Moments of integral-type downward functionals for single death processes
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Moments of integral-type downward functionals for single death processes Jing WANG1,2 , Yuhui ZHANG1 1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China 2 School of Mathematics and Statistics, Yili Normal University, Yili 835000, China
c Higher Education Press 2020
Abstract We get an explicit and recursive representation for high order moments of integral-type downward functionals for single death processes. Meanwhile, main results are applied to more general integral-type downward functionals. Keywords Single death process, integral-type functional, moment MSC 60J60 1
Introduction
Consider a continuous-time homogeneous Markov chain {X(t) : t > 0} on a probability space (Ω, F , P), with transition probability matrix P (t) = (pij (t)) on a countable state space E := {0, 1, . . . }. We call {X(t) : t > 0} a single death process if its transition rate matrix Q = (qij : i, j ∈ E) is irreducible and satisfies that qi,i−1 > 0 for all i > 1 and qi,i−j = 0 for all i > j > 2. Such a matrix Q = (qij ) is called a single death Q-matrix. Assume that Q is P conservative and totally stable (i.e., qi := −qii = j6=i qij < ∞ for all i ∈ E). Define the first hitting time for all i > 0 : τi := inf{t > 0 : X(t) = i}. Let V be a non-negative function and not identically equal to zero on E. Fix i0 ∈ E. We consider the integral-type functional for single death processes in this paper: Z τi 0
ξi0 =
V (X(t))dt.
(1.1)
0
Rt It is well known that the integral functional Y (t) := 0 f (X(s), s)ds, where f is a non-negative function, has attracted lots of attention, due to their Received December 11, 2019; accepted June 28, 2020 Corresponding author: Yuhui ZHANG, E-mail: [email protected]
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Jing WANG, Yuhui ZHANG
importance in practical applications. The integrals arise naturally in the theory of inventories and storage (see [6,7]). In an inventory system or storage reservoir, the integral represents the holding cost associated with the stock in the system over a particular period of time. The moments and distributions of the integrals were investigated. For queueing theory, particularly those involving automobile traffic such as traffic jams and intersection bottlenecks, the integrals up to dissipation are related to the cost of the the flow-stopping incident. See [2], where f (X(s)) = sX(s). Some limit theorems were established. For biology, in the study of response of host to injection of virulent bacteria, f (X(s)) = bX(s), with b > 0, could be regarded as a measure of the total amount of toxins produced by the bacteria during (0, t), assuming a constant toxin-excretion rate per bacterium. Here, X(t) denotes the number of live bacteria at time t, the growth of which is governed by a birth and death process. In [8], where f (X(s)) = X(s), it was concerned with the joint distribution of X(t) and Y (t) and with its limiting forms. In [9], it established some limit theorems on branching processes concerning the behavior of the vec
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