Asymptotic behavior of maxima of independent random variables. Discrete case (Q2044283)

Let \(\xi_1, \xi_2,\ldots\) be independent and identically distributed discrete random variables given by \(\mathsf{P}\{\xi_1=i\}=p_i\), \(\sum_{i=0}^{\infty}p_i=1\). Denote \[ R(n)=-\ln\mathsf{P}\{\xi_1\geq n\}=-\ln\left(\sum_{i=n}^{\infty}p_i\right), \quad r(n)=R(n)-R(n-1). \] Under the certain assumptions on the behaviour of \(r(n)\) at infinity, the authors proves theorems on the asymptotic behaviour of \[ z_n=\max_{1\leq i\leq n}\xi_i \] as \(n\) increases to infinity. A number of examples are provided. They include geometrically distributed random variables, the maximum queue-length distribution in Markovian multi-server queueing systems and recurrent birth-and-death processes.