Boundary Value Problems for Some Two-Dimensional Random Walks
Alexandr Alekseevich Borovkov, Boris Alekseevich Rogozin · Theory of Probability and Its Applications · 1964
Let $\xi _1^{(i)} ,\xi _2^{(i)} , \cdots $, $i = 1,2$, be two sequences of independent random variables, $\xi _k^{(2)} > 0$, $k = 1,2, \cdots $, $s_0^{(i)} = 0$, $s_n^{(i)} = \sum olimits_{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, \cdots $, are assumed to have lattice distributions. The method used in this study is similar to [1].