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 Fundam. Prikl. Mat., 1995, Volume 1, Issue 1, Pages 81–107 (Mi fpm45)

On high-level crossing for a class of discrete-time stochastic processes

E. V. Bulinskaya

M. V. Lomonosov Moscow State University

Abstract: The aim of this paper is to study the asymptotic behaviour of the first passage time for some discrete-time stochastic processes arising in applied probability. The paper is organized as follows. The systems' description is given in § 1 along with the main results. The integer-valued random walks with impenetrable (as well as reflecting) barrier at origin
$$W_{k}=\max(0,W_{k-1}+X_{k}), k \geq 1, W_{0}=x$$
are treated in § 2. The main object of investigation is $N_{x,n}=\inf\{k\colon W_{k}=n\}$, the first overflow time in terms of inventory theory. The limit distribution of normalized random variable $\tau_{x,n}=N_{x,n}(\mathrm{E}N_{x,n})^{-1}$ is obtained for all the initial states $x$ and possible values of $\mathrm{E}X_{k}$ for the case of the three-valued i. i. d. random variables $X_{k}$ (demand and supply in batches of fixed volume). The domain of model's stability with respect to initial state and system's parameters is established as well. The influence of two-level control policy on system's behaviour is dealt with in § 3. It is proved, in particular, that $\tau_{x,n}$ is asymptotically exponential if $\mathrm{E}X_{k}<0$ in a sufficiently wide band in the neighbourhood of the absorbing boundary $n$. The directions of further investigations and various possibilities of application are given in § 4.

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Citation: E. V. Bulinskaya, “On high-level crossing for a class of discrete-time stochastic processes”, Fundam. Prikl. Mat., 1:1 (1995), 81–107

Citation in format AMSBIB
\Bibitem{Bul95} \by E.~V.~Bulinskaya \paper On high-level crossing for a class of discrete-time stochastic processes \jour Fundam. Prikl. Mat. \yr 1995 \vol 1 \issue 1 \pages 81--107 \mathnet{http://mi.mathnet.ru/fpm45} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1789353} \zmath{https://zbmath.org/?q=an:0870.60068} 

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This publication is cited in the following articles:
1. Bulinskaya E., “Multi-level control and asymptotic behaviour of some inventory systems”, International Journal of Production Economics, 59:1–3 (1999), 289–295
2. Bulinskaya E., “Small random perturbations of just-in-time systems”, International Journal of Production Economics, 71:1–3 (2001), 287–294
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