On the continuity of the distribution of a~sum of dependent variables connected with independent walks on the lines

A. K. Grincevičius

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 1, pp. 163-168

Résumé

Let $ \begin{pmatrix} \xi_j\eta_j \\ 01 \end{pmatrix} $, $j=1,2,\dots,$ be independent identically distributed random matrices and $$ \begin{pmatrix} \varphi_n\psi_n \\ 00 \end{pmatrix} =\prod_{j=1}^n \begin{pmatrix} \xi_j\eta_j \\ 01 \end{pmatrix}. $$ Then $\psi_n=\eta_1+\eta_2\xi_1+\dots+\eta_n\xi_1\dots\xi_{n-1}$. Convergence and continuity of the limit distribution of $\psi_n$ as $n\to\infty$ are studied.