Process Level Large Deviations for a Class of Piecewise Homogeneous Random Walks

Abstract

The theory of large deviations is one of the most quickly developing branches of modern probability theory (see books [2] and [3] and the references given there). It is possible to distinguish the theorems on large deviations of several levels. Theorems of the first level describe the asymptotics for k → ∞ of large deviations of the sums s _k = ξ₁ + … + ξ _k of random vectors ξ _i ∈ ℝ^l. Theorems of the process level describe the asymptotics for k → ∞ of large deviations of the probability distributions of trajectories (s ₁, …, s _k ) of the random process generated by these sums. Strongly simplifying, it is possible to say (see details in Section 2) that these theorems state that, if we consider the random polygon line S = S(t), t ∈ [0, T], such that $S(\tfrac{j}{k}) = \tfrac{{{s_j}}}{k}$ and the function S(t) is linear on the intervals $[\tfrac{j}{k},\tfrac{{j + 1}}{k}]$, then for the Borel subsets B of the space C _[0,T](ℝ^l) of continuous functions on [0,T] with values in R ^l, the probability

$$Pr(S \in B) \approx exp\{ - k{\text{ }}\mathop {inf}\limits_{S \in B} N(S)\} $$

(1.1)

for large k, where N(S) is a function which is called the rate function.