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

Let $\begin{pmatrix} \xi_j&\eta_j \\ 0&1 \end{pmatrix}$, $j=1,2,\dots,$ be independent identically distributed random matrices and
$$\begin{pmatrix} \varphi_n&\psi_n \\ 0&0 \end{pmatrix} =\prod_{j=1}^n \begin{pmatrix} \xi_j&\eta_j \\ 0&1 \end{pmatrix}.$$
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.