# Takacs' asymptotic theorem and its applications: A survey

Vyacheslav M. Abramov

Arxiv ID: 0712.2480•Last updated: 6/30/2021

The book of Lajos Tak\'acs \emph{Combinatorial Methods in the Theory of
Stochastic Processes} has been published in 1967. It discusses various problems
associated with $$ P_{k,i}=\mathrm{P}{\sup_{1\leq
n\leq\rho(i)}(N_n-n)<k-i},\leqno(*) $$ where $N_n=\nu_1+\nu_2...+\nu_n$ is a
sum of mutually independent, nonnegative integer and identically distributed
random variables, $\pi_j=\mathrm{P}\{\nu_k=j\}$, $j\geq0$, $\pi_0>0$, and
$\rho(i)$ is the smallest $n$ such that $N_n=n-i$, $i\geq1$. (If there is no
such $n$, then $\rho(i)=\infty$.)
(*) is a discrete generalization of the classic ruin probability, and its
value is represented as $P_{k,i}={Q_{k-i}}/{Q_k}$, where the sequence
$\{Q_k\}_{k\geq0}$ satisfies the recurrence relation of convolution type:
$Q_0\neq0$ and $Q_k=\sum_{j=0}^k\pi_jQ_{k-j+1}$.
Since 1967 there have been many papers related to applications of the
generalized classic ruin probability. The present survey concerns only with one
of the areas of application associated with asymptotic behavior of $Q_k$ as
$k\to\infty$. The theorem on asymptotic behavior of $Q_k$ as $k\to\infty$ and
further properties of that limiting sequence are given on pages 22-23 of the
aforementioned book by Tak\'acs. In the present survey we discuss applications
of Tak\'acs' asymptotic theorem and other related results in queueing theory,
telecommunication systems and dams. Many of the results presented in this
survey have appeared recently, and some of them are new. In addition, further
applications of Tak\'acs' theorem are discussed.

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