Abstract
In this note we propose a two-spine decomposition of the critical Galton-Watson tree and use this decomposition to give a probabilistic proof of Yaglom’s theorem.
Highlights
Consider a critical Galton-Watson process (Zn)n≥0 with Z0 = 1 and offspring distribution μ on N0 := {0, 1, . . . } which has mean 1 and finite variance σ2 > 0, i.e., (1.1)
We propose a k(k − 1)-type size-biased μ-Galton-Watson tree equipped with a two-spine skeleton, which serves as a change-of-measure of the original μ-Galton-Watson tree; and with the help of this two-spine technique, we give a new probabilistic proof of Theorem 1.1(2), i.e. Yaglom’s theorem
In our follow-up paper [8], we show that, in a similar spirit, a two-spine structure can be constructed for a class of critical superprocesses, and a probabilistic proof of a Yaglom type theorem can be obtained for those processes
Summary
In our follow-up paper [8], we show that, in a similar spirit, a two-spine structure can be constructed for a class of critical superprocesses, and a probabilistic proof of a Yaglom type theorem can be obtained for those processes Another aspect of our new proof is that we take advantage of a fact that the exponential distribution can be characterized by a particular x2-type size-biased distributional equation. In [7], Lyons, Pemantle and Peres characterize the exponential distribution by a different but well-known x-type size-biased distributional equation: A nonnegative random variable Y with positive finite mean is exponentially distributed if and only if it satisfies that (1.9). A similar type of argument is used in our follow-up paper [8] for critical superprocesses
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