### Abstract

We are concerned with exploring the probabilities of first order statements for Galton-Watson trees with Poisson(c) offspring distribution. Fixing a positive integer k, we exploit the k-move Ehrenfeucht game on rooted trees for this purpose. Let Σ, indexed by 1 ≤ j ≤ m, denote the finite set of equivalence classes arising out of this game, and D the set of all probability distributions over Σ. Let x_{j}(c) denote the true probability of the class j ∈ Σ under Poisson(c) regime, and x(c) the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function Γ, and a map Ψ = Ψ_{c}: D → D such that x(c) is a fixed point of Ψ_{c}, and starting with any distribution x ∈ D, we converge to this fixed point via Ψ because it is a contraction. We show this both for c ≤ 1 and c > 1, though the techniques for these two ranges are quite different.

Original language | English (US) |
---|---|

Article number | 20 |

Journal | Electronic Communications in Probability |

Volume | 22 |

DOIs | |

State | Published - 2017 |

### Fingerprint

### Keywords

- Almost sure theory
- First order logic
- Galton-Watson trees

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Electronic Communications in Probability*,

*22*, [20]. https://doi.org/10.1214/17-ECP47

**Galton-watson probability contraction.** / Podder, Moumanti; Spencer, Joel.

Research output: Contribution to journal › Article

*Electronic Communications in Probability*, vol. 22, 20. https://doi.org/10.1214/17-ECP47

}

TY - JOUR

T1 - Galton-watson probability contraction

AU - Podder, Moumanti

AU - Spencer, Joel

PY - 2017

Y1 - 2017

N2 - We are concerned with exploring the probabilities of first order statements for Galton-Watson trees with Poisson(c) offspring distribution. Fixing a positive integer k, we exploit the k-move Ehrenfeucht game on rooted trees for this purpose. Let Σ, indexed by 1 ≤ j ≤ m, denote the finite set of equivalence classes arising out of this game, and D the set of all probability distributions over Σ. Let xj(c) denote the true probability of the class j ∈ Σ under Poisson(c) regime, and x(c) the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function Γ, and a map Ψ = Ψc: D → D such that x(c) is a fixed point of Ψc, and starting with any distribution x ∈ D, we converge to this fixed point via Ψ because it is a contraction. We show this both for c ≤ 1 and c > 1, though the techniques for these two ranges are quite different.

AB - We are concerned with exploring the probabilities of first order statements for Galton-Watson trees with Poisson(c) offspring distribution. Fixing a positive integer k, we exploit the k-move Ehrenfeucht game on rooted trees for this purpose. Let Σ, indexed by 1 ≤ j ≤ m, denote the finite set of equivalence classes arising out of this game, and D the set of all probability distributions over Σ. Let xj(c) denote the true probability of the class j ∈ Σ under Poisson(c) regime, and x(c) the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function Γ, and a map Ψ = Ψc: D → D such that x(c) is a fixed point of Ψc, and starting with any distribution x ∈ D, we converge to this fixed point via Ψ because it is a contraction. We show this both for c ≤ 1 and c > 1, though the techniques for these two ranges are quite different.

KW - Almost sure theory

KW - First order logic

KW - Galton-Watson trees

UR - http://www.scopus.com/inward/record.url?scp=85016152017&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85016152017&partnerID=8YFLogxK

U2 - 10.1214/17-ECP47

DO - 10.1214/17-ECP47

M3 - Article

AN - SCOPUS:85016152017

VL - 22

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

SN - 1083-589X

M1 - 20

ER -