The Annals of Probability

Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations

Dimitrios G. Konstantinides and Thomas Mikosch

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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.

Article information

Source
Ann. Probab., Volume 33, Number 5 (2005), 1992-2035.

Dates
First available in Project Euclid: 22 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1127395879

Digital Object Identifier
doi:10.1214/009117905000000350

Mathematical Reviews number (MathSciNet)
MR2165585

Zentralblatt MATH identifier
1085.60017

Subjects
Primary: 60F10: Large deviations
Secondary: 91B30: Risk theory, insurance 60G70: Extreme value theory; extremal processes 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

Keywords
Stochastic recurrence equation large deviations regular variation ruin probability

Citation

Konstantinides, Dimitrios G.; Mikosch, Thomas. Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Probab. 33 (2005), no. 5, 1992--2035. doi:10.1214/009117905000000350. https://projecteuclid.org/euclid.aop/1127395879


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