non-arithmeticity condition A5 is only used in the bound of the term I 2 , following the proof of Theorem 5.3 we show that under the conditions of Theorem 5.1, the terms I 1 and I 3 defined in (5.7) are bounded by c ϕ γ / √ n, uniformly in s ∈ (−η, η), x ∈ S and ϕ ∈ B γ . We omit the details of the rest of the proof.

Proof of moderate deviation expansions
In this section we prove Theorem 2.3. The proof is based on the Berry-Esseen bound in Theorem 5.1 and follows the standard techniques in Petrov [30], and therefore some details will be left to the reader.

Proof of a local limit theorem
In this section we prove Theorem 2.4. By a general result on the narrow convergence of measures, it is enough to prove the theorems for intervals B = [a 1 , a 2 ], where a 1 , a 2 ∈ R. Without loss of generality, we assume that ϕ is non-negative. Denote y i = y + a i √ nσ , i = 1, 2. From Theorem 2.3, we get that uniformly in x ∈ S, y ∈ [0, o( √ n)] and ϕ ∈ B γ , as n → ∞, Substituting the above asymptotic expansions into (7.1), we get the result.