This article is cited in 29 papers

On large deviation principles in metric spaces

A. A. Borovkov^ab, A. A. Mogul'skiĭ<sup>ab

a</sup> Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^b Novosibirsk State University, Novosibirsk

Abstract: Many articles deal with large deviation principles (LDPs) (see [1–4] for instance and the references in [3, 4]), studying mainly the LDP for the sums of random elements or for various stochastic models and dynamical systems. For a sequence of random elements in a metric space, in studying LDPs it turns out natural to introduce the concepts of the local LDP and extended LDP. They enable us to state and prove LDP-type statements in those cases when the usual LDP (cf. [3, 4]) fails (see [5, 6] and Section 6 of this article). We obtain conditions for the fulfillment of the extended LDP in metric spaces. The main among these conditions is the fulfillment of the local LDP. The latter is usually much simpler to prove than the extended LDP.

Keywords: large deviation principle, extended large deviation principle, local large deviation principle, deviation function, totally bounded set, compact set.

UDC: 519.21

Received: 01.02.2010

 English version: Siberian Mathematical Journal, 2010, 51:6, 989–1003

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