Large deviations for solutions to stochastic recurrence equations under Kesten's condition

TL;DR

This paper establishes large deviation principles for solutions to stochastic recurrence equations with power law tails under Kesten's condition, revealing how extreme value clusters influence tail behavior and ruin probabilities.

Contribution

It extends large deviation results to dependent sequences from stochastic recurrence equations, highlighting the role of extreme value clustering under Kesten's condition.

Findings

Large deviations are governed by the tail of the maximum summand.

Clusters of extreme values significantly influence tail asymptotics.

Results apply to ruin probability analysis in stochastic models.

Abstract

In this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation. We assume Kesten's condition [Acta Math. 131 (1973) 207-248] under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails, while the noise sequence of the equations can have light tails. The results of the paper are analogs to those obtained by A. V. Nagaev [Theory Probab. Appl. 14 (1969) 51-64; 193-208] and S. V. Nagaev [Ann. Probab. 7 (1979) 745-789] in the case of partial sums of i.i.d. random variables. In the latter case, the large deviation probabilities of the partial sums are essentially determined by the largest step size of the partial sum. For the solution to a stochastic recurrence equation, the magnitude of the large deviation probabilities is again given by the tail of the maximum summand, but the exact asymptotic tail behavior is also influenced by clusters of extreme values, due to dependencies in the sequence. We apply the large deviation results to study the asymptotic behavior of the ruin probabilities in the model.