Penalization of Galton–Watson processes

Abstract

We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton-Watson processes with a penalizing function of the form P (x)s x where P is a polynomial of degree p and s ∈ [0, 1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s = 1 (or s → 1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton-Watson tree with p distinguished infinite spines. and the associated process Z is distributed, if 0 < κ < 1, as the size-biased process of the GW conditioned on extinction.

Spine Decompositions and Limit Theorems for a Class of Critical Superprocesses

“…As mentioned earlier in Subsection 1.1, this 2-spine decomposition theorem for superprocesses is an analog of the 2-spine decomposition theorem for Galton-Watson trees in [37], and is closely related to the multi-spine theory appeared in [20], [21], [24] and [1]. Of course, depend on the choice of F , there are many versions of Theorem 1.2.…”

“…An analogous k-spine decomposition theorem also appeared in [21] and [24] in the context of continuous time Galton-Watson processes. The k-th size-biased transform of Galton-Watson trees is also considered in [1]. A closely related infinite spine decomposition is also established in [1] for the supercritical Galton-Watson tree.…”

“…There is another decomposition theorem for supercritical Galton-Watson trees with infinite spines which is first introduced in [3, Section 12] and is now known as the skeleton decomposition. The infinite spines in [1] and the skeleton decomposition in [3,Section 12] are two different decomposition theorems. Our 2-spine methods for Galton-Watson trees [37] and for superprocesses in this paper are more relevant to [1].…”

“…The infinite spines in [1] and the skeleton decomposition in [3,Section 12] are two different decomposition theorems. Our 2-spine methods for Galton-Watson trees [37] and for superprocesses in this paper are more relevant to [1].…”

Penalization of Galton Watson trees with marked vertices

“…where j t1•••tz is the multinomial coefficient and ξ j := lim p→+∞ E[M j p ] for all j ≥ 0. The two martingales obtained for µ ≥ 1 are already known and the study under Q is, as a result, unnecessary (for µ = 1, we obtain Kesten's tree and see [9] for µ > 1).…”