Large Deviation Principles for Random Walk Trajectories. II

Abstract

The present paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 56 (2012), pp. 538--561]. It consists of three sections. Section 3 presents an example showing that it is necessary to extend the problem setup and the very concept of the “large deviations principle” (l.d.p.). We introduce a new, extended function space, a metric in it, and a deviation functional (integral) of a more general form (compared to the usual one) that will be used to construct an “extended'' l.d.p. In section 4 we present and prove the main results of the paper for trajectories of univariate random walks in the space ${\mathbb D}$ of functions without discontinuities of the second kind: the local and extended l.d.p.'s. Section 5 extends all the results from section 4 to the multivariate case.