Precise large deviation asymptotics for products of random matrices

TL;DR

This paper derives precise large deviation asymptotics for the norms of products of i.i.d. random matrices, extending existing results and providing new limit theorems for matrix norms and related quantities.

Contribution

It establishes exact asymptotics for large deviation probabilities of matrix products, including invertible and positive matrices, and improves prior large deviation principles.

Findings

Precise asymptotics for large deviations of $\log | G_n x |\

Enhanced large deviation principles for matrix norms

Derived a local limit theorem with large deviations

Abstract

Let $(g_{n})_{n\geq 1}$ be a sequence of independent identically distributed $d\times d$ real random matrices with Lyapunov exponent $\gamma$. For any starting point $x$ on the unit sphere in $\mathbb R^d$, we deal with the norm $ | G_n x | $, where $G_{n}:=g_{n} \ldots g_{1}$. The goal of this paper is to establish precise asymptotics for large deviation probabilities $\mathbb P(\log | G_n x | \geq n(q+l))$, where $q>\gamma $ is fixed and $l$ is vanishing as $n\to \infty$. We study both invertible matrices and positive matrices and give analogous results for the couple $(X_n^x,\log | G_n x |)$ with target functions, where $X_n^x= G_n x /| G_n x |$. As applications we improve previous results on the large deviation principle for the matrix norm $\|G_n\|$ and obtain a precise local limit theorem with large deviations.

Equations

$$\mathbb{P}(S_n⩾n(q+l))\sim \frac{\exp (-n{\Lambda }^{\ast }(q+l))}{s{\sigma }_s\sqrt{2\pi n}},n\to \infty ,$$

$$\mathbb{P}(\log |G_{n}x|⩾nq)\sim {\bar{r}}_s(x)\frac{\exp \left(-n{\Lambda }^{\ast }(q)\right)}{s{\sigma }_s\sqrt{2\pi n}},n\to \infty ,$$

$$\mathbb{P}(\log |G_{n}x|⩾n(q+l))\sim {\bar{r}}_s(x)\frac{\exp \left(-n{\Lambda }^{\ast }(q+l)\right)}{s{\sigma }_s\sqrt{2\pi n}},n\to \infty .$$

$$\mathbb{P}(\log |G_{n}x|\in [nq,nq+\Delta ))\sim \Delta \frac{{\bar{r}}_s(x)}{s{\sigma }_s\sqrt{2\pi n}}{e}^{-n{\Lambda }^{\ast }(q)},n\to \infty .$$

$$c_s⩽\underset{n\to \infty }{\mathrm{liminf}}\frac{\mathbb{P}(\log |G_{n}x|⩾nq)}{\frac{1}{\sqrt{n}}\text{leavevmode}{e}^{-n{\Lambda }^{\ast }(q)}}⩽\underset{n\to \infty }{\mathrm{limsup}}\frac{\mathbb{P}(\log |G_{n}x|⩾nq)}{\frac{1}{\sqrt{n}}\text{leavevmode}{e}^{-n{\Lambda }^{\ast }(q)}}⩽C_s.$$

$$\displaystyle\mathbb{E}\Big{[}\varphi(X_{n}^{x})\psi(\log|G_{n}x|-n(q+l))\Big{]}$$

$$\sim {\bar{r}}_s(x){\nu }_s(\phi ){\int }_{\mathbb{R}}{e}^{-sy}\psi (y)dy\frac{\exp \left(-n{\Lambda }^{\ast }(q+l)\right)}{{\sigma }_s\sqrt{2\pi n}},n\to \infty .$$

$$\displaystyle\mathbb{P}\big{(}\log|G_{n}x|\leqslant n(q+l)\big{)}=\bar{r}_{s}(x)\frac{\exp\left(-n\Lambda^{*}(q+l)\right)}{-s\sigma_{s}\sqrt{2\pi n}}(1+o(1)).$$

$$\frac{e^{n{\Lambda }^{\ast }(q+l)}}{r_s(x)}$$

$$\displaystyle=e^{nh_{s}(l)}\mathbb{E}_{\mathbb{Q}_{s}^{x}}\Big{(}\frac{\psi_{s}(\log|G_{n}x|-n(q+l))}{r_{s}(X_{n}^{x})}\Big{)},$$

$$\displaystyle\mathbb{E}_{\mathbb{Q}_{s}^{x}}\Big{(}\frac{\psi_{s}(\log|G_{n}x|-n(q+l))}{r_{s}(X_{n}^{x})}\Big{)}$$

$$⩽\frac{1}{2\pi }{\int }_{\mathbb{R}}{e}^{-itln}{R}_{s,it}^n(r_s^{-1})(x){\hat{\psi }}_{s,\epsilon }^{+}(t){\hat{\rho }}_{{\epsilon }^2}(t)dt,$$

$$\displaystyle e^{nh_{s}(l)}\Big{\{}\int_{|t|<\delta}+\int_{|t|\geqslant\delta}\Big{\}}e^{-itln}R^{n}_{s,it}(r_{s}^{-1})(x)\widehat{\psi}_{s,\varepsilon}^{+}(t)\widehat{\rho}_{\varepsilon^{2}}(t)dt.$$

$$e^{n{h}_s(l)}{\int }_{-\delta }^{\delta }{e}^{-itln}{\lambda }_{s,it}^n{\Pi }_{s,it}(r_s^{-1})(x){\hat{\psi }}_{s,\epsilon }^{+}(t){\hat{\rho }}_{{\epsilon }^2}(t)dt.$$

$$\Vert \phi {\Vert }_{\infty }:=\underset{x\in \mathcal{S}}{\sup }|\phi (x)|\text{mbox}and\Vert \phi {\Vert }_{\gamma }:=\Vert \phi {\Vert }_{\infty }+\underset{x,y\in \mathcal{S}}{\sup }\frac{|\phi (x)-\phi (y)|}{\mathbf{d}(x,y{)}^{\gamma }},$$

$$I_{\mu }=\{s⩾0:\mathbb{E}(\Vert g_1{\Vert }^s)<+\infty \}.$$

$$V({\Gamma }_{\mu })=\overline{\{\pm v_g\in {\mathbb{S}}^{d-1}:g\in {\Gamma }_{\mu },g\text{mbox}isproximal\}}.$$

$$V({\Gamma }_{\mu })=\overline{\{v_g\in {\mathbb{S}}_{+}^{d-1}:g\in {\Gamma }_{\mu },g\in {\mathcal{G}}_{+}^{\circ }\}}.$$

$$\kappa (s)=\underset{n\to \infty }{\lim }{\left(\mathbb{E}\Vert G_n{\Vert }^s\right)}^{\frac{1}{n}}.$$

$$P_s\phi (x)={\int }_{{\Gamma }_{\mu }}|g_{1}x{|}^s\phi (g_1\cdot x)\mu (d{g}_1),P_s^{\ast }\phi (x)={\int }_{{\Gamma }_{\mu }}|g_1^{\mathrm{T}}x{|}^s\phi (g_1^{\mathrm{T}}\cdot x)\mu (d{g}_1),$$

$$r_s(x)={\int }_{\mathcal{S}}|⟨x,y⟩{|}^s{\nu }_s^{\ast }(dy),r_s^{\ast }(x)={\int }_{\mathcal{S}}|⟨x,y⟩{|}^s{\nu }_s(dy).$$

$$\displaystyle\mathbb{P}\big{(}\log|G_{n}x|\geqslant n(q+l)\big{)}=\bar{r}_{s}(x)\frac{\exp\left(-n\Lambda^{*}(q+l)\right)}{s\sigma_{s}\sqrt{2\pi n}}(1+o(1)).$$

$$\displaystyle\mathbb{P}\big{(}\log|G_{n}x|\geqslant nq\big{)}=\bar{r}_{s}(x)\frac{\exp\left(-n\Lambda^{*}(q)\right)}{s\sigma_{s}\sqrt{2\pi n}}(1+o(1)).$$

$$\displaystyle\Lambda^{*}(q+l)=\Lambda^{*}(q)+sl+\frac{l^{2}}{2\sigma_{s}^{2}}-\frac{l^{3}}{\sigma_{s}^{3}}\zeta_{s}\Big{(}\frac{l}{\sigma_{s}}\Big{)},$$

$$\displaystyle\mathbb{E}\Big{[}\varphi(X_{n}^{x})\psi(\log|G_{n}x|-n(q+l))\Big{]}$$

$$\displaystyle\qquad\qquad=\bar{r}_{s}(x)\frac{\exp\left(-n\Lambda^{*}(q+l)\right)}{\sigma_{s}\sqrt{2\pi n}}\Big{[}\nu_{s}(\varphi)\int_{\mathbb{R}}e^{-sy}\psi(y)dy+o(1)\Big{]}.$$

$$\displaystyle\mathbb{P}\big{(}\log|G_{n}x|\leqslant n(q+l)\big{)}=\bar{r}_{s}(x)\frac{\exp\left(-n\Lambda^{*}(q+l)\right)}{-s\sigma_{s}\sqrt{2\pi n}}(1+o(1)).$$

$$\displaystyle\mathbb{P}\big{(}\log|G_{n}x|\leqslant nq\big{)}=\bar{r}_{s}(x)\frac{\exp\left(-n\Lambda^{*}(q)\right)}{-s\sigma_{s}\sqrt{2\pi n}}(1+o(1)).$$

$$\displaystyle\mathbb{E}\Big{[}\varphi(X_{n}^{x})\psi(\log|G_{n}x|-n(q+l))\Big{]}$$

$$\displaystyle\qquad\qquad\quad=\bar{r}_{s}(x)\frac{\exp\left(-n\Lambda^{*}(q+l)\right)}{\sigma_{s}\sqrt{2\pi n}}\Big{[}\nu_{s}(\varphi)\int_{\mathbb{R}}e^{-sy}\psi(y)dy+o(1)\Big{]}.$$

$$\displaystyle\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}\big{(}\log\|G_{n}\|\geqslant n(q+l)\big{)}=-\Lambda^{*}(q).$$

$$\displaystyle\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}\big{(}\log\|G_{n}\|\leqslant n(q+l)\big{)}=-\Lambda^{*}(q).$$

$$\displaystyle\mathbb{E}\Big{[}\varphi(X_{n}^{x})\mathbbm{1}_{\{\log|G_{n}x|\in n(q+l)+[a,a+\Delta)\}}\Big{]}$$

Full text

1. Introduction

1.1. Background and main objectives

One of the fundamental results in the probability theory is the law of large numbers. The large deviation theory describes the rate of convergence in the law of large numbers. The most important results in this direction are the Bahadur-Rao and the Petrov precise large deviation asymptotics that we recall below for independent and identically distributed (i.i.d.) real-valued random variables $(X_{i})_{i\geqslant 1}$. Let $S_{n}=\sum_{i=1}^{n}X_{i}.$ Denote by $I_{\Lambda}$ the set of real numbers $s\geqslant 0$ such that $\Lambda(s):=\log\mathbb{E}[e^{sX_{1}}]<+\infty$ and by $I_{\Lambda}^{\circ}$ the interior of $I_{\Lambda}$. Let $\Lambda^{*}$ be the Frenchel-Legendre transform of $\Lambda$. Assume that $s\in I_{\Lambda}^{\circ}$ and $q$ are related by $q=\Lambda^{\prime}(s)$. Set $\sigma^{2}_{s}=\Lambda^{\prime\prime}(s).$ From the results of Bahadur and Rao [1] and Petrov [31] it follows that if the law of $X_{1}$ is non-lattice, then the following large deviation asymptotic holds true:

[TABLE]

where $\Lambda^{*}(q+l)=\Lambda^{*}(q)+sl+\frac{l^{2}}{2\sigma_{s}^{2}}+O(l^{3})$ and $l$ is a vanishing perturbation as $n\to\infty.$ Bahadur and Rao [1] have established the equivalence (1.1) with $l=0$. Petrov improved it by showing that (1.1) holds uniformly in $|l|\leqslant l_{n}\to 0$ as $n\to\infty.$ Actually, Petrov’s result is also uniform in $q$ and is therefore stronger than Bahadur-Rao’s theorem even with $l=0.$ The relation (1.1) with $l=0$ and its extension to $|l|\leqslant l_{n}\to 0$ have multiple implications in various domains of probability and statistics. The main goal of the present paper is to establish an equivalence similar to (1.1) for products of i.i.d.  random matrices.

