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We consider dynamic random walks where the nearest neighbour jump rates are determined by an underlying supercritical contact process in equilibrium. This has previously been studied by den Hollander and dos Santos and den Hollander, dos Santos, Sidoravicius. We show the CLT for such a random walk, valid for all supercritical infection rates for the contact process environment.
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In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ${{mathbb{Z}}^d}$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster ${{mathcal{C}}_{infty}}$ of a supercritical Bernoulli percolation. When no more than $k$ individuals per site are allowed, we obtain the $k$-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already $k$ individuals are present. We prove that local survival of the branching random walk on ${{mathbb{Z}}^d}$ also implies that for $k$ sufficiently large the associated $k$-type contact process survives on ${{mathcal{C}}_{infty}}$. This implies that the strong critical parameters of the branching random walk on ${{mathbb{Z}}^d}$ and on ${{mathcal{C}}_{infty}}$ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated $k$-type contact processes. These results are extended to a family of restrained branching random walks, that is, branching random walks where the success of the reproduction trials decreases with the size of the population in the target site.
In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,tin[0,1]}$ with begin{equation*} W_n(s,t):=sum_{k=1}^{lfloor ntrfloor}big(1_{{xi_{S_k}leq s}}-sbig) end{equation*} where $(xi_x, xinmathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{ninmathbb N}$ is a random walk evolving in $mathbb{Z}^d$, independent of the $xi$s. In Wendler (2016), the case where $(S_n)_{ninmathbb N}$ is a recurrent random walk in $mathbb{Z}$ such that $(n^{-frac 1alpha}S_n)_{ngeq 1}$ converges in distribution to a stable distribution of index $alpha$, with $alphain(1,2]$, has been investigated. Here, we consider the cases where $(S_n)_{ninmathbb N}$ is either: a) a transient random walk in $mathbb{Z}^d$, b) a recurrent random walk in $mathbb{Z}^d$ such that $(n^{-frac 1d}S_n)_{ngeq 1}$ converges in distribution to a stable distribution of index $din{1,2}$.
A classical result for the simple symmetric random walk with $2n$ steps is that the number of steps above the origin, the time of the last visit to the origin, and the time of the maximum height all have exactly the same distribution and converge when scaled to the arcsine law. Motivated by applications in genomics, we study the distributions of these statistics for the non-Markovian random walk generated from the ascents and descents of a uniform random permutation and a Mallows($q$) permutation and show that they have the same asymptotic distributions as for the simple random walk. We also give an unexpected conjecture, along with numerical evidence and a partial proof in special cases, for the result that the number of steps above the origin by step $2n$ for the uniform permutation generated walk has exactly the same discrete arcsine distribution as for the simple random walk, even though the other statistics for these walks have very different laws. We also give explicit error bounds to the limit theorems using Steins method for the arcsine distribution, as well as functional central limit theorems and a strong embedding of the Mallows$(q)$ permutation which is of independent interest.
Let $U_1,U_2,ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $ntoinfty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$-vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Barany, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone $C_n$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $|x|^{-(d+gamma)}$, where $gamma=1$. We compute the expected number of facets, the expected intrinsic volumes and the expected $T$-functional of this random convex hull for arbitrary $gamma>0$.
In this paper, we analyse a sub-class of two-dimensional homogeneous nearest neighbour (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions [30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions [13]. Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product-forms and, for this class of random walks, we calculate the eigenvalues and the corresponding eigenvectors of the infinite matrix $mathbf{R}$ appearing in the matrix geometric approach. This result is obtained by connecting and extending three existing approaches available for such an analysis: the matrix geometric approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral properties of the infinite matrix $mathbf{R}$.
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