Related papers: Limit theorems for random walks

We introduce random walks in a sparse random environment on $\mathbb Z$ and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent…

We prove a shape theorem and derive a variational formula for the limiting quenched Lyapunov exponent and the Green's function of random walk in a random potential on a square lattice of arbitrary dimension and with an arbitrary finite set…

We consider a certain sequence of random walks. The state space of the n-th random walk is the set of all strict partitions of n (that is, partitions without equal parts). We prove that, as n goes to infinity, these random walks converge to…

We prove that the Birkhoff sum S(n)/n = (1/n) sum_(k=1)^(n-1) g(k A) with g(x) = cot(Pi x) and golden ratio A converges in the sense that the sequence of functions s(x) = S([ x q(2n)])/q(2n) with Fibonacci numbers q(n) converges to a self…

Let $(X_k,\xi_k)_{k\in \mathbb {N}}$ be a sequence of independent copies of a pair $(X,\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $\xi$ is a positive random variable. The random process with…

We map the problem of diffusion in the quenched trap model onto a new stochastic process: Brownian motion which is terminated at the coverage "time" ${\cal S}_\alpha=\sum_{x=-\infty} ^\infty (n_x)^\alpha$ with $n_x$ being the number of…

We consider a specific random graph which serves as a disordered medium for a particle performing biased random walk. Take a two-sided infinite horizontal ladder and pick a random spanning tree with a certain edge weight $c$ for the…

We consider integer-valued random walks with independent but not identically distributed increments, and extend to this context several classical estimates, including a local limit theorem, precise small-ball estimates (both conditional on…

First, we prove a \emph{local almost sure central limit theorem} for lattice random walks in the plane. The corresponding version for random walks in the line was considered by the author in \cite{5}. This gives us a quantitative version of…

We prove that, after suitable rescaling, the simple random walk on the trace of a large critical branching random walk converges to the Brownian motion on the integrated super-Brownian excursion.

Non-trivial linear bounds are obtained for the displacement of a random walk in a dynamic random environment given by a one-dimensional simple symmetric exclusion process in equilibrium. The proof uses an adaptation of multiscale…

We study the asymptotic behavior of the simple random walk on oriented versions of $\mathbb{Z}^2$. The considered lattices are not directed on the vertical axis but unidirectional on the horizontal one, with random orientations whose…

We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on $\mathbb{Z}^d$. There exist variational formulae for the quenched and averaged rate functions $I_q$ and $I_a$, obtained by…

We consider a modification of classical branching random walk, where we add i.i.d. perturbations to the positions of the particles in each generation. In this model, which was introduced and studied by Bandyopadhyay and Ghosh (2023),…

A class of discrete time random walks has recently been introduced to provide a stochastic process based numerical scheme for solving fractional order partial differential equations, including the fractional subdiffusion equation. Here we…

We study products of random isometries acting on Euclidean space. Building on previous work of the second author, we prove a local limit theorem for balls of shrinking radius with exponential speed under the assumption that a Markov…

Given a random walk $(S_n)$ with typical step distributed according to some fixed law and a fixed parameter $p \in (0,1)$, the associated positively step-reinforced random walk is a discrete-time process which performs at each step, with…

In this note, we prove without using Fourier analysis that the symmetric square integrable random walks in $\Z^{2}$ are recurrent.

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version…

A crinkled subordinator is an $\ell^2$-valued random process which can be thought of as a version of the usual one-dimensional subordinator with each out of countably many jumps being in a direction orthogonal to the directions of all other…