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Penalization of Galton Watson trees with marked vertices

Romain Abraham, Sonia Boulal, Pierre Debs

Abstract

We consider a Galton-Watson tree where each node is marked independently of each others with a probability depending on itsout-degree. Using a penalization method, we exhibit new martingales where the number of marks up to level n -- 1 appears. Then, we use these martingales to define new probability measures via a Girsanov transformation and describe the distribution of the random trees under these new probabilities.

Penalization of Galton Watson trees with marked vertices

Abstract

We consider a Galton-Watson tree where each node is marked independently of each others with a probability depending on itsout-degree. Using a penalization method, we exhibit new martingales where the number of marks up to level n -- 1 appears. Then, we use these martingales to define new probability measures via a Girsanov transformation and describe the distribution of the random trees under these new probabilities.
Paper Structure (15 sections, 23 theorems, 155 equations, 1 figure)

This paper contains 15 sections, 23 theorems, 155 equations, 1 figure.

Table of Contents

  1. introduction
  2. Technical background
  3. The set of marked trees
  4. Masses
  5. Marked Galton-Watson trees
  6. Polynomial penalization
  7. The sub-critical case $\mu<1$
  8. A new martingale
  9. Change of probability and the new random tree
  10. The super-critical case $\mu > 1$
  11. The critical case $\mu=1$
  12. Exponential penalization
  13. Case $\mathbf{p}(0)>0$
  14. Case $\mathbf{p}(0)=0$
  15. Case $s=0$

Key Result

Theorem 1.1

Let $\mathbf{p}$ be an offspring distribution satisfying condp and that admits a moment of order $\ell\in\mathbb{N}^*$, and let $\mathbf{q}$ be a mark function satisfying condq. Then, for every $n\in\mathbb{N}$ and every $\Lambda_n\in\mathscr{F}_n$, we have where $P_\ell$ is an explicit polynomial function of degree $\ell$ and $f_\ell$ is defined for all $s,z\in\mathbb{N}$ by: where $\binom{j}{t

Figures (1)

  • Figure 1: A marked tree and the associated masses.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.1
  • proof
  • ...and 39 more