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Un proceso crítico de Galton-Watson con xn on partículas iniciales se sujeta a la condición de tener yn on partículas en el tiempo n, para ciertos valores x,y  ≥ 0. Se obtienen entonces leyes y difusiones en el límite cuando n → ∞., A critital Galton-Watson process initiated by xn on particles is conditioned on having the size yn on at time n, for some  x,y  ≥ 0. As n → ∞ limit laws and limiting diffusions are obtained.

Tipo de documento: Artículo - Article

Palabras clave: Critical Galton-Watson; initial particles; condition, time; laws; broadcasts; boundary, Proceso crítico de Galton-Watson, partículas iniciales, condición, tiempo, leyes, difusiones, límite





Fuente: http://www.bdigital.unal.edu.co


Introducción



Rev~ta Colomb~ana de Mat~e~ VoL XVI (7982) pag-6.
125 - 140 DIFFUSION CRITICAL LIMITS OF CONDITIONED GALTON-WATSON PROCESSES by Bernhard * MELLEIN RESUMEN. Un proceso critico de Galton-Watson con [xn o(n)] particulas iniciales se sujeta a la condicion de tener [yn o(n)] particulas en el tiempo n, para ciertos valores x,y ~ O.
Se obtienen entonces leyes y difusiones en el limite cuando n 00. ABSTRACT. A critital Galton-Watson process initiated by [xn o(n)] particles is conditioned on having the size [yn o(n)] at time n, for some x,y ~ O.
As n 00 limit laws and limiting diffusions are obtained. 1. In trod uc t ion. in a critical function n Galton-Watson of its offspring Lamperti * Let Z be the number process at time n (GW) and f the generating distribution. and Ney (1968) of particles Assume have proved Research carried out at Johannes Gutenberg Mainz, Federal Republic of Germany. f-O-) = 2a that the finite-diUniversitat, 125 00. mensional J·ointdistributions of {l-z .0~ em [nt] t ~ 1}, condi- tioned on Zo = [axn o(n)] and Zn 0, converge, as n 00, to those of a diffusion process {YLN,x(t) ; 0 ~ t ~ 1} with initial state x ~ o.
Imposing the further condition Z[ J= 0, c 1, i.e. nc. conditioning on extinction in the interval (n,cn], Esty (1976) obtained a limiting diffusion {YE (t); 0 ~ t ~ c} which in ,x,c turn has been shown to converge, as c~l, to another one, which we will denote by {YE (t) ; 0 ~ t ~ 1}.
Letting c~l ,x thought of as conditioning on extinction at time n. might be In this paper we shall show that {~Z[ t]; 0 ~ t ~ 1}, an n conditioned on Zo = [axn o(n)] and Zn = [ayn o(n)], converges in finite-dimensional distributions, as n 00, to a diffusion proc- ess {y (t); 0 ~ t ~ 1}.
It turns out that {YE (t)} and x,y ,x {y o(t)} coincide (in distribution) and that {YLN (t)} may be x, ,x obtained from {Y (t)} by randomization, treating y as a random x,y variable having the distribution of YLN,x(l).
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