Let $(g_{n})_{n\geqslant 1}$ be a sequence of i.i.d. $d\times d$ real random matrices defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with common law $\mu$. Denote by $\|\cdot\|$ the operator norm of a matrix and by $|\cdot|$ the Euclidean norm in $\mathbb{R}^{d}$. Set for brevity $G_{n}:=g_{n}\ldots g_{1},$ $n\geqslant 1$. The study of asymptotic behavior of the product $G_{n}$ attracted much attention, since the fundamental work of Furstenberg and Kesten [15], where the strong law of large numbers for $\log\|G_{n}\|$ has been established. Under additional assumptions, Furstenberg [14] extended it to $\log|G_{n}x|$, for any starting point $x$ on the unit sphere $\mathbb{S}^{d-1}=\{x\in\mathbb{R}^{d}:|x|=1\}.$ A number of noteworthy results in this area can be found in Kesten [28], Kingman [29], Le Page [30], Guivarc’h and Raugi [22], Bougerol and Lacroix [5], Goldsheid and Guivarc’h [17], Hennion [24], Furman [13], Hennion and Hervé [26], Guivarc’h [20], Guivarc’h and Le Page [21], Benoist and Quint [2, 3] to name only a few.

In this paper we are interested in asymptotic behaviour of large deviation probabilities for $\log|G_{n}x|$ where $x\in\mathbb{S}^{d-1}$. Set $I_{\mu}=\{s\geqslant 0:\mathbb{E}(\|g_{1}\|^{s})<+\infty\}.$ For $s\in I_{\mu}$, let $\kappa(s)=\lim_{n\to\infty}\left(\mathbb{E}\|G_{n}\|^{s}\right)^{\frac{1}{n}}.$ Define the convex function $\Lambda(s)=\log\kappa(s)$, $s\in I_{\mu}$, and consider its Fenchel-Legendre transform $\Lambda^{\ast}(q)=\sup_{s\in I_{\mu}}\{sq-\Lambda(s)\},$ $q\in\Lambda^{\prime}(I_{\mu}).$ Our first objective is to establish the following Bahadur-Rao type precise large deviation asymptotic:

[TABLE]

where $\sigma_{s}>0,$ $\bar{r}_{s}=\frac{r_{s}}{\nu_{s}(r_{s})}>0,$ $r_{s}$ and $\nu_{s}$ are, respectively, the unique up to a constant eigenfunction and unique probability eigenmeasure of the transfer operator $P_{s}$ corresponding to the eigenvalue $\kappa(s)$ (see Section 2.2 for precise statements). In fact, to enlarge the area of applications in (1.2) it is useful to add a vanishing perturbation for $q$. In this line we obtain the following Petrov type large deviation expansion: under appropriate conditions, uniformly in $|l|\leqslant l_{n}\to 0$ as $n\to\infty,$

[TABLE]

As an consequence of (1.3) we are able to infer new results, such as large deviation principles for $\log\|G_{n}\|$, see Theorem 2.5. From (1.3) we also deduce a local large deviation asymptotic: there exists a sequence $\Delta_{n}>0$ converging to [math] such that, uniformly in $\Delta\in[\Delta_{n},o(n)]$,

[TABLE]

Our results are established for both invertible matrices and positive matrices. For invertible matrices, Le Page [30] has obtained (1.2) for $s>0$ small enough under more restrictive conditions, such as the existence of exponential moments of $\|g_{1}\|$ and $\|g_{1}^{-1}\|$. The asymptotic (1.2) clearly implies a large deviation result due to Buraczewski and Mentemeier [8] which holds for invertible matrices and positive matrices: for $q=\Lambda^{\prime}(s)$ and $s\in I_{\mu}^{\circ}$, there exist two constants $0<c_{s}<C_{s}<+\infty$ such that

[TABLE]

Consider the Markov chain $X_{n}^{x}:=G_{n}x/|G_{n}x|$. Our second objective is to give precise large deviations for the couple $(X_{n}^{x},\log|G_{n}x|)$ with target functions. We prove that for any Hölder continuous target function $\varphi$ on $X_{n}^{x}$, and any target function $\psi$ on $\log|G_{n}x|$ such that $y\mapsto e^{-sy}\psi(y)$ is directly Riemann integrable, it holds that

[TABLE]

As a special case of (1.1) with $l=0$ and $\psi$ compactly supported we obtain Theorem 3.3 of Guivarc’h [20]. With $l=0$, $\psi$ the indicator function of the interval $[0,\infty)$ and $\varphi=r_{s}$, we get the main result in [8].

Our third objective is to establish asymptotics for lower large deviation probabilities: we prove that for $q=\Lambda^{\prime}(s)$ with $s<0$ sufficiently close to [math], it holds, uniformly in $|l|\leqslant l_{n}$,

[TABLE]

This sharpens the large deviation principle established in [5, Theorem 6.1] for invertible matrices. Moreover, we extend the large deviation asymptotic (1.7) to the couple $(X_{n}^{x},\log|G_{n}x|)$ with target functions.

1.2. Proof outline

Our proof is different from the standard approach of Dembo and Zeitouni [11] based on the Edgeworth expansion, which has been employed for instance in [8]. In contrast to [8], we start with the identity

[TABLE]

where $\mathbb{Q}_{s}^{x}$ is the change of measure defined in Section 3 for the norm cocycle $\log|G_{n}x|$, $\psi_{s}(y)=e^{-sy}\mathbbm{1}_{\{y\geqslant 0\}}$ and $h_{s}(l)=\Lambda^{*}(q+l)-\Lambda^{*}(q)-sl$. Usually the expectation in the right-hand side of (1.2) is handled via the Edgeworth expansion for the distribution function {\mathbb{Q}_{s}^{x}}\big{(}\frac{\log|G_{n}x|-nq}{\sqrt{n}\sigma_{s}}\leqslant t\big{)}; however, the presence of the multiplier $r_{s}(X^{x}_{n})^{-1}$ makes this impossible. Our idea is to replace the function $\psi_{s}$ with some upper and lower smoothed bounds using a technique from Grama, Lauvergnat and Le Page [18]. For simplicity we deal only with the upper bound $\psi_{s}\leqslant{\psi}_{s,\varepsilon}^{+}*{\rho}_{\varepsilon^{2}}$, where $\psi^{+}_{s,\varepsilon}(y)=\sup_{y^{\prime}:|y^{\prime}-y|\leqslant\varepsilon}\psi_{s}(y^{\prime})$, for some $\varepsilon>0$, and $\rho_{\varepsilon^{2}}$ is a density function on the real line satisfying the following properties: the Fourier transform $\widehat{\rho}_{\varepsilon^{2}}$ is supported on $[-\varepsilon^{-2},\varepsilon^{-2}]$, has a continuous extension in the complex plane and is analytic in the domain $\{z\in\mathbb{C}:|z|<\varepsilon^{-2},\Im z\neq 0\}$, see Lemma 4.2. Let $R_{s,it}$ be the perturbed operator defined by $R_{s,it}(\varphi)(x)=\mathbb{E}_{\mathbb{Q}_{s}^{x}}[\varphi(X_{1})e^{it(\log|g_{1}x|-q)}],$ for any Hölder continuous function $\varphi$ on the unit sphere $\mathbb{S}^{d-1}.$ Using the inversion formula we obtain the following upper bound:

[TABLE]

where $R^{n}_{s,it}$ is the $n$-th iteration of $R_{s,it}$. The integral in the right-hand side of (1.2) is decomposed into two parts:

[TABLE]

Since $\widehat{\rho}_{\varepsilon^{2}}$ is compactly supported on $\mathbb{R}$ and $\mu$ is non-arithmetic, the second integral in (1.10) decays exponentially fast to [math]. To deal with the first integral in (1.10), we make use of spectral gap decomposition for the perturbed operator $R_{s,it}$: $R^{n}_{s,it}=\lambda^{n}_{s,it}\Pi_{s,it}+N^{n}_{s,it}.$ Taking into account the fact that the remainder term $N^{n}_{s,it}$ decays exponentially fast to [math], the main difficulty is to investigate the integral:

[TABLE]

To find the exact asymptotic of this integral, we can apply the saddle point method (see Fedoryuk [12]). This is possible, since by the analyticity of the functions $\widehat{\psi}_{s,\varepsilon}^{+}$ and $\widehat{\rho}_{\varepsilon^{2}}$, one can apply Cauchy’s integral theorem to change the integration path so that it passes through the saddle point $z_{0}=z_{0}(l)$, which is the unique solution of the saddle point equation $\log\lambda_{s,z}=zl$.

The lower bound of the integral in (1.2) is a little more delicate, but can be treated in a similar way. The passage to the targeted version is done by using approximation techniques.

We end this section by fixing some notation, which will be used throughout the paper. We denote by $c$, $C$, eventually supplied with indices, absolute constants whose values may change from line to line. By $c_{\alpha}$, $C_{\alpha}$ we mean constants depending only on the index $\alpha.$ The interior of a set $A$ is denoted by $A^{\circ}$. Let $\mathbb{N}=\{1,2,\ldots\}$. For any integrable function $\psi:\mathbb{R}\to\mathbb{C}$, define its Fourier transform by $\widehat{\psi}(t)=\int_{\mathbb{R}}e^{-ity}\psi(y)dy$, $t\in\mathbb{R}$. For a matrix $g$, its transpose is denoted by $g^{\mathrm{T}}.$ For a measure $\nu$ and a function $\varphi$ we write $\nu(\varphi)=\int\varphi d\nu.$

2. Main results

2.1. Notation and conditions

The space $\mathbb{R}^{d}$ is equipped with the standard scalar product $\langle\cdot,\cdot\rangle$ and the Euclidean norm $|\cdot|$. For $d\geqslant 1$, let $M(d,\mathbb{R})$ be the set of $d\times d$ matrices with entries in $\mathbb{R}$ equipped with the operator norm $\|g\|=\sup_{x\in\mathbb{S}^{d-1}}|gx|$, for $g\in M(d,\mathbb{R})$, where $\mathbb{S}^{d-1}=\{x\in\mathbb{R}^{d},|x|=1\}$ is the unit sphere.

We shall work with products of invertible or positive matrices (all over the paper we use the term positive in the wide sense, i.e. each entry is non-negative). Denote by $\mathscr{G}=GL(d,\mathbb{R})$ the general linear group of invertible matrices of $M(d,\mathbb{R}).$ A positive matrix $g\in M(d,\mathbb{R})$ is said to be allowable, if every row and every column of $g$ has a strictly positive entry. Denote by $\mathscr{G}_{+}$ the multiplicative semigroup of allowable positive matrices of $M(d,\mathbb{R})$. We write $\mathscr{G}_{+}^{\circ}$ for the subsemigroup of $\mathscr{G}_{+}$ with strictly positive entries.

