Functional Erdd Os-renyi Laws for Semiexponential Random Variables

For an i.i.d. sequence of random variables with a semiexponential distribution, we give a functional form of the Erdd os-Renyi law for partial sums. In contrast to the classical case, i.e. the case where the random variables have exponential moments of all orders, the set of limit points is not a subset of the continuous functions. This reeects the bigger innuence 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 diiers 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.