Journal ArticleDOI
The Galton-Watson process conditioned on the total progeny
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In this paper, the limiting behavior of the Galton-Watson process conditioned on the event n = n was studied. And the results obtained are of exactly the same form for the subcritical, critical and supercritical cases.Abstract:
Let Zk denote the number in the kth generation of a Galton-Watson process initiated by one individual and let N be the total progeny, i.e., As n → ∞ the limiting behaviour of the process {Zk, 0 ≦ k ≦ n} conditioned on the event {N =n} is studied. The results obtained are of exactly the same form for the subcritical, critical and supercritical cases. This is in marked contrast to the analogous situation got by conditioning on non-extinction by the nth generation and letting n → ∞. In the latter case the limiting results differ in form for the critical and non-critical cases.read more
Citations
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Journal ArticleDOI
Convergence of Probability Measures
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Book ChapterDOI
Stochastic Analysis: The Continuum random tree II: an overview
TL;DR: In this paper, the authors discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes, aimed at theoretical probabilists.
Journal ArticleDOI
An Introduction to Probability Theory and Its Applications. Volume II By William Feller. Pp. xviii, 626. 90s. 1966. (Wiley)
Journal ArticleDOI
Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation
TL;DR: In this paper, the authors give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case and similar results in the case when no equivalent critical Galton-Watson tree exists.
Journal ArticleDOI
Tree-valued Markov chains derived from Galton-Watson processes
David Aldous,Jim Pitman +1 more
TL;DR: In this article, the authors studied the tree-valued Markov process (G u, 0 ≤ u ≤ 1) and an analogous process in which G 1 ∗ is a critical or subcritical Galton-Watson tree conditioned to be infinite.
References
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Book
Convergence of Probability Measures
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Journal ArticleDOI
Convergence of Probability Measures
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.