Boundary Problems in Some Two-Dimensional Random Walks

Let $\xi _1^{(i)},\xi_2^{(i)},\dots$, $i=1,2$, be two sequences of independent random variables, $\xi_k^{(2)}>0$, $k=1,2,\dots$, $s_0^{(i)}=0$, $s_n^{(i)}=\sum\nolimits_{k=1}^n{\xi_k^{(i)}}$, $i=1,2$, $\bar s_n=\max_{0\leqq k\leqq n}s_k^{(1)}$, $\eta_t=\max\{{k:s_k^{(2)}<t}\}$. We study the joint distribution of the random variables $\bar s_{\eta_t}$, $s_{\eta_t+1}^{(1)}$, $s_{\eta_t+1}^{(2)}$ including asymptotic expansions, and all the domains of deviations in which limit theorems of Cramer type hold. The random variables $\xi_k^{(1)}$, $k=1,2,\dots$, are assumed to have lattice distributions. The method used in this study is similar to [1].