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

We consider a continuous time version of Cramer’s theorem with nonnegative summands $S_t=\frac{1}{t}{\sum }_{i:{\tau }_i\le t}{\xi }_i,t\to \mathrm{\infty },$ where $({\tau }_i,{\xi }_i{)}_{i\ge 1}$ is a sequence of random variables such that $t{S}_t$ is a random process with independent increments.