Large deviations for solutions to stochastic recurrence equations under Kesten's condition - Danish National Research Database-Den Danske Forskningsdatabase

Buraczewski, Dariusz^{2}; Damek, Ewa^{2}; Mikosch, Thomas Valentin^{4}; Zienkiewicz, J.^{3}

Affiliations:

^{1} Department of Mathematical Sciences, Faculty of Science, Københavns Universitet^{2} Wroclaw University, Department of Mathematics^{3} Wroclaw University^{4} Department of Mathematical Sciences, Faculty of Science, Københavns Universitet

DOI:

10.1214/12-AOP782

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 [17] 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 of those obtained by A.V. and S.V. Nagaev [21, 22] in the case of partial sums of iid 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. (1.1)

Type:

Journal article

Language:

English

Published in:

Annals of Probability, 2013, Vol 41, Issue 4, p. 2755-2790