Denote by $\mathbb{S}^{d-1}_{+}=\{x\geqslant 0:|x|=1\}$ the intersection of the unit sphere with the positive quadrant. To unify the exposition, we use the symbol $\mathcal{S}$ to denote $\mathbb{S}^{d-1}$ in the case of invertible matrices, and $\mathbb{S}^{d-1}_{+}$ in the case of positive matrices. The space $\mathcal{S}$ is equipped with the metric $\mathbf{d}$ which we proceed to introduce. For invertible matrices, the distance $\mathbf{d}$ is defined as the angular distance (see [21]), i.e., for any $x,y\in\mathbb{S}^{d-1}$, $\mathbf{d}(x,y)=|\sin\theta(x,y)|$, where $\theta(x,y)$ is the angle between $x$ and $y$. For positive matrices, the distance $\mathbf{d}$ is the Hilbert cross-ratio metric (see [24]) defined by $\mathbf{d}(x,y)=\frac{1-m(x,y)m(y,x)}{1+m(x,y)m(y,x)}$, where $m(x,y)=\sup\{\lambda>0:\ \lambda y_{i}\leqslant x_{i},\ \forall i=1,\ldots,d\}$, for any two vectors $x=(x_{1},\ldots,x_{d})$ and $y=(y_{1},\ldots,y_{d})$ in $\mathbb{S}_{+}^{d-1}$.

Let $\mathcal{C}(\mathcal{S})$ be the space of continuous functions on $\mathcal{S}$. We write $\mathbf{1}$ for the identity function $1(x)$, $x\in\mathcal{S}$. Throughout this paper, let $\gamma>0$ be a fixed small constant. For any $\varphi\in\mathcal{C(S)}$, set

[TABLE]

and introduce the Banach space $\mathcal{B}_{\gamma}:=\{\varphi\in\mathcal{C(S)}:\|\varphi\|_{\gamma}<+\infty\}.$

For $g\in M(d,\mathbb{R})$ and $x\in\mathcal{S}$, write $g\cdot x=\frac{gx}{|gx|}$ for the projective action of $g$ on $\mathcal{S}$. For any $g\in M(d,\mathbb{R})$, set $\iota(g):=\inf_{x\in\mathcal{S}}|gx|.$ For both invertible matrices and allowable positive matrices, it holds that $\iota(g)>0.$ Note that for any invertible matrix $g$, we have $\iota(g)=\|g^{-1}\|^{-1}$.

Let $(g_{n})_{n\geqslant 1}$ be a sequence of i.i.d.  random matrices of the same probability law $\mu$ on $M(d,\mathbb{R})$. Set $G_{n}=g_{n}\ldots g_{1},$ for $n\geqslant 1.$ Our goal is to establish, under suitable conditions, a large deviation equivalence similar to (1.1) for the norm cocycle $\log|G_{n}x|$ for invertible matrices and positive matrices. In both cases, we denote by $\Gamma_{\mu}:=[\operatorname{supp}\mu]$ the smallest closed semigroup of $M(d,\mathbb{R})$ generated by $\operatorname{supp}\mu$ (the support of $\mu$), that is, $\Gamma_{\mu}=\overline{\cup_{n=1}^{\infty}\{\operatorname{supp}\mu\}^{n}}$.

Set

[TABLE]

Applying Hölder’s inequality to $\mathbb{E}(\|g_{1}\|^{s})$, it is easily seen that $I_{\mu}$ is an interval. We make use of the following exponential moment condition:

A1**.**

There exist $s\in I_{\mu}^{\circ}$ and $\alpha\in(0,1)$ such that $\mathbb{E}\|g_{1}\|^{s+\alpha}\iota(g_{1})^{-\alpha}<+\infty.$

For invertible matrices, we introduce the following strong irreducibility and proximality conditions, where we recall that a matrix $g$ is said to be proximal if it has an algebraic simple dominant eigenvalue.

A2**.**

(i)(Strong irreducibility)* No finite union of proper subspaces of $\mathbb{R}^{d}$ is $\Gamma_{\mu}$-invariant.*

(ii)(Proximality)* $\Gamma_{\mu}$ contains at least one proximal matrix. *

The conditions of strong irreducibility and proximality are always satisfied for $d=1$. If $g$ is proximal, denote by $\lambda_{g}$ its dominant eigenvalue and by $v_{g}$ the associated normalized eigenvector ($|v_{g}|=1$). In fact, $g$ is proximal iff the space $\mathbb{R}^{d}$ can be decomposed as $\mathbb{R}^{d}=\mathbb{R}\lambda_{g}\oplus V^{\prime}$ such that $gV^{\prime}\subset V^{\prime}$ and the spectral radius of $g$ on the invariant subspace $V^{\prime}$ is strictly less than $|\lambda_{g}|$. For invertible matrices, condition 2 implies that the Markov chain $X_{n}^{x}$ has a unique $\mu$-stationary measure, which is supported on

[TABLE]

For positive matrices, introduce the following condition:

A3**.**

(i) (Allowability)* Every $g\in\Gamma_{\mu}$ is allowable.*

(ii) (Positivity)* $\Gamma_{\mu}$ contains at least one matrix belonging to $\mathscr{G}_{+}^{\circ}$.*

It can be shown (see [7, Lemma 4.3]) that for positive matrices, condition 3 ensures the existence and uniqueness of the invariant measure for the Markov chain $X_{n}^{x}$ supported on

[TABLE]

In addition, $V(\Gamma_{\mu})$ is the unique minimal $\Gamma_{\mu}$-invariant subset (see [7, Lemma 4.2]). According to the Perron-Frobenius theorem, a strictly positive matrix always has a unique dominant eigenvalue, so condition 3(ii) implies condition 2(ii) for $d>1$.

For any $s\in I_{\mu}$, for invertible matrices and for positive matrices, the following limit exists (see [21] and [8]):

[TABLE]

The function $\Lambda=\log\kappa:I_{\mu}\to\mathbb{R}$ is convex and analytic on $I_{\mu}^{\circ}$ (it plays the same role as the $\log$-Laplace transform of $X_{1}$ in the real i.i.d. case). Introduce the Fenchel-Legendre transform of $\Lambda$ by $\Lambda^{\ast}(q)=\sup_{s\in I_{\mu}}\{sq-\Lambda(s)\},$ $q\in\Lambda^{\prime}(I_{\mu}).$ We have that $\Lambda^{*}(q)=sq-\Lambda(s)$ if $q=\Lambda^{\prime}(s)$ for some $s\in I_{\mu}$, which implies $\Lambda^{*}(q)\geqslant 0$ on $\Lambda^{\prime}(I_{\mu})$ since $\Lambda(0)=0$ and $\Lambda(s)$ is convex on $I_{\mu}$.

We say that the measure $\mu$ is arithmetic, if there exist $t>0$, $\beta\in[0,2\pi)$ and a function $\vartheta:\mathcal{S}\to\mathbb{R}$ such that for any $g\in\Gamma_{\mu}$ and any $x\in V(\Gamma_{\mu})$, we have $\exp[it\log|gx|-i\beta+i\vartheta(g\!\cdot\!x)-i\vartheta(x)]=1.$ For positive matrices, we need the following condition:

A4**.**

(Non-arithmeticity)* The measure $\mu$ is non-arithmetic.*

A simple sufficient condition established in [28] for the measure $\mu$ to be non-arithmetic is that the additive subgroup of $\mathbb{R}$ generated by the set $\{\log\lambda_{g}:g\in\Gamma_{\mu},\ g\in\mathscr{G}_{+}^{\circ}\}$ is dense in $\mathbb{R}$ (see [8, Lemma 2.7]).

Note that for positive matrices, condition 4 is used to ensure that $\sigma_{s}^{2}=\Lambda^{\prime\prime}(s)>0$. For invertible matrices, condition 2 implies the non-arithmeticity of the measure $\mu$, hence, $\sigma_{s}$ is also strictly positive (for a proof see Guivarc’h and Urban [23, Proposition 4.6]).

For any $s\in I_{\mu}$, the transfer operator $P_{s}$ and the conjugate transfer operator $P_{s}^{*}$ are defined, for any $\varphi\in\mathcal{C(S)}$ and $x\in\mathcal{S}$, by

[TABLE]

which are bounded linear on $\mathcal{C(S)}$. Under condition 2 for invertible matrices, or condition 3 for positive matrices, the operator $P_{s}$ has a unique probability eigenmeasure $\nu_{s}$ on $\mathcal{S}$ corresponding to the eigenvalue $\kappa(s)$: $P_{s}\nu_{s}=\kappa(s)\nu_{s}.$ Similarly, the operator $P_{s}^{*}$ has a unique probability eigenmeasure $\nu^{*}_{s}$ corresponding to the eigenvalue $\kappa(s)$: $P_{s}^{*}\nu^{*}_{s}=\kappa(s)\nu^{*}_{s}.$ Set, for $x\in\mathcal{S}$,

[TABLE]

Then, $r_{s}$ is the unique, up to a scaling constant, strictly positive eigenfunction of $P_{s}$: $P_{s}r_{s}=\kappa(s)r_{s}$; similarly $r^{*}_{s}$ is the unique, up to a scaling constant, strictly positive eigenfunction of $P_{s}^{*}$: $P_{s}^{*}r^{*}_{s}=\kappa(s)r^{*}_{s}$. We refer for details to Section 3.

Below we shall also make use of normalized eigenfunction $\bar{r}_{s}$ defined by $\bar{r}_{s}(x)=\frac{r_{s}(x)}{\nu_{s}(r_{s})}$, $x\in\mathcal{S}$, which is strictly positive and Hölder continuous on the projective space $\mathcal{S}$, see Proposition 3.1.

2.2. Large deviations for the norm cocycle

The following theorem gives the exact asymptotic behavior of the large deviation probabilities for the norm cocycle.

Theorem 2.1**.**

Assume that $\mu$ satisfies either conditions 1, 2 for invertible matrices, or conditions 1, 3, 4 for positive matrices. Let $q=\Lambda^{\prime}(s)$, where $s\in I_{\mu}^{\circ}$. Then for any positive sequence $(l_{n})_{n\geqslant 1}$ satisfying $\lim_{n\to\infty}l_{n}=0$, we have, as $n\to\infty$, uniformly in $x\in\mathcal{S}$ and $|l|\leqslant l_{n}$,

[TABLE]

In particular, with $l=0$, as $n\to\infty$, uniformly in $x\in\mathcal{S}$,

[TABLE]

The rate function $\Lambda^{*}(q+l)$ admits the following expansion: for $q=\Lambda^{\prime}(s)$ and $l$ in a small neighborhood of [math], we have

[TABLE]

where $\zeta_{s}(t)$ is the Cramér series, $\zeta_{s}(t)=\sum_{k=3}^{\infty}c_{s,k}t^{k-3}=\frac{\Lambda^{\prime\prime\prime}(s)}{6\sigma_{s}^{3}}+O(t),$ with $\Lambda^{\prime\prime\prime}(s)$ and $\sigma_{s}$ defined in Proposition 3.3. We refer for details to Lemma 4.1, where the coefficients $c_{s,k}$ are given in terms of the cumulant generating function $\Lambda=\log\kappa$.

For invertible matrices, a point-wise version of (2.3), without $\sup_{x\in\mathcal{S}}$ and with $l=0$, namely the asymptotic (1.2), has been first established by Le Page [30, Theorem 8] for small enough $s>0$ under a stronger exponential moment condition. For positive matrices, the asymptotic (2.3) is new and implies the large deviation bounds (1.5) established in Buraczewski and Mentemeier [8, Corollary 3.2]. We note that there is a misprint in [8], where $e^{nsq}$ should be replaced by $e^{\Lambda^{*}(q)}$.

Now we consider the precise large deviations for the couple $(X_{n}^{x},\log|G_{n}x|)$ with target functions $\varphi$ and $\psi$ on $X_{n}^{x}:=G_{n}\!\cdot\!x$ and $\log|G_{n}x|$, respectively.

