We establish a Perron-Frobenius type theorem for products of stationary and ergodic nonnegative random matrices, which provides a precise asymptotic description of their coefficients, and reveals a close connection with the ergodic theory for stationary and ergodic sequences of real random variables. As applications, we improve upon existing laws of large numbers for products of nonnegative random matrices by relaxing moment conditions. We then derive limit theorems for a multi-type branching process (Z(n)) in a stationary and ergodic environment, offering a precise description of the growth rate of the population size. In particular, we establish a Kesten-Stigum type theorem for the scalar product < Z(n) ,y > with any nonnegative vector y, revealing a key link between the branching process and products of random matrices. We also prove the convergence of the direction Z(n)/vertical bar vertical bar Z(n vertical bar vertical bar), a complement to the Kesten-Stigum type theorem, and establish new laws of large numbers and central limit theorems for < Z(n) ,y >.

Our objective is to explore random walks on the general linear group, constrained to a specific domain, with a primary focus on establishing the conditioned local limit theorem. This paper represents the first step toward achieving this goal, specifically entailing the construction of a novel entity – the target harmonic measure. This measure, together with the harmonic function, serves as a pivotal component in establishing the conditioned local limit theorem. Using a reversal identity, we introduce a reversed sequence characterized as a dual random walk with a perturbation depending on future observations. The investigation of such walks, which rely on future information, lies at the heart of this paper. To carry out this study, we develop an approach grounded in the finite-size approximation of perturbations, enabling us to simplify the investigation to an array of Markov chains with increasing dimensions.

We prove a local limit theorem for the quantity g_n ⋯ g_1, where g_1, g_2, … is a sequence of independent and identically distributed random invertible square matrices. The norm · is arbitrary and we make no proximality assumption.

We prove a non-arithmeticity property for the complex eigenvalues of a Zariski dense sub-semigroups of real reductive algebraic groups. This property will be used in [7] in order to establish a local limit theorem for the operator norm of products of random matrices.

We consider a matrix branching random walk on the semi-group of nonnegative matrices, where we are able to derive, under general assumptions, an analogue of Biggins' martingale convergence theorem for the additive martingale W_n, a spinal decomposition theorem, convergence of the derivative martingale D_n, and finally, the Seneta-Heyde scaling stating that in the boundary case c √(n) W_n → D_∞ a.s., where D_∞ is the limit of the derivative martingale and c is a positive constant. As an important tool that is of interest in its own right, we provide explicit duality results for the renewal measure of centered Markov random walks, relating the renewal measure of the process, killed when the random walk component becomes negative, to the renewal measure of the ascending ladder process.

Consider the matrix products G_n: = g_n ⋯ g_1 , where (g_n)_n⩾ 1 is a sequence of independent and identically distributed positive random d× d matrices. Under the optimal third moment condition, we first establish a Berry–Esseen theorem and an Edgeworth expansion for the (i, j)-th entry G_n^i,j of the matrix G_n , where 1 ⩽ i, j ⩽ d . Utilizing the Edgeworth expansion for G_n^i,j under the changed probability measure, we then prove precise upper and lower large deviation asymptotics for the entries G_n^i,j subject to an exponential moment assumption. As applications, we deduce local limit theorems with large deviations for G_n^i,j and establish upper and lower large deviations bounds for the spectral radius ρ (G_n) of G_n . A byproduct of our approach is the local limit theorem for G_n^i,j under the optimal second moment condition. In the proofs we develop a spectral gap theory for both the norm cocycle and the coefficients, which is of independent interest.

We study the random walk (S_n)_n≥ 1 with independent and identically distributed real-valued increments having zero mean and an absolute moment of order 2 + δ for some δ> 0. For any starting point x ∈ℝ, let τ_x = inf{k ≥ 1 : x + S_k < 0} denote the first exit time of the random walk x + S_n from the half-line [0, ∞). In the previous work [25], we established a Gaussian heat kernel approximation for both the persistence probability ℙ(τ_x > n) and the joint distribution ℙ(x + S_n ≤·, τ_x > n), uniformly over x ∈ℝ as n →∞. In this paper, we leverage these results to establish a novel conditioned local limit theorem for the walk (x + S_n)_n ≥ 1. For ℤ-valued random walks, we prove that the joint probability ℙ(x + S_n = y, τ_x > n) is uniformly approximated by a distribution governed by the Gaussian heat kernel over all x, y ∈ℤ as n →∞. Our new asymptotic unifies into a single comprehensive formula the classical local limit theorem by Caravenna [6], as well as various results relying on specific assumptions on x and y. As a corollary, we obtain a new uniform-in-x asymptotic formula for the local probability ℙ(τ_x = n). We also extend our analysis to non-lattice random walks.

