Cramer's theorem for nonnegative multivariate point processes with independent increments

Abstract

We consider a continuous time version of Cramer's theorem with nonnegative summands $ S_t=\frac{1}{t}\sum_{i:τ_i\le t}ξ_i, t \to\infty, $ where $(τ_i,ξ_i)_{i\ge 1}$ is a sequence of random variables such that $tS_t$ is a random process with independent increments.