Theorem 2.2**.**

Assume the conditions of Theorem 2.1 and let $q=\Lambda^{\prime}(s)$ for $s\in I_{\mu}^{\circ}$. Then, for any $\varphi\in\mathcal{B}_{\gamma}$, any measurable function $\psi$ on $\mathbb{R}$ such that $y\mapsto e^{-sy}\psi(y)$ is directly Riemann integrable, and any positive sequence $(l_{n})_{n\geqslant 1}$ satisfying $\lim_{n\to\infty}l_{n}=0$, we have, as $n\to\infty$, uniformly in $x\in\mathcal{S}$ and $|l|\leqslant l_{n}$,

[TABLE]

With $\varphi=\mathbf{1}$ and $\psi(y)=\mathbbm{1}_{\{y\geqslant 0\}}$ for $y\in\mathbb{R},$ we obtain Theorem 2.1. For invertible matrices and with $l=0$, Theorem 2.2 strengthens the point-wise large deviation result stated in Theorem 3.3 of Guivarc’h [20], since we do not assume the function $\psi$ to be compactly supported and our result is uniform in $x\in\mathcal{S}$. By the way we would like to remark that in Theorem 3.3 of [20] $\kappa^{n}(s)$ should be replaced by $\kappa^{-n}(s)$, and $\nu_{s}(\varphi r_{s}^{-1})$ should be replaced by $\frac{\nu_{s}(\varphi)}{\nu_{s}(r_{s})}$. For positive matrices, Theorem 2.2 is new. Since $r_{s}$ is a strictly positive and Hölder continuous function on $\mathcal{S}$ (see Proposition 3.1), taking $\varphi=r_{s}$ and $\psi(y)=\mathbbm{1}_{\{y\geqslant 0\}}$, $y\in\mathbb{R}$ in Theorem 2.2, we get the main result of [8] (Theorem 3.1).

Unlike the case of i.i.d. real-valued random variables, Theorems 2.1 and 2.2 do not imply the similar asymptotic for lower large deviation probabilities $\mathbb{P}(\log|G_{n}x|\leqslant n(q+l))$, where $q<\Lambda^{\prime}(0)$. To formulate our results, we need an exponential moment condition, as in Le Page [30]. For $g\in\Gamma_{\mu}$, set $N(g)=\max\{\|g\|,\iota(g)^{-1}\}$, which reduces to $N(g)=\max\{\|g\|,\|g^{-1}\|\}$ for invertible matrices.

A5**.**

There exists a constant $\eta\in(0,1)$ such that $\mathbb{E}[N(g_{1})^{\eta}]<+\infty$.

Under condition 5, the functions $s\mapsto\kappa(s)$ and $s\mapsto\Lambda(s)=\log\kappa(s)$ can be extended analytically in a small neighborhood of [math] of the complex plane; in this case the expansion (2.4) still holds and we have $\sigma_{s}=\Lambda^{\prime\prime}(s)>0$ for $s<0$ small enough. We also need to extend the function $\overline{r}_{s}$ for small $s<0$, which is positive and Hölder continuous on the projective space $\mathcal{S}$, as in the case of $s>0$: we refer to Proposition 3.2 for details.

Theorem 2.3**.**

Assume that $\mu$ satisfies either conditions 2, 5 for invertible matrices or conditions 3, 4, 5 for positive matrices. Then, there exists $\eta_{0}<\eta$ such that for any $s\in(-\eta_{0},0)$ and $q=\Lambda^{\prime}(s)$, for any positive sequence $(l_{n})_{n\geqslant 1}$ satisfying $\lim_{n\to\infty}l_{n}=0$, we have, as $n\to\infty$, uniformly in $x\in\mathcal{S}$ and $|l|\leqslant l_{n}$,

[TABLE]

In particular, with $l=0$, as $n\to\infty$, uniformly in $x\in\mathcal{S}$,

[TABLE]

For invertible matrices, this result sharpens the large deviation principle established in [5]. For positive matrices, our result is new, even for the large deviation principle.

More generally, we also have the precise large deviations result for the couple $(X_{n}^{x},\log|G_{n}x|)$ with target functions.

Theorem 2.4**.**

Assume the conditions of Theorem 2.3. Then, there exists $\eta_{0}<\eta$ such that for any $s\in(-\eta_{0},0)$ and $q=\Lambda^{\prime}(s)$, for any $\varphi\in\mathcal{B}_{\gamma}$, any measurable function $\psi$ on $\mathbb{R}$ such that $y\mapsto e^{-sy}\psi(y)$ is directly Riemann integrable, and any positive sequence $(l_{n})_{n\geqslant 1}$ satisfying $\lim_{n\to\infty}l_{n}=0$, we have, as $n\to\infty$, uniformly in $x\in\mathcal{S}$ and $|l|\leqslant l_{n}$,

[TABLE]

With $\varphi=\mathbf{1}$ and $\psi(y)=\mathbbm{1}_{\{y\leqslant 0\}}$ for $y\in\mathbb{R},$ we obtain Theorem 2.3.

2.3. Applications to large deviation principle for the matrix norm

We use Theorems 2.1 and 2.3 to deduce large deviation principles for the matrix norm $\|G_{n}\|$. Our first result concerns the upper tail and the second one deals with lower tail.

Theorem 2.5**.**

Assume the conditions of Theorem 2.1. Let $q=\Lambda^{\prime}(s)$, where $s\in I_{\mu}^{\circ}$. Then, for any positive sequence $(l_{n})_{n\geqslant 1}$ with $l_{n}\to 0$ as $n\to\infty$, we have, uniformly in $|l|\leqslant l_{n}$,

[TABLE]

For invertible matrices, with $l=0$, Theorem 2.5 improves the large deviation bounds in Benoist and Quint [3, Theorem 14.19], where the authors consider general groups, but without giving the rate function. For positive matrices, the result is new for $l=0$ and $l=O(l_{n})$.

Theorem 2.6**.**

Assume the conditions of Theorem 2.3. Then, there exists $\eta_{0}<\eta$ such that for any $s\in(-\eta_{0},0)$ and $q=\Lambda^{\prime}(s)$, for any positive sequence $(l_{n})_{n\geqslant 1}$ with $l_{n}\to 0$ as $n\to\infty$, we have, uniformly in $|l|\leqslant l_{n}$,

[TABLE]

This result is new for both invertible matrices and positive matrices.

2.4. Local limit theorems with large deviations

Local limit theorems and large and moderate deviations for sums of i.i.d. random variables have been studied by Gnedenko [16], Sheep [34], Stone [35], Breuillard [6], Borovkov and Borovkov [4]. Moderate deviation results in the local limit theorem for products of invertible random matrices have been obtained in [3, Theorems 17.9 and 17.10].

Taking $\varphi=\mathbf{1}$ and $\psi=\mathbbm{1}_{[a,a+\Delta]},$ where $a\in\mathbb{R}$ and $\Delta>0$ do not depend on $n$, it is easy to understand that Theorem 2.2 becomes, in fact, a statement on large deviations in the local limit theorem. It turns out that with the Petrov type extension (2.2) we can derive the following more general statement where $\Delta$ can increase with $n.$

Theorem 2.7**.**

Assume conditions of Theorem 2.1 and let $q=\Lambda^{\prime}(s)$. Then there exists a sequence $\Delta_{n}>0$ converging to [math] as $n\to\infty$ such that, for any $\varphi\in\mathcal{B}_{\gamma}$, for any positive sequence $(l_{n})_{n\geqslant 1}$ with $l_{n}\to 0$ as $n\to\infty$ and any fixed $a\in\mathbb{R}$, we have, as $n\to\infty,$ uniformly in $\Delta\in[\Delta_{n},o(n)]$, $x\in\mathcal{S}$ and $|l|\leqslant l_{n}$,

[TABLE]

Taking $\varphi=\mathbf{1}$, as $n\to\infty,$ uniformly in $\Delta\in[\Delta_{n},o(n)]$, $x\in\mathcal{S}$ and $|l|\leqslant l_{n}$,

[TABLE]

We can compare this result with Theorem 3.3 in [20], from which the above equivalence can be deduced for $l=0$ and $\Delta$ fixed.

It is easy to see that, under additional assumption 5, the assertion of Theorem 2.7 remains true for $s<0$ small enough. This can be deduced from Theorem 2.4: the details are left to the reader.

3. Spectral gap theory for the norm

3.1. Properties of the transfer operator

Recall that the transfer operator $P_{s}$ and the conjugate operator $P_{s}^{*}$ are defined by (2.1). Below $P_{s}\nu_{s}$ stands for the measure on $\mathcal{S}$ such that $P_{s}\nu_{s}(\varphi)=\nu_{s}(P_{s}\varphi),$ for continuous functions $\varphi$ on $\mathcal{S}$, and $P^{*}_{s}\nu^{*}_{s}$ is defined similarly. The following result was proved in [7, 8] for positive matrices, and in [21] for invertible matrices.

Proposition 3.1**.**

Assume that $\mu$ satisfies either conditions 1, 2 for invertible matrices, or conditions 1, 3 for positive matrices. Let $s\in I_{\mu}$. Then the spectral radii $\varrho(P_{s})$ and $\varrho(P_{s}^{*})$ are both equal to $\kappa(s)$, and there exist a unique, up to a scaling constant, strictly positive Hölder continuous function $r_{s}$ and a unique probability measure $\nu_{s}$ on $\mathcal{S}$ such that

[TABLE]

Similarly, there exist a unique strictly positive Hölder continuous function $r_{s}^{\ast}$ and a unique probability measure $\nu_{s}^{*}$ on $\mathcal{S}$ such that

[TABLE]

Moreover, the functions $r_{s}$ and $r_{s}^{*}$ are given by

[TABLE]

It is easy to see that the family of kernels $q_{n}^{s}(x,g)=\frac{|gx|^{s}}{\kappa^{n}(s)}\frac{r_{s}(g\cdot x)}{r_{s}(x)},$ $n\geqslant 1$ satisfies the following cocycle property:

[TABLE]

The equation $P_{s}r_{s}=\kappa(s)r_{s}$ implies that, for any $x\in\mathcal{S}$ and $s\in I_{\mu}$, the probability measures $\mathbb{Q}_{s,n}^{x}(dg_{1},\ldots,dg_{n})=q_{n}^{s}(x,g_{n}\hbox to10.00002pt{.\hss.\hss.}g_{1})\mu(dg_{1})\hbox to10.00002pt{.\hss.\hss.}\mu(dg_{n}),$ $n\geqslant 1,$ form a projective system on $M(d,\mathbb{R})^{\mathbb{N}}$. By the Kolmogorov extension theorem, there is a unique probability measure $\mathbb{Q}_{s}^{x}$ on $M(d,\mathbb{R})^{\mathbb{N}}$, with marginals $\mathbb{Q}_{s,n}^{x}$; denote by $\mathbb{E}_{\mathbb{Q}_{s}^{x}}$ the corresponding expectation.

If $(g_{n})_{n\in\mathbb{N}}$ denotes the coordinate process on the space of trajectories $M(d,\mathbb{R})^{\mathbb{N}}$, then the sequence $(g_{n})_{n\geqslant 1}$ is i.i.d. with the common law $\mu$ under $\mathbb{Q}_{0}^{x}.$ However, for any $s\in I_{\mu}^{\circ}$ and $x\in\mathcal{S}$, the sequence $(g_{n})_{n\geqslant 1}$ is Markov-dependent under the measure $\mathbb{Q}_{s}^{x}$. Let

[TABLE]

By the definition of $\mathbb{Q}_{s}^{x}$, for any bounded measurable function $f$ on $(\mathcal{S}\times\mathbb{R})^{n}$, it holds that

[TABLE]

Under the measure $\mathbb{Q}_{s}^{x}$, the process $(X_{n}^{x})_{n\in\mathbb{N}}$ is a Markov chain with the transition operator given by

[TABLE]

It has been proved in [7] for positive matrices, and in [21] for invertible matrices, that $Q_{s}$ has a unique invariant probability measure $\pi_{s}$ supported on $V(\Gamma_{\mu})$ and that, for any $\varphi\in\mathcal{C(S)}$,

[TABLE]

Moreover, letting $\mathbb{Q}_{s}=\int\mathbb{Q}_{s}^{x}\pi_{s}(dx),$ from the results of [7, 21], it follows that, under the assumptions of Theorem 2.1, for any $s\in I_{\mu}$, we have $\lim_{n\to\infty}\frac{\log|G_{n}x|}{n}=\Lambda^{\prime}(s),$ $\mathbb{Q}_{s}$-a.s. and $\mathbb{Q}_{s}^{x}$-a.s., where $\Lambda^{\prime}(s)=\frac{\kappa^{\prime}(s)}{\kappa(s)}$.

