相关论文: Conditioned limit theorems for products of positiv…

Consider the product $G_{n}=g_{n} ... g_{1}$ of the random matrices $g_{1},...,g_{n}$ in $GL(d,\mathbb{R}) $ and the random process $ G_{n}v=g_{n}... g_{1}v$ in $\mathbb{R}^{d}$ starting at point $v\in \mathbb{R}^{d}\smallsetminus \{0\} .$…

In this short note, we study the behaviour of a product of matrices with a simultaneous renormalization. Namely, for any sequence $(A\_n)\_{n\in \mathbb{N}}$ of $d\times d$ complex matrices whose mean $A$ exists and whose norms' means are…

We consider the products $G_n = A_n \cdots A_1$ of independent and identical distributed nonnegative $d \times d$ matrices $(A_i)_{i \geq 1}$. For any starting point $x \in \mathbb{R}_+^d$ with unit norm, we establish the convergence to a…

Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices, where $d\geq 2$ is an integer. For any starting point $x \in \mathbb{R}_+^d$ with $|x| = 1$ and $y \in \mathbb R$, we…

Let $(S_n)_n$ be the random process on $\mathbb R$ driven by the product of i.i.d. non-negative random matrices and $\tau$ its exit time from $]0, +\infty[$. By using the adapted strategy initiated by D. Denisov and V. Wachtel, we obtain an…

In this paper we survey and further study partial sums of a stationary process via approximation with a martingale with stationary differences. Such an approximation is useful for transferring from the martingale to the original process the…

Under reasonable algebraic assumptions and under an infinite second order moment assumption, we show that the logarithm of the norm (log-norm) of a product of random i.i.d. matrices with entries in $\mathbb{R}$ or in any other local field…

Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices and consider the matrix product $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish the Berry-Esseen bounds…

We extend our recent result [Cipolloni, Erd\H{o}s, Schr\"oder 2019] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices $X$ with independent, identically distributed complex entries to the real…

Let A(n) be a sequence of i.i.d. topical (i.e. isotone and additively homogeneous) operators. Let $x(n,x_0)$ be defined by $x(0,x_0)=x_0$ and $x(n,x_0)=A(n)x(n-1,x_0)$. This can modelize a wide range of systems including, task graphs, train…

Consider the 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. Under the optimal third moment condition, we first establish a…

We study sums of a random multiplicative function; this is an example, of number-theoretic interest, of sums of products of independent random variables (chaoses). Using martingale methods, we establish a normal approximation for the sum…

Central limit theorems are established for the sum, over a spatial region, of observations from a linear process on a $d$-dimensional lattice. This region need not be rectangular, but can be irregularly-shaped. Separate results are…

Consider a $N\times n$ matrix $\Sigma_n=\frac{1}{\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear…

Let $\mu$ be a probability measure on $\text{GL}_d(\mathbb R)$ and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$'s are i.i.d.'s with law $\mu$. We study statistical properties of random variables of the…

For the partial sums $(S_n)$ of independent random variables we define a stochastic process $s_n(t):=(1/d_n)\sum_{k \le [nt]} ({S_k}/{k}-\mu)$ and prove that $$(1/{\log N})\sum_{n\le N}(1/n)\mathbf {I}\left\{s_n(t)\le x\right\} \to…

Fix a positive integer $d$ and let $(G_n)_{n\geq1}$ be a sequence of finite abelian groups with orders tending to infinity. For each $n \geq 1$, let $C_n$ be a uniformly random $G_n$-circulant matrix with entries in $\{0,1\}$ and exactly…

We develop a martingale approximation approach to studying the limiting behavior of quadratic forms of Markov chains. We use the technique to examine the asymptotic behavior of lag-window estimators in time series and we apply the results…

Let $\log^{2+\varepsilon} n \le d \le n/2$ for some fixed $\varepsilon \in (0,1)$, and let $M_n$ be an $n\times n$ random matrix with entries in ${0,1}$, where each row is independently and uniformly sampled from the set of all vectors in…

Consider the product $X = X_{1}\cdots X_{m}$ of $m$ independent $n\times n$ iid random matrices. When $m$ is fixed and the dimension $n$ tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of $X$ for…