For an i.i.d. sequence of random variables with a semiexponential distribution, we give a functional form of the Erdös–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 limit points is not a subset of 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.
"Functional Erdős-Renyi laws for semiexponential random variables." Ann. Probab. 26 (3) 1356 - 1369, July 1998. https://doi.org/10.1214/aop/1022855755