When $s\in(-\eta_{0},0)$ for small enough $\eta_{0}>0$, define the transfer operator $P_{s}$ as follows: for any $\varphi\in\mathcal{C(S)}$,

[TABLE]

which is well-defined under condition 5. The following proposition is proved in [36].

Proposition 3.2**.**

Assume that $\mu$ satisfies either conditions 2, 5 for invertible matrices, or conditions 3, 5 for positive matrices. Then there exists $\eta_{0}<\eta$ such that for any $s\in(-\eta_{0},0)$, the spectral radius $\varrho(P_{s})$ of the operator $P_{s}$ is equal to $\kappa(s)$. Moreover there exist a unique, up to a scaling constant, strictly positive Hölder continuous function $r_{s}$ and a unique probability measure $\nu_{s}$ on $\mathcal{S}$ such that

[TABLE]

Based on Proposition 3.2, in the same way as for $s>0$, one can define the measure $\mathbb{Q}_{s}^{x}$ for negative values $s<0$ sufficiently close to [math], and one can extend the change of measure formula (3.1) to $s<0$. Under the measure $\mathbb{Q}_{s}^{x}$, the process $(X_{n}^{x})_{n\in\mathbb{N}}$ is a Markov chain with the transition operator $Q_{s}$ and the assertion (3.3) holds true. We refer to [36] for details.

3.2. Spectral gap of the perturbed operator

Recall that the Banach space $B_{\gamma}$ consists of all $\gamma$-Hölder continuous function on $\mathcal{S}$, where $\gamma>0$ is a fixed small constant. Denote by $\mathcal{L(B_{\gamma},B_{\gamma})}$ the set of all bounded linear operators from $\mathcal{B}_{\gamma}$ to $\mathcal{B}_{\gamma}$ equipped with the operator norm $\left\|\cdot\right\|_{\mathcal{B}_{\gamma}\to\mathcal{B}_{\gamma}}$. For $s\in I_{\mu}^{\circ}$ and $z\in\mathbb{C}$ with $s+\Re z\in I_{\mu}$, define a family of perturbed operators $R_{s,z}$ as follows: for any $\varphi\in\mathcal{B}_{\gamma}$,

[TABLE]

It follows from the cocycle property (3.1) that

[TABLE]

The following proposition collects useful assertions that we will use in the proofs of our results. Denote $B_{\delta}(0):=\{z\in\mathbb{C}:|z|\leqslant\delta\}$.

Proposition 3.3**.**

Assume that $\mu$ satisfies either conditions 1, 2 for invertible matrices, or conditions 1, 3 for positive matrices. Then, there exists $\delta>0$ such that for any $z\in B_{\delta}(0)$,

[TABLE]

Moreover, for any $s\in I_{\mu}^{\circ}$, the following assertions hold:

• (i)

$\Pi_{s,z}$* is a rank-one projection for $|z|\leqslant\delta$, with $\Pi_{s,0}(\varphi)(x)=\pi_{s}(\varphi)$ for any $\varphi\in\mathcal{B}_{\gamma}$ and $x\in\mathcal{S}$, $\Pi_{s,z}N_{s,z}=N_{s,z}\Pi_{s,z}=0$ and*

[TABLE]

For any fixed $k\geqslant 1$, there exist $\varkappa_{s}\in(0,1)$ and $c_{s}$ such that

[TABLE]

In addition, the mappings $z\mapsto\Pi_{s,z}:B_{\delta}(0)\to\mathcal{L(B_{\gamma},B_{\gamma})}$ and $z\mapsto N_{s,z}:B_{\delta}(0)\to\mathcal{L(B_{\gamma},B_{\gamma})}$ are analytic in the strong operator sense.

• (ii)

For any compact set $K\subseteq\mathbb{R}\backslash\{0\}$, there exists a constant $C_{K}>0$ such that for any $n\geqslant 1$ and $\varphi\in\mathcal{B}_{\gamma}$, we have

[TABLE]

• (iii)

*The mapping $z\mapsto\lambda_{s,z}:B_{\delta}(0)\to\mathbb{C}$ is analytic, and *

[TABLE]

where

[TABLE]

and

[TABLE]

In addition, if the measure $\mu$ is non-arithmetic, then the asymptotic variance $\sigma_{s}^{2}$ is strictly positive.

The assertions (i), (ii), (iii) of Proposition 3.3, except (3.6), have been proved in [8] for imaginary-valued $z\in(-i\delta,i\delta)$, based on the perturbation theory (see [25]). The assertions (i), (iii) can be extended to the complex-valued $z\in B_{\delta}(0)$ without changes in the proof in [8].

The identity (3.6) is not proved in [8], but can be obtained by using the arguments from [36]. By the perturbation theory, the operator $P_{s}$ and its spectral radius $\kappa(s)$ can be extended to $P_{s+z}$ and the eigenvalue $\kappa(s+z)$, respectively, with $z$ in the small neighborhood of [math], see [21]. By the definitions of $R_{s,z}$ and $P_{z}$ using the change of measure (3.1), we obtain for any $\varphi\in\mathcal{B}_{\gamma}$, $n\geqslant 1$, $s\in I_{\mu}^{\circ}$ and $z\in B_{\delta}(0)$,

[TABLE]

Since $r_{s}$ is uniformly bounded, using (3.7) and the fact that $\kappa(s+z)$ is the unique eigenvalue of $P_{s+z}$, we deduce (3.6).

For negative values $s<0$ sufficiently close to [math], we can define the perturbed operator $R_{s,z}$ as in (3.4). The following spectral gap property of $R_{s,z}$ is established in [36].

Proposition 3.4**.**

Assume that $\mu$ satisfies conditions 2, 5 for invertible matrices, or conditions 3, 5 for positive matrices. Then, there exist $\eta_{0}<\eta$ and $\delta>0$ such that for any $s\in(-\eta_{0},0)$ and $z\in B_{\delta}(0)$,

[TABLE]

Moreover, for any $s\in(-\eta_{0},0)$, the assertions (i), (ii), (iii) of Proposition 3.3 hold true.

4. Proof of Theorems 2.1 and 2.3

4.1. Auxiliary results

We need some preliminary statements. Following Petrov [32], under the changed measure $\mathbb{Q}_{s}^{x}$, define the Cramér series $\zeta_{s}$ by

[TABLE]

where $\gamma_{s,k}=\Lambda^{(k)}(s)$ and $\Lambda(s)=\log\kappa(s)$. The following lemma gives a full expansion of $\Lambda^{*}(q+l)$ in terms of power series in $l$ in a neighborhood of [math], for $q=\Lambda^{\prime}(s)$ and $s\in I_{\mu}^{\circ}\cup(\eta_{0},0)$, where $\eta_{0}$ is from Proposition 3.4.

Lemma 4.1**.**

Assume conditions of Theorem 2.1 or Theorem 2.3. Let $q=\Lambda^{\prime}(s)$. Then, there exists $\delta>0$ such that, for any $|l|\leqslant\delta,$

[TABLE]

*where $h_{s}$ is linked to the Cramér series $\zeta_{s}$ by the identity *

[TABLE]

Proof.

Let $(\Lambda^{\prime})^{-1}$ be the inverse function of $\Lambda^{\prime}.$ With the notation $l_{s}=(\Lambda^{\prime})^{-1}(q+l)-s$, we have $\Lambda^{\prime}(s+l_{s})=q+l$. By the definition of $\Lambda^{*}$, it follows that $\Lambda^{*}(q+l)=(s+l_{s})(q+l)-\Lambda(s+l_{s})$. This, together with $\Lambda^{*}(q)=sq-\Lambda(s)$ and Taylor’s formula, gives

[TABLE]

From $\Lambda^{\prime}(s+l_{s})=q+l$ and $\Lambda^{\prime}(s)=q$, we deduce that $l=\Lambda^{\prime}(s+l_{s})-\Lambda^{\prime}(s)$, so that, by Taylor’s formula,

[TABLE]

The rest of the proof is similar to that in Petrov [32] (chapter VIII, section 2). For $|l|$ small enough, the equation (4.3) has a unique solution $l_{s}$ given by

[TABLE]

Together with (4.2) and (4.3), this implies

[TABLE]

∎

Let us fix a non-negative Schwartz function $\rho$ on $\mathbb{R}$ with $\int_{\mathbb{R}}\rho(y)dy=1$, whose Fourier transform $\widehat{\rho}$ is supported on $[-1,1]$ and has a continuous extension in the complex plane. Moreover, $\widehat{\rho}$ is analytic in the domain $D:=\{z\in\mathbb{C}:|z|<1,\Im z\neq 0\}$. Such a function can be constructed as follows. On the real line define $\widehat{\varsigma}(t)=e^{-\frac{1}{1-t^{2}}}$ if $t\in[-1,1]$, and $\widehat{\varsigma}=0$ elsewhere. The function $\widehat{\varsigma}$ is compactly supported and has finite derivatives of all orders. Its inverse Fourier transform $\varsigma$, however, is not non-negative. Let $\widehat{\rho}_{0}=\widehat{\varsigma}\ast\widehat{\varsigma}$ be the convolution of $\widehat{\varsigma}$ with itself. It is supported by $[-2,2]$ and its inverse Fourier transform $\rho_{0}$ satisfies $\rho_{0}=2\pi\varsigma^{2}\geqslant 0$. We show below that $\widehat{\rho}_{0}$ has a continuous extension in the complex plane, and $\widehat{\rho}_{0}$ is analytic in the domain $D$. Finally we rescale and renormalize $\rho_{0}$ by setting $\rho(y)=\rho_{0}(y/2)/[2\widehat{\rho}_{0}(0)]$ for $y\in\mathbb{R}$.

Lemma 4.2**.**

$\widehat{\rho}_{0}$* has a continuous extension in the complex plane, and $\widehat{\rho}_{0}$ is analytic in the domain $D$.*

Proof.

The function $\widehat{\varsigma}$ can be extended to the complex plane as follows:

[TABLE]

It is easily verified that $\widehat{\varsigma}$ is continuous in the interior of the unit disc and outside it, but is not continuous at any point on the unit circle $|z|=1$. Note also that $\widehat{\varsigma}$ is uniformly bounded on $\mathbb{C}$. Recall that the function $\widehat{\rho}_{0}=\widehat{\varsigma}\ast\widehat{\varsigma}$ is defined on the real line. We extend it to the complex plane by setting $\widehat{\rho}_{0}(z)=\int_{-1}^{1}\widehat{\varsigma}(t)\widehat{\varsigma}(z-t)\mathbbm{1}_{\{|z-t|<1\}}dt.$ The latter integral is well defined for any $z\in\mathbb{C}$, since $\widehat{\varsigma}$ is bounded. We are going to show that $\widehat{\varsigma}$ is continuous in $\mathbb{C}$. For any fixed $z\in\mathbb{C}$ and $h\in\mathbb{C}$ with $|h|$ small, we write

[TABLE]

The set $T_{z}=\{t:|z-t|=1\}$ of points of discontinuity of the function $t\mapsto\widehat{\varsigma}(z-t)$ consists of at most two points. For any $t\in[-1,1]$, $t\not\in T_{z}$, by the definition of $\widehat{\varsigma}$, we have that $|\widehat{\varsigma}(z-t+h)-\widehat{\varsigma}(z-t)|\to 0$ as $|h|\to 0$. Since the Lebesgue measure of $T_{z}$ is [math], applying the Lebesgue dominated convergence theorem and taking into account the boundedness of the function $\widehat{\varsigma}$ on $\mathbb{C}$, we see that $\widehat{\rho}_{0}$ is continuous in the complex plane.

