Convex large deviation rate functions under mixtures of linear transformations, with an application to ruin theory

Abstract

Let X 1 , X 2 , … be a sequence of random vectors taking values in R d . Let A be a random d ′ × d matrix which is independent of the process { X n } . Suppose that { X n } satisfies the large deviations upper or lower bounds with a convex rate function. Starting with this, we derive large deviations statements for the mixture { A X n } . The case where A is deterministic is studied in more detail in the framework of the Gärtner–Ellis theorem. The results are applied to a ruin problem.