FUNCTIONAL ERDOS-RENYI LAWS FOR SEMIEXPONENTIAL RANDOM-VARIABLES

For an i.i.d. sequence of random variables with a semiexponential dist ribution, we give a functional form of the Erdos-Renyi law for partial sums. In contrast to the classical case, that is, the case where the random variables have exponential moments of all orders, the set of li mit points is not a subset elf the continuous functions. This reflects the bigger influence of extreme values. The proof is based on a large deviation principle for the trajectories of the corresponding random walk. The normalization in this large deviation principle differs from the usual normalization and depends on the tail of the distribution. In the same way, we prove a functional limit law for moving averages.