The general theory of stochastic population processes

Moyal, J. E.

doi:10.1007/bf02545761

Cited by 205 publications

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“…Next by Moyal [32], there exists a jump process x, e R+, with values in (Z, e) defined on a family (, x PT) such that (i) , PT) is isomorphic to (f',ff^T, P}) and (ii) the counting processes W(A, t) corresponding to xt are "isomorphic" to the processes P'(A, t) constructed above. Furthermore, pX(A, t) =/(A [0, A T]).…”

mentioning

confidence: 99%

Martingales on Jump Processes. II: Applications

Boel¹,

Varaiya²,

Wong³

1975

SIAM Journal on Control

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confidence: 99%

Martingales on Jump Processes. II: Applications

Boel¹,

Varaiya²,

Wong³

1975

SIAM Journal on Control

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“…Although such a process can be constructed by continuation of sample paths, 29) we shall here construct them by an analytic method originated by J.E. Moyal [10].…”

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On Signed Branching Markov Processes with Age

Sirao

1968

Nagoya Mathematical Journal

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“…The equilibrium process. Let (£,5, A) be a tr-finite measure space, where 5 is a <r-field which contains all one point subsets of E. A Poisson process Ai • ), as defined by Moyal [5], is a stochastic counting measure (or equivalently, a symmetric random point process) on the sets of g, which is uniquely determined by its probability generating functional, (1) E (exp\-(six)dAix) \ =exp f (e~s{x) -1 \ ¿A (x) where s(x) ^ 0 is a measurable function on E. The salient facts about this process (and the only facts we shall need) are the following:…”

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“…Consequently from (5.7) we obtain £ r2EM"(B;r) = VarSn(B) ~ n\x(B) + 2 f EyN(B)dX(y)] (5)(6)(7)(8) r = 1 = na2(B).…”

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confidence: 99%

Equilibrium Processes

Port¹

1966

Transactions of the American Mathematical Society

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1. Introduction. Let (£,$) be a set with a c-field of subsets \J containing all one point sets, and let P(x, B) be a transition function of a Markov process with states in £. Assume that F has at least one cr-finite invariant measure X which we take as fixed throughout the discussion. In §2 we describe precisely how to construct a system of denumerably many independent Markov processes all having the same transition law P and whose initial positions are given by the Poisson process on £ with mean X(B) on B. There we establish the fundamental fact that this system maintains itself in macroscopic equilibrium; thus we call such a process an equilibrium process. This property was first established for systems of this type by Doob for Brownian motion and by Derman for countable state space Markov chains. Our purpose in this paper will be to investigate (1) the number of processes, Mn(B; r), whose occupation time in B is exactly r by time n; (2) the number of distinct processes, Ln(B), which are in B at least once by time n; (3) the number of processes, Jn(B), which are in B for a last time at time n ; and (4) the number of processes, A"(B), which are in B at time n, where B is always a transient set of finite, positive X measure. Previously these quantities were investigated for this system in the countable state space case by the author in [6], and the results we obtain here will be extensions of those in [6] to the more general setting of this paper(x).In summary, then, we do the following. In §2 we describe the construction of the basic system(2), and in §3 we give some preliminary material on Markov processes having nontrivial dissipative part which is needed in the sequel. In §4 we show that Mn(B; r)/n converges with probability one to a constant Cr(B) and that if Cr(B)>0, then [M"(B,r) -EMn(B,r)](nCr(B))~1'2 is asymptotically normally distributed. As a corollary we show that L"(B)/n converges with probability one to a constant C(B) > 0, and that

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The general theory of stochastic population processes

Cited by 205 publications

References 5 publications

Martingales on Jump Processes. II: Applications

Martingales on Jump Processes. II: Applications

On Signed Branching Markov Processes with Age

Equilibrium Processes

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