Long Strange Segments, Ruin Probabilities and the Effect of Memory on Moving Average Processes

TL;DR

This paper analyzes how the memory effect in infinite moving average processes influences the growth of long segments and ruin probabilities, with explicit rate calculations showing similarity to i.i.d. processes under certain conditions.

Contribution

It provides explicit rates for long segment growth and ruin probabilities in moving average processes, highlighting the impact of coefficient decay on memory effects.

Findings

Rates are similar to i.i.d. processes when coefficients decay fast.

Slower decay leads to significantly different rates.

Memory effects are linked to the decay rate of moving average coefficients.

Abstract

We obtain the rate of growth of long strange segments and the rate of decay of infinite horizon ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as the moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients.