Large deviations for solutions to stochastic recurrence equations under Kesten's condition
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Large deviations for solutions to stochastic recurrence equations under Kesten's condition. / Buraczewski, Dariusz; Damek, Ewa ; Mikosch, Thomas Valentin; Zienkiewicz, J.
I: Annals of Probability, Bind 41, Nr. 4, 2013, s. 2755-2790.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Large deviations for solutions to stochastic recurrence equations under Kesten's condition
AU - Buraczewski, Dariusz
AU - Damek, Ewa
AU - Mikosch, Thomas Valentin
AU - Zienkiewicz, J.
PY - 2013
Y1 - 2013
N2 - 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)
AB - 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)
U2 - 10.1214/12-AOP782
DO - 10.1214/12-AOP782
M3 - Journal article
VL - 41
SP - 2755
EP - 2790
JO - Annals of Probability
JF - Annals of Probability
SN - 0091-1798
IS - 4
ER -
ID: 94844134