Stationary sequences with simple joint Poisson distributions
type: Item
instance of: scholarly article
title: Stationary sequences with simple joint Poisson distributions
zbMATH Open document ID: 0966.60047
DOI: 10.1016/S0378-3758(00)00106-3
author: Gérard Letac
published in: Journal of Statistical Planning and Inference
publication date: 2001-08-06T00:00:00Z
review text: A sequence \(X = \{X_n\}_{n\geq 1}\) of random variables is simply Poisson distributed if there exists a map \(P \mapsto [0, + \infty]\) such that \(E(z_1^{x_1}...z_n^{x_n}...) = \exp{\sum_T a_T (z^T - 1)}\), where \(z^T\) denotes \(\prod_{i\in T} z_i\) and \(P\) is the family of non-empty finite subsets of naturals, \(T\in P\). A sequence \(X = \{X_n\}_{n\geq 1}\) of random variables taking values in the same measurable space is said to be stationary if the sequences \(X\) and \(X' = \{X_{n+1}\}_{n\geq 1}\) have the same distributions. The most famous example is given when the \(X_n\) are real-valued and the joint distribution of \((X_1, ..., X_n)\) is centered Gaussian for all \(n\). One would like to solve the similar problem of the description of all stationary sequences when the \(X_n\) are valued in \(\{0, 1, ..., n, ...\}\) and when the joint distribution of \(X\) is Poisson. The author finds all stationary sequences which are also simply Poisson distributed. Essentially, their distributions are convolutions of \(\mu_{\varepsilon, \lambda_{\varepsilon}}\) in the distribution of \(\{\sum_{i\geq 0} \varepsilon_i Y_{n+i}\}_{n\geq 1}\), \(\{Y_n\}_{n\geq 1}\) being a sequence of independent ordinary Poisson variables with mean \(\lambda\) and sequence \(\varepsilon = (\varepsilon_0, ...\varepsilon_n, ...)\) being valued in \(\{0, 1\}\) with \(\varepsilon_n=0\) for \(n\) large enough. A sequence \(X = \{X_n\}_{n\geq 1}\) is exchangeable if for all \(n\) and for all permutations \(\sigma\) of \(\{1, ..., n\}\) the distributions of \((X_1, ..., X_n)\) and of \((X_{\sigma(1)}, ..., X_{\sigma(n)})\) coincide. Exchangeable Poisson sequences are also characterized by an integral formula.
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60J05
zbMATH DE Number: 1532893
zbMATH Keywords: simple Poisson distribution
zbMATH Keywords: ergodic properties
zbMATH Keywords: second-order interactions
zbMATH Keywords: exchangeable Poisson sequences
reviewed by: Viktor Ohanyan
MaRDI profile type: Publication
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full work available at URL: s0378-3758(00)00106-3
OpenAlex ID: W2026687524
description: scientific article; zbMATH DE number 1532893