A Limit Theorem for Integral Functionals of an Extremum of Independent Random Processes

Abstract

We prove a theorem on the convergence of integral functionals of an extremum of independent stochastic processes to a degenerate law of distributions.space ( Ω, A, P ) ; here, T is a measurable set on the real straight line R.We define an integral functional of a measurable function x t ( ) according to the formulawhere h t s ( , ) is a continuous function on T × R and μ is the Lebesgue measure. Let H (s) be a certain fixed continuous function such that H s ( ) > 0. By ᑠ H we denote the class of integral functionals of the form (1) for whichAssume that finite-dimensional distributions of the random processes Y t n ( ) converge to a finite-dimensional distribution of the random process Y t ( ). It is natural to pose the problem of conditions for the conver-In this general statement, the problem of the convergence of integral functionals of random processes is well studied (see [1][2][3]). Note that there exist important examples of integral functionals that are not involved in the scheme presented above. These are, e.g., integral functionals of an extremum of independent random functions under condition (6). The specific feature of this case is the fact that the limit process has independent values at every point and is not measurable in the corresponding space. This fact does not allow one to use the traditional approach, namely the investigation of conditions for the weak convergence of measures in functional spaces [1].It is known that the theory of extreme values in the one-dimensional case is a well-developed part of probability theory (see, e.g., [4 -6]). At the same time, the number of publications related to multidimensional extrema is much smaller [5], and the infinite-dimensional case has not yet been systematically considered.In view of the significance of the maximum scheme, it seems important to study limit theorems for extreme values of a sequence of independent random functions.To author's knowledge, there are few works devoted to the investigation of extrema in the infinite-dimensional case. One of the first of them was the paper [7], where the weak convergence of a maximum of Brownian-motion processes normalized in a special way to an extreme process in the space C [ , ] 0 1 was established.