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

Let X1,X2,…X1,X2,… be a sequence of random vectors taking values in Rd. Let AA be a random d′×dd′×d matrix which is independent of the process {Xn}{Xn}. Suppose that {Xn}{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 {AXn}{AXn}. The case where AA is deterministic is studied in more detail in the framework of the Gärtner–Ellis theorem. The results are applied to a ruin problem.