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A spine approach to branching diffusions with applications to $${\mathcal L}^p$$-convergence of martingales. (English) Zbl 1193.60100
Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLII. Berlin: Springer (ISBN 978-3-642-01762-9/pbk; 978-3-642-01763-6/ebook). Lecture Notes in Mathematics 1979, 281-330 (2009).
One of the central elements of the ‘spine’ approach is to interpret the behaviour of a branching process under a certain change of measure. This approach is crystallized within the last twenty years. In the paper, a modified formalization of the ‘spine’ change of measure approach for branching diffusions in the spirit of those found by Kyprianou and Lyons is presented. The approach leads to a proof of a general form of the Many-to-One theorem that enables expectations of sums over particles in the branching process to be calculated in terms of an expectation of a single particle. The authors also exemplify spine proofs of the $${\mathcal L}^p$$-convergence $$(p\geq 1)$$ of some key ‘additive’ martingales for three distinct models of branching diffusions, including new results for a multi-type branching Brownian motion and discussion of left-most particle speeds.
For the entire collection see [Zbl 1166.60002].

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60F25 $$L^p$$-limit theorems
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