The counting process and summation of a random number of random variables

The behavior of the tail of the sum of a random number of random variables $\overline{F(x)}=\pr\{\sum_{i=1}^{\nu}\xi_i>x\}$ is considered as $x \to \infty$. Estimates of the convergence of $\overline{F(x)}$ to the limit function are constructed in terms of renewal theory. The estimates are based on the variance $\operatorname{Var}\nu(t)$ of the counting process $\nu(t)=\min\bigl\{n\colon\sum_{i=1}^n \xi_i>t\bigr\}$. A survey of bounds for $\operatorname{Var}\nu(t)$ is given for different sequences $\{\xi_i\}$, in particular, for the case where the terms of the sequence $\{\xi_i\}$ are not identically distributed.