Ion Grama
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View article: Local limit theorems for conditioned random walks by the heat kernel approximation
Local limit theorems for conditioned random walks by the heat kernel approximation Open
We study the random walk $(S_n)_{n\geq 1}$ 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 $x \in \mathbb{R}$, let $τ_x = …
View article: Spinal decomposition, martingale convergence and the Seneta-Heyde scaling for matrix branching random walks
Spinal decomposition, martingale convergence and the Seneta-Heyde scaling for matrix branching random walks Open
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 spina…
View article: Limit theorems for critical branching processes in a finite state space Markovian environment
Limit theorems for critical branching processes in a finite state space Markovian environment Open
Let $(Z_n)_{n\geq 0}$ be a critical branching process in a random environment defined by a Markov chain $(X_n)_{n\geq 0}$ with values in a finite state space $\mathbb X$. Let $ S_n = \sum_{k=1}^n \ln f_{X_k}'(1)$ be the Markov walk associa…
View article: Asymptotic Equivalence for Nonparametric Generalized Linear Models
Asymptotic Equivalence for Nonparametric Generalized Linear Models Open
We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance $Δ$; the models are then asymptotical…
View article: Asymptotic Equivalence for Nonparametric Regression
Asymptotic Equivalence for Nonparametric Regression Open
We consider a nonparametric model $\mathcal{E}^{n},$ generated by independent observations $X_{i},$ $i=1,...,n,$ with densities $p(x,θ_{i}),$ $i=1,...,n,$ the parameters of which $θ_{i}=f(i/n)\in Θ$ are driven by the values of an unknown f…
View article: A functional Hungarian construction for sums of independent random variables
A functional Hungarian construction for sums of independent random variables Open
We develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions $f$ from a class $\mathcal{H}$, but the supremum over $f\in $ $\mathcal{H}$ is…
View article: PRODUCTS OF POSITIVE RANDOM MATRICES AND MULTI-TYPE BRANCHING PROCESSES IN RANDOM ENVIRONMENTS: MOMENTS AND LARGE DEVIATIONS
PRODUCTS OF POSITIVE RANDOM MATRICES AND MULTI-TYPE BRANCHING PROCESSES IN RANDOM ENVIRONMENTS: MOMENTS AND LARGE DEVIATIONS Open
International audience
View article: Gaussian heat kernel asymptotics for conditioned random walks
Gaussian heat kernel asymptotics for conditioned random walks Open
Consider a random walk $S_n=\sum_{i=1}^n X_i$ 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 $x\in \mathbb R$, let $τ_x…
View article: Conditioned random walks on linear groups II: local limit theorems
Conditioned random walks on linear groups II: local limit theorems Open
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 …
View article: Conditioned random walks on linear groups I: construction of the target harmonic measure
Conditioned random walks on linear groups I: construction of the target harmonic measure Open
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 marks the initial stride toward achieving this goa…
View article: Construction d'un espace de Banach pour le produit de matrices aléatoires
Construction d'un espace de Banach pour le produit de matrices aléatoires Open
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 o…
View article: LIMIT THEOREMS FOR MULTITYPE BRANCHING PROCESSES IN RANDOM ENVIRONMENTS AND PRODUCTS OF POSITIVE RANDOM MATRICES
LIMIT THEOREMS FOR MULTITYPE BRANCHING PROCESSES IN RANDOM ENVIRONMENTS AND PRODUCTS OF POSITIVE RANDOM MATRICES Open
Let $Z_n^x =(Z^x_n (1), \cdots, Z^x_n (d))$ be a supercritical d-type branching process in an independent and identically distributed random environment $(\xi_n)$, starting with $Z_0=x \in \bb N^d\setminus \{0\}$, whose offspring distribut…
View article: Conditioned local limit theorems for products of positive random matrices
Conditioned local limit theorems for products of positive random matrices Open
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 de…
View article: Conditioned limit theorems for hyperbolic dynamical systems
Conditioned limit theorems for hyperbolic dynamical systems Open
Let $({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$ . Consider the Birkhoff sums $S_n f = \sum _{k=0}^{n-…
View article: Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group
Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group Open
Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq 1$, where $(g_n)_{n\geq 1}$ is a sequence of independent and identically distributed random elements with law $μ$ on the general linear group ${\rm GL}(V)$ with $V=\mathbb R^d$. Un…
View article: Edgeworth expansion for the coefficients of random walks on the general linear group
