Long range dependence is a very important phenomenon that has been observed in many real life situations. The large deviation principle is a very important probabilistic tool for dealing with rare events. The interaction between the two topics is investigated. We study the effect of the dependence structure of the process on large deviations in the perspective of the moving average process and the infinitely divisible process with exponentially light tails.

The large deviations of an infinite moving average process with exponentially light tails are very similar to those of an i.i.d. sequence as long as the coefficients decay fast enough. If they do not, the large deviations change dramatically. We study this phenomenon in the context of functional large, moderate and huge deviation principles. We apply the results to study the rate of growth of long strange segments and the rate of decay of ruin probabilities and the effect of memory on those.

We study the functional large deviation principle for a general class of long range dependent infinitely divisible processes driven by a null recurrent Markov chain. We also apply the principle to obtain the rate of decay of the probability of ruin for these models.