The Exponential Decay of Markov Transition Probabilities

Kingmán, J. F. C.

doi:10.1112/plms/s3-13.1.337

Cited by 152 publications

(113 citation statements)

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“…This was observed by Kingman [20,Theorem 8] and, more recently, by Jacka and Roberts [15, Equation (3.1.4)], whose example with strict inequalities in (19) is encompassed in the setting that is described next.…”

Section: Rate Of Convergencementioning

confidence: 76%

“…Since the two limits must be equal, (20) , which concerns pure birth-death processes. When γ i > 0 for infinitely many states i, the situation differs essentially from the pure birth-death setting in that we may simultaneously have both α > 0 and divergence of the series in (20). If either series in (20) converges then the quantities q j of (21) (or (22)) constitute a quasi-stationary distribution (see, for example, [22]).…”

Section: Theorem 2 If α > 0 and Eventual Extinction Is Certain Thenmentioning

confidence: 99%

See 1 more Smart Citation

Extinction Probability in A Birth-Death Process with Killing

Doorn

Zeifman²

2005

J. Appl. Probab.

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We study birth-death processes on the nonnegative integers, where {1, 2, . . . } is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

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Section: Rate Of Convergencementioning

confidence: 76%

Section: Theorem 2 If α > 0 and Eventual Extinction Is Certain Thenmentioning

confidence: 99%

Extinction Probability in A Birth-Death Process with Killing

Doorn

Zeifman²

2005

J. Appl. Probab.

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“…Then, α k = kα. To see this, observe that α k = lim t→∞ −(1/t) log Pr(T > t), where T is the time to first exit of the process from E k (see Kingman (1963)); the limit does not depend on the initial distribution over states. However, T = min{T 1 , .…”

Section: Appendixmentioning

confidence: 99%

“…Furthermore, p j (t)/(1 − p 0 (t)) → π j , where π j = u j / k∈C u k (j ∈ C). The quantity α is called the decay parameter, because p j (t) = O(e −αt ) (see Kingman (1963)). For the (reducible) ensemble model, we must appeal to recent results of Van Doorn and Pollett (2007) to establish the existence of a LCD over the set of non-absorbing states C E := E\E 0 = ∪ n k=1 E k .…”

Section: Quasi-equilibrium Behaviourmentioning

confidence: 99%

Ensemble Behaviour in Population Processes with Applications to Ecological Systems

Pollett

2008

Environ Model Assess

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We study Markovian models for population processes in continuous time, addressing questions concerning the behaviour of ensembles of individuals (equilibrium, quasi-equilibrium and time-dependent behaviour), and in particular what can be deduced from models for individual behaviour. It is self evident that ensemble behaviour is precisely the combined behaviour of individuals, so let me be more precise by way of three examples of populations, which will be used throughout to illustrate our major results. I will make the distinction between the two kinds of models (or processes) by referring to them simply as the "individual model" or the "ensemble model" (or process). Population 1. Our first example is a population network, frequently called a metapopulation (see for example Gilpin and Hanski (1991)), where a fixed number n of individuals occupies geographically separated regions or patches. Patches may become empty, but can be recolonized through migration from other patches. From the point of view of the individual, it spends a period of time in a given patch and might then emigrate to another patch, spend a period there, and so forth. Assuming individuals do not affect each other's progress through the network, one could model the progress of the individual as a random walk on the patches, and thus evaluate quantities such as the probability p j (t) that the individual occupies patch j at time t. Our intuition tells us that, for the ensemble, the proportion of individuals in patch j should be approximately equal to p j (t). So strong is this intuition that scientists frequently model population proportions using individual-level models.In Section 2 we give a careful examination of whether it is reasonable to approximate random proportions of individuals that share a characteristic using probabilities derived from individual-based models. We are able to make a very precise statement for a very general class of models.Population 2. This is a variant of Population 1 where we allow death or external emigration from any patch. We will study two cases: (i) the open network, where there is external immigration to one or more patches, and (ii) the closed network, where there are n individuals to begin with, each eventually disappearing from the network through death or external emigration. As before, individuals are assumed not to affect one another's progress, but now individuals (perhaps arriving from outside the network) perform a random walk on the patches but then eventually leave. In contrast to Population 1, the total number of individuals is random. Yet, we would expect to be able to draw similar conclusions concerning ensemble proportions. Furthermore, as the population would be expected to settle down to a stable equilibrium, we might ask whether it is also reasonable to approximate the equilibrium proportion of individuals occupying patch j using the equilibrium probability that an individual is in patch j. This is certainly reasonable in the open case, but even closed metapopulations can exhibit "q...

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“…For Markov processes with continuous time parameter and countable state space the analytic properties (in this context called 'a-theory') of the transition pro-babilities are studied by Kingman (1963), and some quasi-stationary results are described by Vere-Jones (1969) and Tweedie (1974b).…”

Section: Introductionmentioning

confidence: 99%

Semi-Markov processes on a general state space: α-theory and quasi-stationarity

Arjas

Nummelin²,

Tweedie

1980

J Aust Math Soc A

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By amalgamating the approaches of Tweedie (1974) and Nummelin (1977), an a-theory is developed for general semi-Markov processes. It is shown that x-transient, 2-recurrent and 2-positive recurrent processes can be denned, with properties analogous to those for transient, recurrent and positive recurrent processes. Limit theorems for a-positive recurrent processes follow by transforming to the probabilistic case, as in the above references: these then give results on the existence and form of quasistationary distributions, extending those of Tweedie (1975) and Nummelin (1976).

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The Exponential Decay of Markov Transition Probabilities

Cited by 152 publications

References 3 publications

Extinction Probability in A Birth-Death Process with Killing

Extinction Probability in A Birth-Death Process with Killing

Ensemble Behaviour in Population Processes with Applications to Ecological Systems

Semi-Markov processes on a general state space: α-theory and quasi-stationarity

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