Journal of Theoretical Probability最新文献

We consider Bernoulli bond and site percolation on Cayley graphs of reflection groups in the three-dimensional hyperbolic space ({mathbb {H}^3}) corresponding to a very large class of Coxeter polyhedra. In such setting, we prove the existence of a non-empty non-uniqueness percolation phase, i.e. that (p_c < p_u). This means that for some values of the Bernoulli percolation parameter there are a.s. infinitely many infinite components in the percolation subgraph. The proof relies on upper estimates for the spectral radius of the graph and on a lower estimate for its growth rate. The latter estimate involves only the number of generators of the group and is proved in the article as well.

Abstract

We introduce and study a fractional version of the Skellam process of order k by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order k (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of FSPoK which are utilized to establish its long-range dependence property. Later, we consider two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent Lévy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes.

We consider a generalization of the classical coupon collector problem, where the set of available coupons consists of standard coupons (which can be part of the collection), and two coupons with special purposes: one that speeds up the collection process and one that slows it down. We obtain several asymptotic results related to the expectation and the variance of the waiting time until a portion of the collection is sampled, as the number of standard coupons tends to infinity.

We consider a variant of the generalized excited random walk (GERW) in dimension (dge 2) where the lower bound on the drift for excited jumps is time-dependent and decays to zero. We show that if the lower bound decays more slowly than (n^{-beta _0}) (n is time), where (beta _0) depends on the transitions of the process, the GERW is transient in the direction of the drift.

In this paper, we consider random variables and stochastic processes from the space ({textbf{F}}_psi (Omega )) and study approximation problems for such processes. The method of series decomposition of a stochastic process from ({textbf{F}}_psi (Omega )) is used to find an approximating process called a model. The rate of convergence of the model to the process in the uniform norm is investigated. We develop an approach for estimating the cut-off level of the model under given accuracy and reliability of the simulation.

Abstract

The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article, we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of p-variationa. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities.

The almost-sure sample path behavior of the operator fractional Brownian motion with exponent D, including multivariate fractional Brownian motion, is investigated. In particular, the global and the local moduli of continuity of the sample paths are established. These results show that the global and the local moduli of continuity of the sample paths are completely determined by the real parts of the eigenvalues of the exponent D, as well as the covariance matrix at some unit vector. These results are applicable to multivariate fractional Brownian motion.

We give rates of convergence in the almost sure invariance principle for sums of dependent random variables with semi-exponential tails, whose coupling coefficients decrease at a sub-exponential rate. We show that the rates in the strong invariance principle are in powers of (log n). We apply our results to iid products of random matrices.

In this paper, we investigate the sojourn times of conditionally Gaussian processes, i.e., the sojourns of (xi (t)+lambda -zeta ,t^beta ) and (xi (t)(lambda -zeta ,t^beta )), (tin [0, T], T>0), where (xi ) is a Gaussian zero-mean stationary process and (lambda ) and (zeta ) are random variables independent of (xi (cdot )), and (beta >0) is a constant.

In this paper, we consider a large class of super-Brownian motions in ({mathbb {R}}) with spatially dependent branching mechanisms. We establish the almost sure growth rate of the mass located outside a time-dependent interval ((-delta t,delta t)) for (delta >0). The growth rate is given in terms of the principal eigenvalue (lambda _{1}) of the Schrödinger-type operator associated with the branching mechanism. From this result, we see the existence of phase transition for the growth order at (delta =sqrt{lambda _{1}/2}). We further show that the super-Brownian motion shifted by (sqrt{lambda _{1}/2},t) converges in distribution to a random measure with random density mixed by a martingale limit.