Large Deviations and Ruin Probabilities for Solutions to Stochastic Recurrence Equations with Heavy-tailed Innovations

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 EAt < 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(supn≥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.