Unimprovable exponential bounds for distributions of sums of a random number of random variables

The basic object of the study is the asymptotic behavior of $\mathbf{P}(Z_\nu>t)$ as $t\to\infty $ for sums $Z_\nu$ of random number $\nu$ of random variables $\zeta_1,\zeta_2,\dots$ . It was established in [1] that, if conditional “with respect to the past” probabilities of the events $\{\zeta_k>t\}$ are dominated by a function $\delta_1(t)$, $\mathbf{P}(\nu>t)<\delta_2(t)$, and the functions $\delta_1$ and $\delta_2$ are close to power functions, then $\mathbf{P}(Z_\nu>t)<c\max(\delta_1(t),\delta_2(t))$, $c=\mathrm{const}$, and this bound cannot be improved. In the present paper, we study the asymptotics of $\mathbf{P}(Z_\nu>t)$ in the case when the functions $\delta_1$ and $\delta_2$ are exponential. The nature of unimprovable bounds for $\mathbf{P}(Z_\nu>t)$ turns out in this case to be different.