Let (gn)n,1 be a sequence of independent and identically distributed random elements with law mu on the general linear group GL(V ), where V = R-d. Consider the random walk G(n) :=g(n) g1, n > 1. Under suitable conditions on mu, we establish the first order Edgeworth expansion for the coefficients (f, Gnv) with v is an element of V and f is an element of V & lowast;. Our result is established under the sole irreducibility and proximality condition, together with finiteness of an exponential moment, thus avoiding a stronger Cram & eacute;r-type assumption which is usual for higher order expansions as, for instance, in [14].

Consider a random walk S-n=& sum;(n)(i=1)X(i )with independent and identically distributed real-valued increments X-i of zero mean and finite variance. Assume that X-i is non-lattice and has a moment of order 2+delta. For any x >= 0, let tau(x)=inf{k >= 1:x+S-k<0} be the first time when the random walk x+S-n leaves the half-line [0,infinity). We study the asymptotic behavior of the probability P (tau(x) >n) and that of the expectation E(f(x+S-n),tau(x)>n) for a large class of target function f and various values of x, y possibly depending on n. This general setting implies limit theorems for the joint distribution P(x+S-n is an element of y+[0,Delta],tau(x)>n) where Delta>0 may also depend on n. In particular, the case of moderate deviations y=sigma root qn log n is considered. We also deduce some new asymptotics for random walks with drift and give explicit constants in the asymptotic of the probability P (tau(x) >n). For the proofs we establish new conditioned integral limit theorems with precise error terms.

Consider a branching random walk $(G_u)_{u\in \mathbb T}$ on the general linear group $\textrm{GL}(V)$ of a finite dimensional space $V$, where $\mathbb T$ is the associated genealogical tree with nodes $u$. For any starting point $v \in V \setminus\{0\}$, let $M^v_n=\max_{|u| = n} \log \| G_u v \|$ denote the maximal position of the walk $\log \| G_u v \|$ in the generation $n$. We first show that under suitable conditions, $\lim_{n \to \infty} \frac{M_n^v }{n} = \gamma_{+}$ almost surely, where $\gamma_{+}\in \mathbb R$ is a constant. Then, in the case when $\gamma_+=0$, under appropriate boundary conditions, we refine the last statement by determining the rate of convergence at which $M_n^v$ converges to $-\infty$. We prove in particular that $\lim_{n \to \infty} \frac{M_n^v}{\log n} = -\frac{3}{2\alpha}$ in probability, where $\alpha >0$ is a constant determined by the boundary conditions. Similar properties are established for the minimal position. As a consequence we derive the asymptotic speed of the maximal and minimal positions for the coefficients, the operator norm and the spectral radius of $G_u$.

Consider a random walk S_n=∑_i=1^n X_i with independent and identically distributed real-valued increments with zero mean, finite variance and moment of order 2 + δ for some δ>0. For any starting point x∈ℝ, let τ_x = inf{ k≥ 1: x+S_k < 0 } denote the first time when the random walk x+S_n exits the half-line [0,∞). We investigate the uniform asymptotic behavior over x∈ℝ of the persistence probability ℙ (τ_x >n) and the joint distribution ℙ( x + S_n ≤ u, τ_x > n ), for u≥ 0, as n →∞. New limit theorems for these probabilities are established based on the heat kernel approximations. Additionally, we evaluate the rate of convergence by proving Berry-Esseen type bounds.

The purpose of this article is to show that Theorems 2.2-2.5 from [1] apply to the product of random matrices considered by Grama, Le Page, and Peigné [2]. This allows us, in particular, to emphasize the general nature of the formulation of our theorems in [1] by showing that our assumptions are verified for previous models.

We consider a supercritical branching random walk where each particle gives birth to a random number of particles of the next generation, which move on the real line, according to a fixed law. Let Zn be the counting measure which counts the number of particles of n -th generation situated in a given region. Under suitable conditions, we establish a Berry-Esseen bound and a Cramer type moderate deviation expansion for Zn with suitable norming.

