It is known that large deviations of sums of subexponential random variables are most likely realised by deviations of a single random variable. In this article we give a detailed picture of how subexponential random variables are distributed when a large deviation of the sum is observed.

We prove that for any [alpha]-mixing stationary process the hitting time of any n-string An converges, when suitably normalized, to an exponential law. We identify the normalization constant [lambda](An). A similar statement holds also for the return time. To establish this result we prove two...

There exists a wide literature on parametrically or semi-parametrically modelling strongly dependent time series using a long-memory parameter d, including more recent work on wavelet estimation. As a generalization of these latter approaches, in this work we allow the long-memory parameter d to...

A Markov process on a local field which can be projected to a Markov process on a smaller local field is regarded as a lift of the one on the smaller field. The first part of this article is concerned with a Markov process on a local field which is obtained as the one projected from a larger...

In this paper, we prove some limit theorems for the Fourier estimator of multivariate volatility proposed by Malliavin and Mancino (2002, 2009) [14] and [15]. In a general framework of discrete time observations we establish the convergence of the estimator and some associated central limit...

A small world is obtained from the d-dimensional torus of size 2L adding randomly chosen connections between sites, in a way such that each site has exactly one random neighbour in addition to its deterministic neighbours. We study the asymptotic behaviour of the meeting time TL of two random...

The goal of this paper is to show that under some assumptions, for a d-dimensional fractional Brownian motion with Hurst parameter H>1/2, the density of the solution of the stochastic differential equation admits the following asymptotics at small times:

Free quadratic harness is a Markov process from the class of quadratic harnesses, i.e. processes with linear regressions and quadratic conditional variances. The process has recently been constructed for a restricted range of parameters in Bryc et al. (2010) [7] using Askey-Wilson...

We study the contact process on the complete graph on n vertices where the rate at which the infection travels along the edge connecting vertices i and j is equal to [lambda]wiwj/n for some [lambda]0, where wi are i.i.d. vertex weights. We show that when there is a phase transition at...

We establish a multivariate empirical process central limit theorem for stationary -valued stochastic processes (Xi)i=1 under very weak conditions concerning the dependence structure of the process. As an application, we can prove the empirical process CLT for ergodic torus automorphisms. Our...