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Probabilités de présence dans un processus de branchement spatial markovien. (Presence probability in a spatial Markov branching process). (English) Zbl 0611.60082
Given the point processes \(\{M_{t,x}:\) \(t\geq 0\), \(-\infty <x<\infty \}\), a sequence of spatial branching processes, indexed by a small parameter \(\epsilon\) is constructed in the following way. Let \(S_{\epsilon}M\) be obtained by contracting the point process M by \(\epsilon\) ; so, if m is a point of M, \(\epsilon\) m is a point of \(S_{\epsilon}M\). Now if a kth generation person is at x, his children’s positions, relative to his own, are given by an independent copy of \(S_{\epsilon}M_{k\epsilon,x}.\)
Let \(P^{\epsilon}_{k,x}\) be the law of the process resulting from a single kth generation person at x. If \(\zeta_ n\) is the point process of the nth generation people then the main objective of this paper is to establish lower bounds on \[ \liminf \epsilon \log (P^{\epsilon}_{0,x}(\zeta_ n[y-u,y+u]>0), \] as \(\epsilon\to 0\), where \(n=[t/\epsilon]\) and \(u=\sigma \sqrt{\epsilon}\). Obviously interest centres on y values for which the probability here is small (and these, in a sense, constitute the subcritical region of the process). Techniques from branching processes and large deviation theory are used; the lower bounds obtained arise from ’action integrals’ defined through the Cramér transform of the intensity measure of \(M_{t,x}\).
Reviewer: J.D.Biggins

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F10 Large deviations
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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