Laws of Large Numbers for Non-Homogeneous Markov Systems
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Laws of Large Numbers for Non-Homogeneous Markov Systems P.-C. G. Vassiliou1
Received: 4 August 2017 / Revised: 27 November 2017 / Accepted: 10 December 2017 © The Author(s) 2018. This article is an open access publication
Abstract In the present we establish Laws of Large Numbers for Non-Homogeneous Markov Systems and Cyclic Non-homogeneous Markov systems. We start with a theorem, where we establish, that for a NHMS under certain conditions, the fraction of time that a membership is in a certain state, asymptotically converges in mean square to the limit of the relative population structure of memberships in that state. We continue by proving a theorem which provides the conditions under which the mode of covergence is almost surely. We continue by proving under which conditions a Cyclic NHMS is Cesaro strongly ergodic. We then proceed to prove, that for a Cyclic NHMS under certain conditions the fraction of time that a membership is in a certain state, asymptotically converges in mean square to the limit of the relative population structure in the strongly Cesaro sense of memberships in that state. We then proceed to establish a founding Theorem, which provides the conditions under which, the relative population structure asymptotically converges in the strongly Cesaro sense with geometrical rate. This theorem is the basic instrument missing to prove, under what conditions the Law of Large Numbers for a Cycl-NHMS is with almost surely mode of convergence. Finally, we present two applications firstly for geriatric and stroke patients in a hospital and secondly for the population of students in a University system. Keywords Non-homogeneous Markov systems · Cyclic non homogeneous Markov systems · Laws of large numbers Mathematics Subject Classification (2010) 60J10 · 60J20
P.-C. G. Vassiliou
[email protected] 1
Department of Statistical Sciences, University College London, Gower Street, London WC1E 6BT, UK
Methodol Comput Appl Probab
1 Introductory Notes One of the most celebrated theorems in probability theory is the Law of Large Numbers (Grimmett and Stirzaker 2001). The Law of Large Numbers were also studied for finite Markov chains (Kemeny and Snell 1981). The Law of Large Numbers for a regular homogeneous Markov chain states, that if πj is the limiting probability of being in state j independent of the initial state, then also πj represents the fraction of time, that the process can be expected to be in state j for a large number of steps. The Law of Large Numbers for Markov chains is also linked with the Martingale Convergence Theorem (Kemeny et al. 1976). Laws of Large Numbers were also studied for non-homogeneous semi-Markov processes by Vadori and Swishchuk (2015). For Markov chains in general state spaces there exists a chapter on Laws of Large Numbers in Meyn and Tweedie (2009), where the theory of martingales is the main instrument for proving various types of LLN. These laws are of value for Markov chains exactly as they are for all stochastic processes: the LLN and CLT, in particular, pro
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