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

摘要

Let X_1,X_2,… be a sequence of random vectors taking values in . Let be a random d′×d matrix which is independent of the process {Xn}. Suppose that {Xn} satisfies the large deviations upper or lower bounds with a convex rate function. Starting with this, we derive large deviations statements for the mixture . The case where is deterministic is studied in more detail in the framework of the Gortner–Ellis theorem. The results are applied to a ruin problem.