Let $(\mathbb X, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu$ and let $f$ be a real-valued H\"older continuous function on $\mathbb X$ such that $\nu(f) = 0$. Consider the Birkhoff sums $S_n f = \sum_{k=0}^{n-1} f \circ T^{k}$, $n\geq 1$. For any $t \in \mathbb R$, denote by $\tau_t^f$ the first time when the sum $t+ S_n f$ leaves the positive half-line for some $n\geq 1$. By analogy with the case of random walks with independent identically distributed increments, we study the asymptotic as $n\to\infty$ of the probabilities $ \nu(x\in \mathbb X: \tau_t^f(x)>n) $ and $ \nu(x\in \mathbb X: \tau_t^f(x)=n) $. We also establish integral and local type limit theorems for the sum $t+ S_n f(x)$ conditioned on the set $\{ x \in \mathbb X: \tau_t^f(x)>n \}$.

We investigate random walks on the general linear group constrained within a specific domain, with a focus on their asymptotic behavior. In a previous work [38], we constructed the associated harmonic measure, a key element in formulating the local limit theorem for conditioned random walks on groups. The primary aim of this paper is to prove this theorem. The main challenge arises from studying the conditioned reverse walk, whose increments, in the context of random walks on groups, depend on the entire future. To achieve our goal, we combine a Caravenna-type conditioned local limit theorem with the conditioned version of the central limit theorem for the reversed walk. The resulting local limit theorem is then applied to derive the local behavior of the exit time.

Let (g(n))(n >= 1) be a sequence of independent and identically distributed positive random dxd matrices and consider the matrix product G(n) = g(n) ... g(1). Under suitable conditions, we establish the Berry-Esseen bounds on the rate of convergence in the central limit theorem and Cramer-type moderate deviation expansions, for any matrix norm parallel to G(n)parallel to of G(n), its entries G(n)(i,j) and its spectral radius rho(G(n)). Extended versions of their joint law with the direction of the random walk G(n)x are also established, where x is a starting point in the unit sphere of R-d.

Consider the random matrix products $G_n: = g_n \ldots g_1$, where $(g_{n})_{n\geq 1}$ is a sequence of independent and identically distributed positive random $d\times d$ matrices for any integer $d \geq 2$. For any starting point $x \in \mathbb R_+^d$ with $|x| = 1$ and $y \geq 0$, consider the exit time $\tau_{x, y} = \inf \{ k \geq 1: y + \log |G_k x| < 0 \}$. In this paper, we study the conditioned local probability $\mathbb P ( y + \log |G_n x| \in [0, \Delta] + z, \, \tau_{x, y} > n )$ under various assumptions on $y$ and $z$. For $y = o(\sqrt{n})$, we prove precise upper and lower bounds when $z$ is in a compact interval and give exact asymptotics when $z \to \infty$. We also study the case when $y \asymp \sqrt{n}$ and establish the corresponding asymptotics in function of the behaviour of $z$.

Consider (gn)n>1 a sequence of independent and identically distributed random matrices and the left random walk Gn := gn . . . g1, n > 1 on the general linear group GL(d, R). Under suitable conditions we establish a Bahadur-Rao-Petrov type large deviation expansion for the coefficient (f, Gnv) of the product Gn, where v & epsilon; Rd and f & epsilon; (Rd)*. In particular, we obtain an explicit rate function in the large deviation principle, thus im-proving significantly the known large deviation bounds. A local limit theorem with large deviations for the coefficients and large deviation expansions under the change of probability measure are also proved.

Let (Zn) be a supercritical branching process in an independent and identically distributed random environment xi. We deduce the exact decay rate of the probability P(Zn = j|Z0 = k) as n -> infinity, for each j >= k, assuming that P(Z1 = 0) = 0. We also study the existence of harmonic moments of the random variable W = limn ->infinity Zn E(Zn|xi) under a simple moment condition.

Consider a multitype branching process in a random environment, whose reproduction law of generation n depends on the random environment at time n, unlike a constant distribution assumed in the Galton-Watson process. The famous Kesten-Stigum theorem for a supercritical multitype Galton-Watson process gives a precise description of the exponential increasing rate of the population size via a criterion for the nondegeneracy of the fundamental mar-tingale. Finding the corresponding result in the random environment case is a longstanding problem. For the single-type case the problem has been solved by Athreya and Karlin for the sufficiency (Ann. Math. Stat. 42 (1971) 1499- 1520) and Tanny for the necessity (Stochastic Process. Appl. 28 (1988) 123- 139), but for the multitype case it has been open for 50 years. Here we solve this problem in the typical case, by constructing a suitable martingale which reduces to the fundamental one in the constat environment case, and by es-tablishing a criterion for the nondegeneracy of its limit. The convergence in law of the direction of the branching process is also considered. Our results open ways in establishing other limit theorems, such as law of large numbers, central limit theorems, Berry-Essen bound, and large deviation results.