RENEWAL THEOREM FOR （L, 1）-RANDOM WALK IN RANDOM ENVIRONMENT

We consider a random walk on Z in random environment with possible jumps {-L,…, -1, 1}, in the case that the environment {ωi ： i ∈ Z} are i.i.d.. We establish the renewal theorem for the Markov chain of ＂the environment viewed from the particle＂ in both annealed probability and quenched probability, which generalize partially the results of Kesten （1977） and Lalley （1986） for the nearest random walk in random environment on Z, respectively. Our method is based on （L, 1）-RWRE formulated in Hong and Wang the intrinsic branching structure within the （2013）.     

Light-tailed behavior of stationary distribution for state-dependent random walks on a strip

We consider the state-dependent reflecting random walk on a half- strip. We provide explicit criteria for （positive） recurrence, and an explicit expression for the stationary distribution. As a consequence, the light-tailed behavior of the stationary distribution is proved under appropriate conditions. The key idea of the method employed here is the decomposition of the trajectory of the random walk and the main tool is the intrinsic branching structure buried in the random walk on a strip, which is different from the matrix-analytic method.     

Mean and variance of first passage time of non-homogeneous random walk

The prime concern of this paper is the first passage time of a non-homogeneous random walk, which is nearest neighbor but able to stay at its position. It is revealed that the branching structure of the walk corresponds to a 2-type non-homogeneous branching process and the first passage time of the walk can be expressed by that branching process. Therefore, one can calculate the mean and variance of the first passage time, though its exact distribution is unknown.     

On the distribution density of the supremum of a random walk in the subexponential case

We consider a random walk {S _n } with negative drift and heavy-tailed jumps. We study the asymptotic behavior at infinity of the distribution density of the supremum supn S _n .     

An iterative approximation procedure for the distribution of the maximum of a random walk

Let I(F) be the distribution function (d.f.) of the maximum of a random walk whose i.i.d. increments have the common d.f. F and a negative mean. We derive a recursive sequence of embedded random walks whose underlying d.f.'s F_k converge to the d.f. of the first ladder variable and satisfy FF₁F₂ on [0,∞) and I(F)=I(F₁)=I(F₂)=. Using these random walks we obtain improved upper bounds for the difference of I(F) and the d.f. of the maximum of the random walk after finitely many steps.     

Random walk on the random connection model

We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like ${|x?y|}^{?\alpha }$ with $\alpha >d$, where $d$ denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits similar properties as classical discrete long-range percolation models studied by Berger (2002) with regard to recurrence and transience of the random walk. Moreover, we address a question which is related to a conjecture by Heydenreich, Hulshof and Jorritsma (2017) for this graph.     

On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice

In this paper,we form a method to calculate the probability generating function of the total progeny of multitype branching process.As examples,we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and(2-1) random walk.Consequently,we could give the probability generating functions and the distributions of the first passage time of corresponding random walks.Especially,for recurrent random walk which could stay at its position with probability 0 r 1,we show that the tail probability of the first passage time decays as 2/(π(1-r)~(1/2)) n~(1/1)= when n →∞.     

On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk

Consider a branching random walk on the real line. Madaule (2016) showed the renormalized trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. Besides, Chen (2015) proved that the renormalized trajectory leading to the leftmost individual at time $n$ converges in law to a standard Brownian excursion. In this article, we prove that the renormalized trajectory of an individual selected according to a supercritical Gibbs measure also converges in law toward the Brownian excursion. Moreover, refinements of this results enables to express the probability for the trajectories of two individuals selected according to the Gibbs measure to have split before time $t$, partially answering a question of Derrida and Spohn (1988).     

Moments and distribution of the local time of a two-dimensional random walk

Let ?(n,x)$?(n,x)$ be the local time of a random walk on Z²${\mathbb{Z}}^2$. We prove a strong law of large numbers for the quantity _Ln(α)=_∑x∈Z2?(n,x)^α$L_n\left(\alpha \right)={\sum }_{x\in {\mathbb{Z}}^2}?{(n,x)}^{\alpha }$ for all α≥0$\alpha \ge 0$. We use this result to describe the distribution of the local time of a typical point in the range of the random walk.     

Some sample path properties of a random walk on the cube

Some properties of the set of vertices not visited by a random walk on the cube are considered. The asymptotic distribution of the first timeQ this set is empty is derived. The distribution of the number of vertices not visited is found for times nearEQ. Next the first time all unvisited vertices are at least some distanced apart is explored. Finally the expected time taken by the path to come within a distanced of all points is calculated. These results are compared to similar results for random allocations.