## The Annals of Probability

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

#### 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.

#### Article information

Source
Ann. Probab., Volume 41, Number 4 (2013), 2755-2790.

Dates
First available in Project Euclid: 3 July 2013

https://projecteuclid.org/euclid.aop/1372859766

Digital Object Identifier
doi:10.1214/12-AOP782

Mathematical Reviews number (MathSciNet)
MR3112931

Zentralblatt MATH identifier
1283.60043

Subjects
Primary: 60F10: Large deviations
Secondary: 91B30: Risk theory, insurance 60G70: Extreme value theory; extremal processes

#### Citation

Buraczewski, D.; Damek, E.; Mikosch, T.; Zienkiewicz, J. Large deviations for solutions to stochastic recurrence equations under Kesten’s condition. Ann. Probab. 41 (2013), no. 4, 2755--2790. doi:10.1214/12-AOP782. https://projecteuclid.org/euclid.aop/1372859766

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