Galton-watson probability contraction

Moumanti Podder, Joel Spencer

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Article number20
JournalElectronic Communications in Probability
Volume22
DOIs
StatePublished - 2017

Fingerprint

Contraction
Equivalence class
Siméon Denis Poisson
Fixed point
Game
Galton-Watson Tree
Denote
Rooted Trees
Recursion
Finite Set
Probability Distribution
First-order
Converge
Integer
Range of data
Equivalence
Class
Probability distribution

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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

In: Electronic Communications in Probability, Vol. 22, 20, 2017.

Research output: Contribution to journalArticle

@article{117c187a7c5c4d4cb0a0baff8e150342,
title = "Galton-watson probability contraction",
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 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.",
keywords = "Almost sure theory, First order logic, Galton-Watson trees",
author = "Moumanti Podder and Joel Spencer",
year = "2017",
doi = "10.1214/17-ECP47",
language = "English (US)",
volume = "22",
journal = "Electronic Communications in Probability",
issn = "1083-589X",
publisher = "Institute of Mathematical Statistics",

}

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 -