For a 1-dependent stationary sequence {Xn} we first show that if u satisfies p1=p1(u)=P(X1u)[less-than-or-equals, slant]0.025 and n3 is such that 88np13[less-than-or-equals, slant]1, thenP{max(X1,...,Xn)[less-than-or-equals, slant]u}=[nu]·[mu]n+O{p13(88n(1+124np13)+561)}, n3,where...

In the nonlinear filtering model with signal and observation noise independent, we show that the filter depends continuously on the law of the signal. We do not assume that the signal process is Markov and prove the result under minimal integrability conditions. The analysis is based on...

Consider the following bond percolation process on : each vertex is connected to each of its nearest neighbour in the vertical direction with probability pv=[var epsilon]0; and in the horizontal direction each vertex is connected to each of the vertices x±(i,0) with probability...

A stochastic process on a finite-dimensional real vector space is operator-self-similar if a linear time change produces a new process whose distributions scale back to those of the original process, where we allow scaling by a family of affine linear operators. We prove a spectral decomposition...

In this paper we study the existence and uniqueness of weak solutions of stochastic differential equations on Banach spaces. We also study the existence of invariant measures for the corresponding Markovian semigroups. Our main tool is the factorization of stochastic convolutions. We close the...

We study the tail probability of the local time at the origin of Gaussian processes with stationary increments. The order of infinitesimal is obtained.

We consider the extreme values of fractional Brownian motions, self-similar Gaussian processes and more general Gaussian processes which have a trend -ct[beta] for some constants c,[beta]0 and a variance t2H. We derive the tail behaviour of these extremes and show that they occur mainly in the...

The infinite-dimensional Ornstein-Uhlenbeck process v is constructed from Brownian motion on the infinite-dimensional sphere SN-1(1) (the Wiener sphere) - or equivalently, by rescaling, on - which is defined for infinite N by nonstandard analysis. This gives rigorous sense to the informal idea...

In this paper we study asymptotic behaviour of marked point processes describing failure processes of repairable systems in which repair decisions depend on the past. Under natural conditions on system parameters such processes admit unique time stationary distributions and are ergodic....

We prove a large deviation principle for a class of semilinear stochastic partial differential equations driven by the space-time white noise. This class of equation contains as special cases Burgers equation and the parabolic SPDEs perturbed by the space-time white noise.