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and EAκ t < 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 .W e 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.

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