CONDITIONAL FUNCTIONAL LIMIT THEOREM FOR A RANDOM RECURRENCE SEQUENCE CONDITIONED ON A LARGE DEVIATION EVENT

Let {Z(n), n >= 0} be a branching process in an independent and identically distributed (i.i.d.) random environment and {Sn, n >= 1} be the associated random walk with steps xi(i). Under the Cramer condition on xi(1) and moment assumptions on a number of descendants of one particle, we know the asymptotics of the large deviation probabilities P(ln Z(n) > x), where x/n > mu*. Here, mu*(;) is a parameter depending on the process type. We study the asymptotic behavior of the process trajectory under the condition of a large deviation event. In particular, we obtain a conditional functional limit theorem for the trajectory of (Z([nt]), t is an element of [0,1]) given ln Z(n)>x. This result is obtained in a more general model of linear recurrence sequence Yn+1=A(n)Y(n)+B-n, n >= 0, where {A(i)} is a sequence of i.i.d. random variables, Y-0, B-i, i >= 0, are possibly dependent and have different distributions, and we need only some moment assumptions on them.