Random Series with Time-Varying Discounting

Let xi(1), xi(2), ... be i.i.d. non negative random variables. Consider the series 1 + Sigma(infinity)(n=1) Pi(n)(i=1) xi(i). It turns out that its partial sums are equal to some random determinants. This allows us to represent the original series as an infinite product of random variables that form a Markov chain. The limit behavior of the series depends on the stationary distribution of this chain. Making use of the Mellin transform, we study the relationship between this distribution and the distribution of xi(1).