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 $\overline {F(x)}=\mathsf P\{\sum _{i=1}^{\nu }\xi _i>x\}$ 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.