We next show that $\widehat{\rho}_{0}$ is analytic in the domain $D=\{z^{\prime}\in\mathbb{C}:|z^{\prime}|<1,\Im z^{\prime}\neq 0\}$. Fix $z\in D$. Let $\varepsilon=\Im z/2\in(0,\frac{1}{2})$. Denote $D(\varepsilon)=:\{z^{\prime}\in D:|\Im z^{\prime}|>\varepsilon\}$. One can verify that the derivative $\widehat{\varsigma}^{\prime}(z)$ exists and is uniformly bounded by $\frac{c}{\varepsilon^{4}}$ on the domain $D(\varepsilon)$. For any $h\in\mathbb{C}$ with $|h|$ small enough, we have

[TABLE]

Since for any $t\in[-1,1]$ and $\theta\in[0,1]$, we have $|\Im(z-t+\theta h)|\geqslant\varepsilon$ uniformly in $|h|<\varepsilon$. This implies that $z-t+\theta h\in D(\varepsilon)$ and thus $\widehat{\varsigma}^{\prime}(z-t+\theta h)$ is bounded, uniformly in $|h|<\varepsilon$ and $t\in[-1,1]$. Applying twice the Lebesgue dominated convergence theorem, we obtain that $\widehat{\rho}_{0}^{\prime}(z)$ exists and is given by $\widehat{\rho}_{0}^{\prime}(z)=\int_{[-1,1]\setminus T_{z}}\widehat{\varsigma}(t)\widehat{\varsigma}^{\prime}(z-t)dt$. Hence $\widehat{\rho}_{0}$ is analytic in the domain $D$. ∎

For any $\varepsilon>0$, define the density $\rho_{\varepsilon}(y)=\frac{1}{\varepsilon}\rho(\frac{y}{\varepsilon})$, $y\in\mathbb{R},$ whose Fourier transform has a compact support in $[-\varepsilon^{-1},\varepsilon^{-1}]$ and is analytically extendable in a neighborhood of [math]. For any non-negative integrable function $\psi$, following the paper [19], we introduce two modified functions related to $\psi$ and establish some two-sided bounds. For any $\varepsilon>0$ and $y\in\mathbb{R}$, set $\mathbb{B}_{\varepsilon}(y)=\{y^{\prime}\in\mathbb{R}:|y^{\prime}-y|\leqslant\varepsilon\}$ and

[TABLE]

Lemma 4.3**.**

Suppose that $\psi$ is a non-negative integrable function and that ${\psi}^{+}_{\varepsilon}$ and ${\psi}^{-}_{\varepsilon}$ are measurable for any $\varepsilon>0$, then for sufficiently small $\varepsilon$, there exists a positive constant $C_{\rho}(\varepsilon)$ with $C_{\rho}(\varepsilon)\to 0$ as $\varepsilon\to 0$, such that, for any $x\in\mathbb{R}$,

[TABLE]

The proof of the above lemma, being similar to that of Lemma 5.2 in [18], will not be detailed here.

The next assertion is the key point in establishing Theorem 2.1. Its proof is based on the spectral gap properties of the perturbed operator $R_{s,z}$ (see Proposition 3.3) and on the saddle point method, see Daniels [10], Richter [33], Ibragimov and Linnik [27] and Fedoryuk [12]. Let us introduce the necessary notation. In the following, let $\varphi$ be a $\gamma$-Hölder continuous function on $\mathcal{S}$. Assume that $\psi:\mathbb{R}\mapsto\mathbb{C}$ is a continuous function with compact support in $\mathbb{R}$, and moreover, $\psi$ has a continuous extension in some neighborhood of [math] in the complex plane and can be extended analytically to the domain $D_{\delta}:=\{z\in\mathbb{C}:|z|<\delta,\Im z\neq 0\}$ for some small $\delta>0$. Recall that $\pi_{s}$ is the invariant measure of the Markov chain $X_{n}^{x}$ under the changed measure $\mathbb{Q}_{s}^{x}$, see (3.3).

Proposition 4.4**.**

*Assume conditions of Theorem 2.1. Let $q=\Lambda^{\prime}(s)$, where $s\in I_{\mu}^{\circ}.$ Then, for any positive sequence $(l_{n})_{n\geqslant 1}$ satisfying $l_{n}\to 0$ as $n\to\infty$, we have, uniformly in $x\in\mathcal{S}$, $|l|\leqslant l_{n}$ and $\varphi\in\mathcal{B}_{\gamma}$, *

[TABLE]

Proof.

Denote $c_{s}(\psi)=\frac{\sqrt{2\pi}}{\sigma_{s}}\psi(0)\pi_{s}(\varphi)$. Taking sufficiently small $\delta>0$, we write

[TABLE]

For $I(n)$, since $\psi$ is bounded and compactly supported on the real line, taking into account Proposition 3.3 (ii), the fact $|e^{-itln}|=1$ and equality (4.1), we get

[TABLE]

For $J(n)$, by Proposition 3.3 (i), we have

[TABLE]

Set for brevity $\psi_{s,x}(t)=\Pi_{s,it}(\varphi)(x)\psi(t)$. It follows that

[TABLE]

For the second term $J_{2}(n)$, applying Proposition 3.3 (i), we get that there exist constants $c_{\delta}>0$ and $\varkappa\in(0,1)$ such that

[TABLE]

Combining this with the continuity of the function $\psi$ at the point [math] and the fact $|e^{-itln}|=1$, we obtain that, uniformly in $|l|\leqslant l_{n}$, $x\in\mathcal{S}$ and $\varphi\in\mathcal{B}_{\gamma}$,

[TABLE]

For the first term $J_{1}(n)$, we shall use the method of steepest descends to derive a precise asymptotic expansion. We make a change of variable $z=it$ to rewrite $J_{1}(n)$ as an integral over the complex interval $L_{0}=(-i\delta,i\delta):$

[TABLE]

where $K_{s}(z)=\log\lambda_{s,z}$ (we choose the branch where $K_{s}(0)=0$), which is an analytic function for $|z|\leqslant\delta$ by Proposition 3.3 (iii). Since the function $z\mapsto e^{n(K_{s}(z)-zl)}$ is analytic in the neighborhood of [math], and the function $z\mapsto\psi_{s,x}(-iz)$ has an analytic extension in the domain $D_{\delta}:=\{z\in\mathbb{C}:|z|<\delta,\Im z\neq 0\}$ and has a continuous extension in the domain $\overline{D}_{\delta}:=\{z\in\mathbb{C}:|z|\leqslant\delta\}$, by Cauchy’s integral theorem we can choose a special path of the integration which passes through the saddle point of the function $K_{s}(z)-zl$. From (3.6), we have

[TABLE]

which implies that for $|z|<\delta$,

[TABLE]

where $\gamma_{s,k}=\Lambda^{(k)}(s)$ and $\Lambda(s)=\log\kappa(s)$. From this Taylor’s expansion and the fact that $\Lambda^{(2)}(s)=\sigma_{s}^{2}>0$, it follows that the function $K_{s}(z)-zl$ is convex in the neighborhood of [math]. Consider the saddle point equation

[TABLE]

An equivalent formulation of (4.12) is $l=\sum_{k=2}^{\infty}\gamma_{s,k}\frac{z^{k-1}}{(k-1)!}$, which by simple series inversion techniques gives the following solution:

[TABLE]

From (4.13), it follows that the solution $z_{0}=z_{0}(l)$ is real for sufficiently small $l$ and that $z_{0}=z_{0}(l)\to 0$ as $l\to 0.$ Moreover, $z_{0}>0$ for sufficiently small $l>0$, and $z_{0}<0$ for sufficiently small $l<0$. By Cauchy’s integral theorem, $J_{1}(n)$ can be rewritten as

[TABLE]

where $L_{1}=(-i\delta,z_{0}-i\delta)$, $L_{2}=(z_{0}-i\delta,z_{0}+i\delta)$ and $L_{3}=(z_{0}+i\delta,i\delta)$. By (4.11), we get $K_{s}(it)=-\frac{1}{2}\sigma_{s}^{2}t^{2}+O(t^{3})$, which implies that $|e^{nK_{s}(it)}|\leqslant e^{-\frac{n}{3}\sigma_{s}^{2}t^{2}}$, when $t$ is sufficiently small. Combining this with (4.13) and the continuity of $K_{s}(z)$ in the neighborhood of [math] yields that, for sufficiently small $l$, $|e^{nK_{s}(z)}|\leqslant e^{-\frac{n}{4}\sigma_{s}^{2}\delta^{2}}$, for any $z\in L_{1}\cup L_{3}$. Since, for sufficiently small $l$, $lz_{0}>0$, we get that, for $z\in L_{1}\cup L_{3}$, $|e^{-nzl}|=|e^{-nlz_{0}}|\leqslant 1$. Moreover, using the continuity of the function $z\mapsto\psi_{s,x}(-iz)$ in a small neighborhood of [math] in the complex plane, there exists a constant $C_{s}>0$ such that, on $L_{1}$ and $L_{3}$, we have $\sup_{x\in\mathcal{S}}|\psi_{s,x}(-iz)|\leqslant C_{s}\|\varphi\|_{\gamma}$. Therefore, we obtain, for $n$ sufficiently large, uniformly in $|l|\leqslant l_{n}$ and $x\in\mathcal{S}$,

[TABLE]

It follows that

[TABLE]

Without loss of generality, assume that $n\geqslant 3$. Making a change of variable $z=z_{0}+it$ gives

[TABLE]

From (4.12) and (4.13), we have $K_{s}^{\prime}(z_{0})=l$. By Taylor’s formula, we get that for $|t|<\delta$,

[TABLE]

Using $K_{s}^{\prime}(z_{0})=l$ and (4.11), it follows that

[TABLE]

Combining this with (4.13) and Lemma 4.1 gives $K_{s}(z_{0})-z_{0}l=-h_{s}(l)$. Thus

[TABLE]

Since $K_{s}^{\prime\prime}(z_{0})=\sigma_{s}^{2}+O(z_{0})>\frac{1}{2}\sigma_{s}^{2}$, for small enough $z_{0}$, $\delta$ and $l$, we obtain that $\Re(\sum_{k=2}^{\infty}\frac{K_{s}^{(k)}(z_{0})(it)^{k}}{k!})<-\frac{1}{8}\sigma_{s}^{2}t^{2}$. Therefore, using (4.15) and the fact that uniformly in $x\in\mathcal{S}$, the function $z\mapsto\psi_{s,x}(z)$ is continuous in a neighborhood of [math] in the complex plane, we obtain that, uniformly in $x\in\mathcal{S}$ and $|l|\leqslant l_{n}$,

[TABLE]

This, together with (4.1)-(4.15), implies

[TABLE]

Noting that $\Pi_{s,0}(\varphi)(x)=\pi_{s}(\varphi)$ and $\psi_{s,x}(0)=\psi(0)\pi_{s}(\varphi)$, we write

[TABLE]

We give a control of $J_{11}(n)$. Note that $|\psi_{s,x}(t-iz_{0})|$ is bounded by $C_{s}\|\varphi\|_{\gamma}$, uniformly in $|t|<n^{-\frac{1}{2}}\log n$. Note also that for $|t|<n^{-\frac{1}{2}}\log n$ and for large enough $n$, we have $|e^{n\Re{\sum_{k=3}^{\infty}\frac{K^{(k)}(z_{0})(it)^{k}}{k!}}}|\leqslant e^{cnt^{4}}\leqslant C$. Hence using the inequality $|e^{z}-1|\leqslant e^{\Re{z}}|z|$ yields

[TABLE]

Now we control $J_{12}(n)$. Recalling that $z_{0}=z_{0}(l)\leqslant c_{s}l_{n}$, using the fact that uniformly with respect to $x\in\mathcal{S}$, the map $z\mapsto\psi_{s,x}(z)$ is continuous in the neighborhood of [math] in the complex plane, we get that for $|t|\leqslant n^{-\frac{1}{2}}\log n$,

[TABLE]

We then obtain

[TABLE]

It is easy to see that $J_{13}(n)\leqslant C\|\varphi\|_{\gamma}e^{-c_{s}\log^{2}n}$. This, together with (4.1)-(4.17), proves that $J_{1}(n)\leqslant c_{s}(n^{-\frac{1}{2}}\log n+l_{n})\|\varphi\|_{\gamma}.$ The desired result follows by combining this with (4.1)-(4.9). ∎

Assume that the functions $\varphi$ and $\psi$ satisfy the same properties as in Proposition 4.4. The following result, for $s<0$ small enough, will be used to prove Theorem 2.3.

