Some important problems of probability and statistics can be reduced to the evaluation of supremum of some homogeneous functional defined on the Strasses ball in the space of smooth functions on the square. We give the solution of this extremal problem in two particular cases: when the...

In this article sharp asymptotics for the solution of nonhomogeneous Kolmogorov, Petrovskii and Pisciunov equation depending on a small parameter are considered when the initial condition is the characteristic function of a set . We show how to extend the Ben Arous and Rouault's result that...

Let Xk be a sequence of i.i.d. random variables taking values in a finite set, and consider the problem of estimating the law of X1 in a Bayesian framework. We prove that the sequence of posterior distributions satisfies a large deviation principle, and give an explicit expression for the rate...

Let Un be a second-degree, nondegenerate, zero mean U-statistic with a bounded kernel, scaled so that Un/[radical sign]n => N(0, [sigma]2). Large deviation approximations are developed for tail probabilities P(Un > x[radical sign]n) using a new explicit tilting procedure.

Suppose is a sequence of random variables such that the probability law of Yn/n satisfies the large deviation principle and suppose . Let T(A)=inf{n: Yn[set membership, variant]A} be the first passage time and, to obtain a suitable scaling, let T[var epsilon](A)=[var epsilon]inf{n: Yn[set...

We point out that some general rate function representations in large deviation theory can be proved using Varadhan's integral lemma only.