Edgeworth expansion for the coefficients of random walks on the general linear group Open
Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically distributed random elements with law $μ$ on the general linear group $\textup{GL}(V)$, where $V=\mathbb R^d$. Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq 1$.…
View article: Edgeworth expansion and large deviations for the coefficients of products of positive random matrices
Edgeworth expansion and large deviations for the coefficients of products of positive random matrices Open
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 B…
View article: The extremal position of a branching random walk on the general linear group
The extremal position of a branching random walk on the general linear group Open
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…
View article: Cram\'{e}r's moderate deviations for martingales with applications
Cram\'{e}r's moderate deviations for martingales with applications Open
Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $X_n=\sum_{i=1}^n \xi_i $ and $ \langle X \rangle_n=\sum_{i=1}^n \mathbf{E}(\xi_i^2|\mathcal{F}_{i-1}).$ We prove Cram\'er's moderate deviation expansions fo…
View article: Limit theorems for critical branching processes in a finite-state-space Markovian environment
Limit theorems for critical branching processes in a finite-state-space Markovian environment Open
Let $(Z_n)_{n\geq 0}$ be a critical branching process in a random environment defined by a Markov chain $(X_n)_{n\geq 0}$ with values in a finite state space $\mathbb{X}$ . Let $ S_n = \sum_{k=1}^n \ln f_{X_k}^{\prime}(1)$ be the Markov wa…
View article: Convergence in $$L^p$$ for a Supercritical Multi-type Branching Process in a Random Environment
Convergence in $$L^p$$ for a Supercritical Multi-type Branching Process in a Random Environment Open
International audience
View article: LIMIT THEOREMS FOR THE COEFFICIENTS OF RANDOM WALKS ON THE GENERAL LINEAR GROUP
LIMIT THEOREMS FOR THE COEFFICIENTS OF RANDOM WALKS ON THE GENERAL LINEAR GROUP Open
Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically distributed random elements with law $\mu$ on the general linear group $\textup{GL}(V)$, where $V=\mathbb R^d$. Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq 1…
View article: Limit theorems for the coefficients of random walks on the general linear group
Limit theorems for the coefficients of random walks on the general linear group Open
Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically distributed random elements with law $μ$ on the general linear group $\textrm{GL}(V)$, where $V=\mathbb R^d$. Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq 1$,…
View article: Conditioned limit theorems for hyperbolic dynamical systems
Conditioned limit theorems for hyperbolic dynamical systems Open
Let $(\mathbb X, T)$ be a subshift of finite type equipped with the Gibbs measure $ν$ and let $f$ be a real-valued Hölder continuous function on $\mathbb X$ such that $ν(f) = 0$. Consider the Birkhoff sums $S_n f = \sum_{k=0}^{n-1} f \circ…
View article: Berry–Esseen bound and precise moderate deviations for products of random matrices
Berry–Esseen bound and precise moderate deviations for products of random matrices Open
Let {(g_{n})_{n\geq 1}} be a sequence of independent and identically distributed (i.i.d.) {d\times d} real random matrices. For {n\geq 1} set {G_n = g_n \ldots g_1} . Given any starting point {x=\mathbb R v\in\mathbb{P}^{d-1}} , consider t…
View article: Conditioned local limit theorems for random walks on the real line
Conditioned local limit theorems for random walks on the real line Open
Consider a random walk $S_n=\sum_{i=1}^n X_i$ 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 $2+δ$. For any $x\geq 0$, …
View article: Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices
Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices Open
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 …
View article: Large deviation expansions for the coefficients of random walks on the general linear group
Large deviation expansions for the coefficients of random walks on the general linear group Open
Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically distributed elements of the general linear group $GL(d, \mathbb R)$. Consider the random walk $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish Bahadur-Rao-P…
View article: BERRY -ESSEEN BOUND AND CRAMÉR MODERATE DEVIATION EXPANSION FOR A SUPERCRITICAL BRANCHING RANDOM WALK
BERRY -ESSEEN BOUND AND CRAMÉR MODERATE DEVIATION EXPANSION FOR A SUPERCRITICAL BRANCHING RANDOM WALK Open
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 $Z_ n$ be the counting measure which count…