On the tail of the branching random walk local time

Consider a critical branching random walk on $$\mathbb Z^d$$ Z d , $$d\ge 1$$ d ≥ 1 , started with a single particle at the origin, and let L(x) be the total number of particles that ever visit a vertex x. We study the tail of L(x) under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.

SECOND ORDER SUBEXPONENTIALITY AND INFINITE DIVISIBILITY

We characterize the second order subexponentiality of an infinitely divisible distribution on the real line under an exponential moment assumption. We investigate the asymptotic behaviour of the difference between the tails of an infinitely divisible distribution and its Lévy measure. Moreover, we study the second order asymptotic behaviour of the tail of the $t$ th convolution power of an infinitely divisible distribution. The density version for a self-decomposable distribution on the real line without an exponential moment assumption is also given. Finally, the regularly varying case for a self-decomposable distribution on the half line is discussed.

Exponential Moments and Piecewise Thinning for the Bessel Point Process

We obtain exponential moment asymptotics for the Bessel point process. As a direct consequence, we improve on the asymptotics for the expectation and variance of the associated counting function and establish several central limit theorems. We show that exponential moment asymptotics can also be interpreted as large gap asymptotics, in the case where we apply the operation of a piecewise constant thinning on several consecutive intervals. We believe our results also provide important estimates for later studies of the global rigidity of the Bessel point process.

Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains

This article is motivated by the quantitative study of the exponential growth of Markov-driven bifurcating processes [see Hervé et al., ESAIM: PS 23 (2019) 584–606]. In this respect, a key property is the multiplicative ergodicity, which deals with the asymptotic behaviour of some Laplace-type transform of nonnegative additive functional of a Markov chain. We establish a spectral version of this multiplicative ergodicity property in a general framework. Our approach is based on the use of the operator perturbation method. We apply our general results to two examples of Markov chains, including linear autoregressive models. In these two examples the operator-type assumptions reduce to some expected finite moment conditions on the functional (no exponential moment conditions are assumed in this work).

Strong $L^2$ convergence of time numerical schemes for the stochastic two-dimensional Navier–Stokes equations

We prove that some time discretization schemes for the two-dimensional Navier–Stokes equations on the torus subject to a random perturbation converge in $L^2(\varOmega )$. This refines previous results that established the convergence only in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier–Stokes equations and convergence of a localized scheme we can prove strong convergence of fully implicit and semiimplicit temporal Euler discretizations and of a splitting scheme. The speed of the $L^2(\varOmega )$ convergence depends on the diffusion coefficient and on the viscosity parameter.

Strong convergence properties for arrays of rowwise negatively orthant dependent random variables

Let {Xni , i ≥ 1, n ≥1} be an array of rowwise negatively orthant dependent random variables which is stochastically dominated by a random variable X. Wang et al. [15. Complete convergence for arrays of rowwise negatively orthant dependent random variables, RACSAM, 106 (2012), 235–245] studied the complete convergence for arrays of rowwise negatively orthant dependent random variables under the condition that X has an exponential moment, which seems too strong. We will further study the complete convergence for arrays of rowwise negatively orthant dependent random variables under the condition that X has a moment, which is weaker than exponential moment. Our results improve the corresponding ones of Wang et al. [15].