In general, estimators of the extreme value index of i.i.d. random variables crucially depend on the sample fraction that is used for estimation. In case of the well-known Hill estimator the optimal number knopt of largest order statistics was given by Hall and Welsh (1985) as a function of some...

We show that most random walks in the domain of attraction of a symmetric stable law have a non-trivial almost sure central limit theorem with the normal law as the limit.

We consider Mandelbrot's fractal percolation process, obtained by repeated subdivision of the unit square, and obtain an explicit almost sure lower bound on the lower box-counting dimension of paths within the retained set that cross the square from left to right.

We shall address here the optimization problem of an investor who wants to maximize the expected utility from terminal wealth. The novelty of this paper is that the drift process and the driving Brownian motion appearing in the stochastic differential equation for the security prices are not...

The non-linear filtering problem consists in computing the conditional distributions of a Markov signal process given its noisy observations. The dynamical structure of such distributions can be modelled by a measure valued dynamical Markov process. Several random particle approximations were...

A limit theory was developed in the papers of Davis and Dunsmuir (1996) and Davis et al. (1995) for the maximum likelihood estimator, based on a Gaussian likelihood, of the moving average parameter in an MA(1) model when is equal to or close to 1. Using the local parameterization, , where is the...

In random environments, the most elementary processes are Sinai's simple random walk and Brox's diffusion process, respectively in discrete and continuous time settings. The two processes are often considered as a kind of companions, somewhat in the same way as the usual random walk and Brownian...

Building on and extending the results of Gr, (J. Appl. Probab. 26, 296-303), approximation formulae for solutions of renewal-type equations are derived. These are obtained by finding the first and higher Fréchet derivatives of the functional that has the underlying lifetime density as input and...

The focus in this article is on point processes on a product space that satisfy stochastic differential equations with a Poisson process as one of the driving processes. The questions we address are that of existence and uniqueness of both stationary and non stationary solutions, and convergence...

Large deviation principles are established for the Fleming-Viot processes with neutral mutation and selection, and the corresponding equilibrium measures as the sampling rate goes to 0. All results are first proved for the finite allele model, and then generalized, through the projective limit...