Systems of independent Markov chains

Liggett, Thomas M.; Port, Sidney C.

doi:10.1016/0304-4149(88)90060-9

Cited by 11 publications

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“…Suppose that (m, ξ) ∈ S 2 . By definition of S 2 , we have m = αe 0 for some α > 0, and (13) for all n ∈ N, t 1 , . .…”

Section: Proof Of the Main Resultmentioning

confidence: 99%

“…Statement of the problem. Stationary systems of particles evolving independently of each other according to the law of a Markov process have been extensively studied by many authors (see, e.g., the monographs [5], Chapter 1, [11], Chapter 1, [18], as well as the papers [3,4,7,8,13,14], to cite only a few references). The aim of the present paper is to study systems of particles evolving independently of each other in a Gaussian rather than Markovian way.…”

Section: Introductionmentioning

confidence: 99%

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Stationary systems of Gaussian processes

Kabluchko¹

2010

Ann. Appl. Probab.

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We describe all countable particle systems on $\mathbb{R}$ which have the following three properties: independence, Gaussianity and stationarity. More precisely, we consider particles on the real line starting at the points of a Poisson point process with intensity measure $\mathfrak{m}$ and moving independently of each other according to the law of some Gaussian process $\xi$. We classify all pairs $(\mathfrak{m},\xi)$ generating a stationary particle system, obtaining three families of examples. In the first, trivial family, the measure $\mathfrak{m}$ is arbitrary, whereas the process $\xi$ is stationary. In the second family, the measure $\mathfrak{m}$ is a multiple of the Lebesgue measure, and $\xi$ is essentially a Gaussian stationary increment process with linear drift. In the third, most interesting family, the measure $\mathfrak{m}$ has a density of the form $\alpha e^{-\lambda x}$, where $\alpha >0$, $\lambda\in\mathbb{R}$, whereas the process $\xi$ is of the form $\xi(t)=W(t)-\lambda\sigma ^2(t)/2+c$, where $W$ is a zero-mean Gaussian process with stationary increments, $\sigma ^2(t)=\operatorname {Var}W(t)$, and $c\in\mathbb{R}$.Comment: Published in at http://dx.doi.org/10.1214/10-AAP686 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…Suppose that (m, ξ) ∈ S 2 . By definition of S 2 , we have m = αe 0 for some α > 0, and (13) for all n ∈ N, t 1 , . .…”

Section: Proof Of the Main Resultmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stationary systems of Gaussian processes

Kabluchko¹

2010

Ann. Appl. Probab.

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On the Structure of Entrance Laws in Discrete Spatial Critical Branching Processes

Wakolbinger

1991

Mathematische Nachrichten

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MARKovian branching population-valued stochastic processes in discrete time are considered, in which the individuals live on a discrete space of sites and an individual a t site x produces, independently of the others, in the next generation a random offspring whose distribution depends on x, whose mean total number is assumed to be one and whose mean number a t site y is denoted by J(x, (y}). It is proved that, provided the MARKOV chain associated with the transition matrix J is null-recurrent, exactly those among the entrance laws for the population-vnlued process are extremal and have a finite mean number of individuals a t any site and time, which tire "of POISSON type", i.e. arise in a natural way from a PoIssoNian remote past. This generalizes a result of LICCETT/PORT [3] on the pure motion case to the case of branching, and also comments on a remark of DYNKIN ([I], p. 110). IntroductionLet x be a clustering field on a discrete phase space A , i.e. for any a E A , = qg,) is a probability measure on the space M of counting measures (populations) on A , describing the distribution of the offspring population of an individual 6,. By requiring the branching property x(g+y) = x,@) * qY), @, ! P E M , one obtains a brunching dynamics x ,~) , @ E M . In the terminology of DYNKIN [l] (cf. the addendum in his paper), the correspontling branching process in discrete time (which is an M-valued MARKOV chain with transition probability is a superprocess over the seniigroup generated by the kernel J ( a , .) :=s@(.) x{,)(d@), a E A .In this note, we assume the branching to be criticul, i.e. its intensity kernel J is supposed to obey J ( a , A ) = 1 for all a E A . Hefore giving our main result, let us state some notation:A x-entrcince lnw is a sequence (P,JnCE of distributions on M which obeyThe set of x-entrance laws is convex and admits an integral representation over its extremal elements. Any x-entrance law corresponds to a Markovian sequence ( @ n ) n E E of branching populations, and vice versa.

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