Pointwise estimates for first passage times of perpetuity sequences

We consider first passage times τu=inf{n:Yn>u} for the perpetuity sequence Yn=B1+A1B2+⋯+(A1…An−1)Bn,where (An,Bn) are i.i.d. random variables with values in R+×R. Recently, a number of limit theorems related to τu were proved including the law of large numbers, the central limit theorem and large deviations theorems (see Buraczewski et al., in press). We obtain a precise asymptotics of the sequence P[τu=logu∕ρ], ρ>0, u→∞ which considerably improves the previous results of Buraczewski et al. (in press). There, probabilities P[τu∈Iu] were identified, for some large intervals Iu around ku, with lengths growing at least as loglogu. Remarkable analogies and differences to random walks (Buraczewski and Maślanka, in press; Lalley, 1984) are discussed.