Proposition 4.5**.**

Assume conditions of Theorem 2.3. Then, there exists $\eta_{0}<\eta$ such that for any $s\in(-\eta_{0},0)$, $q=\Lambda^{\prime}(s)$ and for any positive sequence $(l_{n})_{n\geqslant 1}$ satisfying $l_{n}\to 0$ as $n\to\infty$, we have, uniformly in $x\in\mathcal{S}$, $|l|\leqslant l_{n}$ and $\varphi\in\mathcal{B}_{\gamma}$,

[TABLE]

Proof.

Using Propositions 3.2 and 3.4, the proof of Proposition 4.5 can be carried out as the proof of Proposition 4.4. We omit the details. ∎

4.2. Proof of Theorem 2.1

Recall that $q=\Lambda^{\prime}(s)$, $\Lambda^{*}(q+l)=\Lambda^{*}(q)+sl+h_{s}(l)$, $x\in\mathcal{S},$ and $|l|\leqslant l_{n}\to 0$, as $n\to\infty$. Taking into account that $e^{n\Lambda^{*}(q)}=e^{sqn}/\kappa^{n}(s)$ and using the change of measure (3.1), we write

[TABLE]

Setting $T_{n}^{x}=\log|G_{n}x|-nq$ and $\psi_{s}(y)=e^{-sy}\mathbbm{1}_{\{y\geqslant 0\}}$, from (4.2) we get

[TABLE]

Upper bound. Let $\varepsilon\in(0,1)$ and ${\psi}^{+}_{s,\varepsilon}(y)=\sup_{y^{\prime}\in\mathbb{B}_{\varepsilon}(y)}\psi_{s}(y^{\prime})$ be defined as in (4.5) but with $\psi_{s}$ instead of $\psi$. Using Lemma 4.3 leads to

[TABLE]

Denote by $\widehat{{\psi}}^{+}_{s,\varepsilon}$ the Fourier transform of ${\psi}^{+}_{s,\varepsilon}$. Elementary calculations give

[TABLE]

By the inversion formula, for any $y\in\mathbb{R},$

[TABLE]

Substituting $y=T_{n}^{x}-nl$, taking expectation with respect to $\mathbb{E}_{\mathbb{Q}_{s}^{x}}$, and using Fubini’s theorem, we get

[TABLE]

where

[TABLE]

Note that $\widehat{\psi}^{+}_{s,\varepsilon}\widehat{\rho}_{\varepsilon^{2}}$ is compactly supported in $\mathbb{R}$ since $\widehat{\rho}_{\varepsilon^{2}}$ has a compact support. One can verify that $\widehat{\psi}^{+}_{s,\varepsilon}$ has an analytic extension in a neighborhood of [math]. By Lemma 4.2, we see that the function $\widehat{\rho}_{\varepsilon^{2}}$ has a continuous extension in the complex plane, and has an analytic in the domain $D_{\varepsilon^{2}}:=\{z\in\mathbb{C}:|z|<\varepsilon^{2},\Im z\neq 0\}$. Using Proposition 4.4 with $\varphi=r_{s}^{-1}$ and $\psi=\widehat{\psi}^{+}_{s,\varepsilon}\widehat{\rho}_{\varepsilon^{2}}$, it follows that

[TABLE]

Since $\widehat{\rho}_{\varepsilon^{2}}(0)=1$, from (4.19)-(4.23), we have that for sufficiently small $\varepsilon\in(0,1)$,

[TABLE]

Letting $\varepsilon\to 0$ and noting that $C_{\rho}(\varepsilon)\to 0$, we obtain the upper bound:

[TABLE]

Lower bound. For $\varepsilon\in(0,1)$, let ${\psi}^{-}_{s,\varepsilon}(y)=\inf_{y^{\prime}\in\mathbb{B}_{\varepsilon}(y)}\psi_{s}(y^{\prime})$ be defined as in (4.5) with $\psi_{s}$ instead of $\psi$. From (4.19) and Lemma 4.3, we get

[TABLE]

For the first term $B_{n}^{-}(x,l)$, applying (4.22) with ${\psi}^{+}_{s,\varepsilon}\rho_{\varepsilon^{2}}$ replaced by ${\psi}^{-}_{s,\varepsilon}\rho_{\varepsilon^{2}}$, we get

[TABLE]

In the same way as for the upper bound, using $\widehat{\psi}^{-}_{s,\varepsilon}(0)=\frac{e^{-2s\varepsilon}}{s}$ and Proposition 4.4 with $\varphi=r_{s}^{-1}$ and $\psi=\widehat{\psi}^{-}_{s,\varepsilon}\widehat{\rho}_{\varepsilon^{2}}$ (one can check that the functions $\varphi$ and $\psi$ satisfy the required conditions in Proposition 4.4), we obtain the lower bound:

[TABLE]

For the second term $C_{n}^{-}(x,l)$, noting that $\psi^{-}_{s,\varepsilon}\leqslant\psi_{s}$ and applying Lemma 4.3 to $\psi_{s}$, we get $\psi^{-}_{s,\varepsilon}\leqslant\psi_{s}\leqslant(1+C_{\rho}(\varepsilon)){\psi}_{s,\varepsilon}^{+}\!\ast\!\rho_{\varepsilon^{2}}$. We use the same argument as in (4.22) to obtain

[TABLE]

Notice that, from Lemma 4.1, for any fixed $y\in\mathbb{R}$, it holds, uniformly in $l$ satisfying $|l|\leqslant l_{n}$, that $e^{nh_{s}(l)-nh_{s}(l+\frac{y}{n})}\to 1$ as $n\to\infty$. Applying Proposition 4.4 again with $\varphi=r_{s}^{-1}$, $\psi=\widehat{\psi}^{+}_{s,\varepsilon}\widehat{\rho}_{\varepsilon^{2}}$, and using the Lebesgue dominated convergence theorem, we obtain

[TABLE]

since $\rho$ is integrable on $\mathbb{R}$. This, together with (4.2)-(4.26), implies the lower bound:

[TABLE]

as required. We conclude the proof of Theorem 2.1 by combining (4.24) and (4.27).

4.3. Proof of Theorem 2.3

Since the change of measure formula can be extended for small $s<0$, under the conditions of Theorem 2.3, we have, similar to (4.2),

[TABLE]

Applying Proposition 4.5, we can follow the proof of Theorem 2.1 to show Theorem 2.3. We omit the details.

5. Proof of Theorems 2.2 and 2.4

We first establish the following assertion which will be used to prove Theorem 2.2, but which is of independent interest. Let $\psi$ be a measurable function on $\mathbb{R}$ and $\varepsilon>0.$ Denote, for brevity, $\psi_{s}(y)=e^{-sy}\psi(y)$ and

[TABLE]

Introduce the following condition: for any $s\in I_{\mu}^{\circ}$ and $\varepsilon>0,$ the functions $y\mapsto\psi^{+}_{s,\varepsilon}(y)$ and $y\mapsto\psi^{-}_{s,\varepsilon}(y)$ are measurable and

[TABLE]

Theorem 5.1**.**

Suppose the assumptions of Theorem 2.1 hold true. Let $q=\Lambda^{\prime}(s)$, where $s\in I_{\mu}^{\circ}$. Assume that $\varphi$ is a Hölder continuous function on $\mathcal{S}$ and $\psi$ is a measurable function on $\mathbb{R}$ satisfying condition (5.1). Then, for any positive sequence $(l_{n})_{n\geqslant 1}$ satisfying $\lim_{n\to\infty}l_{n}=0$, we have

[TABLE]

Before proceeding with the proof of this theorem, let us give some examples of functions satisfying condition (5.1). It is easy to see that (5.1) holds for increasing non-negative functions $\psi$ satisfying $\int_{\mathbb{R}}e^{-sy}\psi(y)dy<+\infty,$ in particular, for the indicator function $\psi(y)=\mathbbm{1}_{\{y\geqslant c\}}$, $y\in\mathbb{R}$, where $c\in\mathbb{R}$ is a fixed constant. Another example for which (5.1) holds true is when $\psi$ is non-negative, continuous and there exists $\varepsilon>0$ such that

[TABLE]

where the function $\psi_{\varepsilon}^{+}(y)=\sup_{y^{\prime}\in\mathbb{B}_{\varepsilon}(y)}\psi(y^{\prime})$ is assumed to be measurable.

Proof of Theorem 5.1.

Without loss of generality, we assume that both $\varphi$ and $\psi$ are non-negative (otherwise, we decompose the functions $\varphi=\varphi^{+}-\varphi^{-}$ and $\psi=\psi^{+}-\psi^{-}$). Let $T_{n}^{x}=\log|G_{n}x|-nq$. Since $e^{n\Lambda^{*}(q)}=e^{sqn}/\kappa^{n}(s)$, using the change of measure (3.1), we have

[TABLE]

For brevity, set $\Phi_{s}(x)=\left(\varphi r_{s}^{-1}\right)(x),$ $x\in\mathcal{S}$, and $\Psi_{s}(y)=e^{-sy}\psi(y),$ $y\in\mathbb{R}$. Then,

[TABLE]

Upper bound. We wish to write the expectation in (5.4) as an integral of the Fourier transform of $\Psi_{s},$ which, however, may not belong to the space $L^{1}(\mathbb{R})$. As in the proof of Theorem 2.1 (see Section 4.2), we make use of the convolution technique to overcome this difficulty. Applying Lemma 4.3 to $\Psi_{s}$, one has, for sufficiently small $\varepsilon>0$,

[TABLE]

where ${\Psi}^{+}_{s,\varepsilon}(y)=\sup_{y^{\prime}\in\mathbb{B}_{\varepsilon}(y)}\Psi_{s}(y^{\prime})$, $y\in\mathbb{R}$. Using the same arguments as for deducing (4.22), we have

[TABLE]

where $R^{n}_{s,it}\Phi_{s}(x)=\mathbb{E}_{\mathbb{Q}_{s}^{x}}\left[e^{itT_{n}^{x}}\Phi_{s}(X_{n}^{x})\right]$ and $\widehat{\Psi}^{+}_{s,\varepsilon}$ is the Fourier transform of $\Psi^{+}_{s,\varepsilon}$. Note that $\Phi_{s}$ is strictly positive and $\gamma$-Hölder continuous function on $\mathcal{S}$, and $\widehat{\Psi}^{+}_{s,\varepsilon}\widehat{\rho}_{\varepsilon^{2}}$ has a compact support in $\mathbb{R}$. Applying Proposition 4.4 with $\varphi=\Phi_{s}$ and $\psi=\widehat{\Psi}^{+}_{s,\varepsilon}\widehat{\rho}_{\varepsilon^{2}}$ (one can verify that the functions $\varphi$ and $\psi$ satisfy the required conditions in Proposition 4.4), we obtain

[TABLE]

