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The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

Funded by The Danish National Research Foundation

MPS-RR 2004-9
February 2004

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

by: Dimitrios G. Konstantinides , Thomas Mikosch


In this paper we consider the stochastic recurrence equation $Yt = At Yt 1 + Bt$ for an iid sequence of pairs $(A_t,B_t)$ of non-negative random variables, where we assume that $B_t$ is regularly varying with index $\kappa > 0$ and $EA_t^\kappa < 1$. We show that the stationary solution ($Y_t$) to this equation has regularly varying finite-dimensional distributions with index $\kappa$. This implies that the partial sums $S_n = Y_1 +\dots+Y_n$ of this process are regularly varying. In particular, the relation $P(S_n > x) \sim c_1 n P(Y_1 > x)$ as $x \to 1$ holds for some constant $c_1 > 0$. For $\kappa > 1$, we also study the large deviation probabilities $P(S_n ES_n > x)$, $x \geq x_n$, for some sequence $x_n \to 1$ whose growth depends on the heaviness of the tail of the distribution of $Y_1$. We show that the relation $P(S_n -ES_n > x) \sim c_2 nP(Y_1 > x)$ holds uniformly for $x \geq x_n$ and some constant $c_2 > 0$. Then we apply the large deviation results to derive bounds for the ruin probability $\psi(u) = P(\sup_{n\geq 1}((S_n-ES_n)-\mu n) > u)$ for any $\mu > 0$. We show that $\psi(u) \sim c_3 u P(Y_1 > u) \mu^{-1} (\kappa 1)^{-1}$ for some constant $c_3 > 0$. In contrast to the case of iid regularly varying $Y_t$'s, when the above results hold with $c_1 = c_2 = c_3 = 1$, the constants $c_1$, $c_2$ and $c_3$ are different from 1.

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This paper has now been published in Annals of Probablity (to appear)