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July 2013 Large deviations for solutions to stochastic recurrence equations under Kesten’s condition
D. Buraczewski, E. Damek, T. Mikosch, J. Zienkiewicz
Ann. Probab. 41(4): 2755-2790 (July 2013). DOI: 10.1214/12-AOP782


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.


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D. Buraczewski. E. Damek. T. Mikosch. J. Zienkiewicz. "Large deviations for solutions to stochastic recurrence equations under Kesten’s condition." Ann. Probab. 41 (4) 2755 - 2790, July 2013.


Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1283.60043
MathSciNet: MR3112931
Digital Object Identifier: 10.1214/12-AOP782

Primary: 60F10
Secondary: 60G70 , 91B30

Keywords: large deviations , ruin probability , stochastic recurrence equation

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 4 • July 2013
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