On random walks with jumps scaled by cumulative sums of random variables (Q1373958)

Consider a sequence \(\{(X_k, I_k), k\geq 1\}\) of i.i.d. random vectors whose components have finite means \(\mu= EX_1\) and \(a= EI_1\). Let \(\{A_n\}\) be the partial sums of the \(I_k\)-sequence and, for given \(Z_0\), define the time nonhomogeneous random walk \(\{Z_n\}\) recursively by \(Z_n= Z_{n-1}+ A_nX_n\), \(n\geq 1\). It is shown that \(n^{-2}Z_n\to a\mu/2\) a.s. and, if \(\text{Var}(X_1)\) and \(\text{Var}(I_1)\) are finite, that \(n^{-1}Z_n\) converges in distribution to an explicit limit law. A result of \textit{S. N. Ethier} [Ann. Appl. Probab. 6, No. 4, 1248-1259 (1996; Zbl 0876.60051)] is extended by showing that \(P\left[\sup_{n\geq 0} Z_n= 0\right]> 0\) whenever \(a\mu<0\), \(Z_0= 0\), \(P[\mu A_1< 0]> 0\) and \(A_1\) is independent of \(\{(X_k, I_k), k\geq 1\}\). An upper bound for the tail of the distribution of \(\min\{n\geq 1: Z_n> x\}\), \(x>0\), is also presented. The definition of \(\{Z_n\}\) is motivated by earlier work of the author [Sib. Math. J. 37, No. 4, 683-689 (1996); translation from Sib. Mat. Zh. 37, No. 4, 783-789 (1996)].