Since $\widehat{\Psi}^{+}_{s,\varepsilon}(0)=\int_{\mathbb{R}}\sup_{y^{\prime}\in\mathbb{B}_{\varepsilon}(y)}e^{-sy^{\prime}}\psi(y^{\prime})dy$ and $\widehat{\rho}_{\varepsilon^{2}}(0)=1$, letting $\varepsilon$ go to [math], using the condition (5.1) and the fact that $C_{\rho}(\varepsilon)\to 0$ as $\varepsilon\to 0$, we get the upper bound:

[TABLE]

Lower bound. Denote ${\Psi}^{-}_{s,\varepsilon}(y)=\inf_{y^{\prime}\in\mathbb{B}_{\varepsilon}(y)}\Psi_{s}(y^{\prime})$. From (5.4), using Lemma 4.3, we get

[TABLE]

For $B_{n}^{-}(x,l)$, we proceed as for (5) and (5.6), with ${\Psi}^{+}_{s,\varepsilon}$ replaced by ${\Psi}^{-}_{s,\varepsilon}$. Using Proposition 4.4, with $\varphi=\Phi_{s}$ and $\psi=\widehat{\Psi}^{-}_{s,\varepsilon}\widehat{\rho}_{\varepsilon^{2}},$ and the fact that $\widehat{\rho}_{\varepsilon^{2}}(0)=1$ and $\widehat{\Psi}^{-}_{s,\varepsilon}(0)=\int_{\mathbb{R}}\inf_{y^{\prime}\in\mathbb{B}_{\varepsilon}(y)}e^{-sy^{\prime}}\psi(y^{\prime})dy$, in an analogous way as in (5.7), we obtain that

[TABLE]

where the last convergence is due to the condition (5.1). For $C_{n}^{-}(x,l)$, noting that $\Psi^{-}_{s,\varepsilon}\leqslant\Psi_{s}$, applying Lemma 4.3 to $\Psi_{s}$ we get $\Psi^{-}_{s,\varepsilon}\leqslant(1+C_{\rho}(\varepsilon))\widehat{\Psi}^{+}_{s,\varepsilon}\widehat{\rho}_{\varepsilon^{2}}$. Similarly to (5.6), we show that

[TABLE]

From Lemma 4.1, for any fixed $y\in\mathbb{R}$, it holds that $e^{nh_{s}(l)-nh_{s}(l+\frac{y}{n})}\to 1$, uniformly in $|l|\leqslant l_{n}$ as $n\to\infty$. Applying Proposition 4.4 with $\varphi=\Phi_{s}$ and $\psi=\widehat{\Psi}^{+}_{s,\varepsilon}\widehat{\rho}_{\varepsilon^{2}}$, it follows, from the Lebesgue dominated convergence theorem, that

[TABLE]

as $\varepsilon\to 0$. Combining this with (5)-(5), we get the lower bound

[TABLE]

Putting together (5.7) and (5.10), and noting that $\pi_{s}(\Phi_{s})=\pi_{s}(\varphi r_{s}^{-1})=\frac{\nu_{s}(\varphi)}{\nu_{s}(r_{s})}$, the result follows. ∎

In the sequel, we deduce Theorem 2.2 from Theorem 5.1 using approximation techniques.

Proof of Theorem 2.2.

Without loss of generality, we assume that $\varphi\geqslant 0$ and $\psi\geqslant 0$. Let $\Psi_{s}(y)=e^{-sy}\psi(y)$, $y\in\mathbb{R}$. We construct two step functions as follows: for any $\eta\in(0,1)$, $m\in\mathbb{Z}$ and $y\in[m\eta,(m+1)\eta)$, set

[TABLE]

By the definition of the direct Riemann integrability, the following two limits exist and are equal:

[TABLE]

Since $\Psi_{s}$ is directly Riemann integrable, we have $M:=\sup_{y\in\mathbb{R}}\Psi_{s}(y)<+\infty$. Let $\varepsilon\in(0,M\eta)$ be fixed. Denote $I_{m}=[(m-1)\eta,m\eta)$, I_{m}^{-}=\big{(}m\eta-\frac{\varepsilon}{M4^{|m|}},m\eta\big{)}, and I_{m}^{+}=\big{[}m\eta,m\eta+\frac{\varepsilon}{M4^{|m|}}\big{)}, $m\in\mathbb{Z}$. Set $k_{m}^{+}:=M4^{|m|}\frac{\Psi_{s,\eta}^{+}(m\eta)-\Psi_{s,\eta}^{+}((m-1)\eta)}{\varepsilon}$, $m\in\mathbb{Z}$. For the step function $\Psi_{s,\eta}^{+}$, in the neighborhood of every possible discontinuous point $m\eta$, $m\in\mathbb{Z}$, if $\Psi_{s,\eta}^{+}(m\eta)\geqslant\Psi_{s,\eta}^{+}((m-1)\eta)$, then for any $y\in I_{m}\cup I_{m+1}$, $m\in\mathbb{Z}$, we define

[TABLE]

If $\Psi_{s,\eta}^{+}(m\eta)<\Psi_{s,\eta}^{+}((m-1)\eta)$, then we define

[TABLE]

From this construction, the non-negative continuous function $\Psi_{s,\eta,\varepsilon}^{+}$ satisfies $\Psi^{+}_{s,\eta}\leqslant\Psi_{s,\eta,\varepsilon}^{+}$ and $\int_{\mathbb{R}}[\Psi_{s,\eta,\varepsilon}^{+}(y)-\Psi^{+}_{s,\eta}(y)]dy<\varepsilon$. Similarly, for the step function $\Psi_{s,\eta}^{-}$, one can construct a non-negative continuous function $\Psi_{s,\eta,\varepsilon}^{-}$ which satisfies $\Psi_{s,\eta,\varepsilon}^{-}\leqslant\Psi^{-}_{s,\eta}$ and $\int_{\mathbb{R}}[\Psi^{-}_{s,\eta}(y)-\Psi_{s,\eta,\varepsilon}^{-}(y)]dy<\varepsilon$. Consequently, in view of (5.11), we obtain that, for $\eta$ small enough,

[TABLE]

For brevity, set $c_{s,l,n}=\sqrt{2\pi n}\leavevmode\nobreak\ \sigma_{s}e^{n\Lambda^{*}(q+l)}$ and $T_{n,l}^{x}=\log|G_{n}x|-n(q+l)$. Recalling that $\Psi_{s}(y)=e^{-sy}\psi(y)$, we write

[TABLE]

To control $J_{2}$, we shall verify the conditions of Theorem 5.1. Noting that the function $y\mapsto e^{sy}\Psi_{s,\eta,\varepsilon}^{+}(y)$ is non-negative and continuous, it remains to check the condition (5.3). By the construction of $\Psi_{s,\eta,\varepsilon}^{+}$ one can verify that there exists a constant $\varepsilon_{1}\in(0,\min\{M\eta,\eta/3\})$ such that

[TABLE]

where the series is finite since the function $\Psi_{s}$ is directly Riemann integrable. Hence, applying Theorem 5.1 to $y\mapsto e^{sy}\Psi_{s,\eta,\varepsilon}^{+}(y)$, we get

[TABLE]

For $J_{3}(x)$, recall that $\Psi_{s,\eta,\varepsilon}^{-}\leqslant\Psi_{s}\leqslant\Psi_{s,\eta,\varepsilon}^{+}$. Using (5.12) and the fact that $r_{s}$ is uniformly bounded on $\mathcal{S}$, we get that there exists a constant $C_{s}>0$ such that

[TABLE]

For $J_{1}$, note that $e^{sy}\Psi_{s,\eta,\varepsilon}^{-}(y)\leqslant e^{sy}\Psi_{s}(y)\leqslant e^{sy}\Psi_{s,\eta,\varepsilon}^{+}(y)$, $y\in\mathbb{R}$. Combining this with the positivity of $\varphi$, it holds that

[TABLE]

Using (5.15), it holds that, as $n\to\infty$, $J_{11}\to 0$, uniformly in $x\in\mathcal{S}$ and $|l|\leqslant l_{n}$. For $J_{12}$, note that the function $y\mapsto e^{sy}\Psi_{s,\eta,\varepsilon}^{-}(y)$ is non-negative and continuous. By the construction of $\Psi_{s,\eta,\varepsilon}^{-}$, similarly to (5), one can verify that there exists $\varepsilon_{2}>0$ such that $\int_{\mathbb{R}}\sup_{y^{\prime}\in\mathbb{B}_{\varepsilon_{2}}(y)}\Psi_{s,\eta,\varepsilon}^{-}(y^{\prime})dy<+\infty.$ We deduce from Theorem 5.1 that $J_{12}\to 0$ as $n\to\infty$, uniformly in $x\in\mathcal{S}$ and $|l|\leqslant l_{n}$. For $J_{13}$, we use (5.12) to get that $J_{13}\leqslant C_{s}\varepsilon$. Consequently, we obtain that, as $n\to\infty$, $J_{1}\leqslant C_{s}\varepsilon$, uniformly in $x\in\mathcal{S}$ and $|l|\leqslant l_{n}$. This, together with (5), (5.15)-(5.16), implies that

[TABLE]

Since $\varepsilon>0$ is arbitrary, we conclude the proof of Theorem 2.2. ∎

Proof of Theorem 2.4.

Following the proof of Theorem 5.1, one can verify that the asymptotic (5.1) holds true for $s<0$ small enough and for $\psi$ satisfying condition (5.1). The passage to a directly Riemann integrable function $\psi$ can be done by using the same approximation techniques as in the proof of Theorem 2.2. ∎

6. Proof of Theorems 2.5, 2.6 and 2.7

Proof of Theorems 2.5 and 2.6.

We first give a proof of Theorem 2.5. Since $\log|G_{n}x|\leqslant\log\|G_{n}\|$ and the function $\bar{r}_{s}$ is strictly positive and uniformly bounded on $\mathcal{S}$, applying Theorem 2.1 we get the lower bound:

[TABLE]

For the upper bound, since all matrix norms are equivalent, there exists a positive constant $C$ which does not depend on the product $G_{n}$ such that $\log\|G_{n}\|\leqslant\max_{1\leqslant i\leqslant d}\log|G_{n}e_{i}|+C,$ where $(e_{i})_{1\leqslant i\leqslant d}$ is the canonical orthonormal basis in $\mathbb{R}^{d}$. From this inequality, we deduce that

[TABLE]

Using Lemma 4.1, we see that there exists a constant $C_{s}>0$ such that $e^{n[\Lambda^{*}(q+l-C/n)-\Lambda^{*}(q+l)]}\leqslant C_{s}$, uniformly in $|l|\leqslant l_{n}$ and $n\geqslant 1$. Again by Theorem 2.1, we obtain the upper bound:

[TABLE]

This, together with (6.1), proves Theorem 2.5. Using Theorem 2.3, the proof of Theorem 2.6 can be carried out in the same way. ∎

Proof of Theorem 2.7.

Without loss of generality, we assume that the function $\varphi$ is non-negative. From Theorem 2.2, we deduce that there exists a sequence $(r_{n})_{n\geqslant 1}$, determined by the matrix law $\mu$ such that $r_{n}\to 0$ as $n\to\infty$ and, uniformly in $x\in\mathcal{S}$, $|l|\leqslant l_{n}$ and $0\leqslant\Delta\leqslant o(n)$, it holds that

[TABLE]

Taking the difference of (6) with $\Delta=0$ and with $\Delta>0$, we get, as $n\to\infty$,

[TABLE]

where

[TABLE]

An elementary analysis using Lemma 4.1 shows that

[TABLE]

uniformly in $|l|\leqslant l_{n}$ and $\Delta_{n}\leqslant\Delta\leqslant o(n)$, for any $(\Delta_{n})_{n\geqslant 1}$ converging to [math] slowly enough ($\Delta_{n}^{-1}=o(r_{n}^{-1})$). This concludes the proof of Theorem 2.7. ∎