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

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(Sn−ESn>x), x≥xn, for some sequence xn→∞ whose growth depends on the heaviness of the tail of the distribution of Y1. We show that the relation P(Sn−ESn>x)∼c2nP(Y1>x) holds uniformly for x≥xn 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((Sn−ESn)−μn)>u) for any μ>0. We show that ψ(u)∼c3uP(Y1>u)μ−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.