Stationary sequences with simple joint Poisson distributions
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.
scientific article; zbMATH DE number 1532893