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September 2005 Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations
Dimitrios G. Konstantinides, Thomas Mikosch
Ann. Probab. 33(5): 1992-2035 (September 2005). DOI: 10.1214/009117905000000350

Abstract

In this paper we consider the stochastic recurrence equation Yt=AtYt−1+Bt for an i.i.d. sequence of pairs (At,Bt) of nonnegative random variables, where we assume that Bt is regularly varying with index κ>0 and EAtκ<1. We show that the stationary solution (Yt) to this equation has regularly varying finite-dimensional distributions with index κ. This implies that the partial sums Sn=Y1+⋯+Yn of this process are regularly varying. In particular, the relation P(Sn>x)∼c1nP(Y1>x) as x→∞ holds for some constant c1>0. For κ>1, we also study the large deviation probabilities P(SnESn>x), xxn, for some sequence xn→∞ whose growth depends on the heaviness of the tail of the distribution of Y1. We show that the relation P(SnESn>x)∼c2nP(Y1>x) holds uniformly for xxn and some constant c2>0. Then we apply the large deviation results to derive bounds for the ruin probability ψ(u)=P(sup n≥1((SnESn)−μn)>u) for any μ>0. We show that ψ(u)∼c3uP(Y1>u−1(κ−1)−1 for some constant c3>0. In contrast to the case of i.i.d. regularly varying Yt’s, when the above results hold with c1=c2=c3=1, the constants c1, c2 and c3 are different from 1.

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Dimitrios G. Konstantinides. Thomas Mikosch. "Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations." Ann. Probab. 33 (5) 1992 - 2035, September 2005. https://doi.org/10.1214/009117905000000350

Information

Published: September 2005
First available in Project Euclid: 22 September 2005

zbMATH: 1085.60017
MathSciNet: MR2165585
Digital Object Identifier: 10.1214/009117905000000350

Subjects:
Primary: 60F10
Secondary: 60G35 , 60G70 , 91B30

Keywords: large deviations , regular variation , ruin probability , stochastic recurrence equation

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 5 • September 2005
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