On the distribution of the absorption moment for semimarkov multiplication process

Let $\{\tau_i\}_{i=1}^{\infty}$ and $\{\gamma_i\}_{i=1}^{\infty}$ be independent sequences of independent positive random variables. For the process
$$ Y_n=\gamma_1\gamma_2\dots\gamma_n(x-\xi_n),\quad\text{where}\quad \xi_n=\sum_{i=1}^n\tau_i/\gamma_1\gamma_2\dots\gamma_{i-1}, $$
we consider a random variable $\zeta(x)=\inf\{n\colon Y_n\le 0\ (Y_0=x)\}$ and investigate its limit distributions when $x\to\infty$.