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Generalized Poisson distributions as limits of sums for arrays of dependent random vectors. (English) Zbl 0821.60032
Let \(\{X_{k,n}, k \in Z, n \in N\}\), where \(Z\) is the set of integers and \(N\) that of natural numbers, be arrays of random vectors in \(R^ d\) and let \((k_ n)\) be a sequence of integers with \(k_ n \to \infty\) as \(n \to \infty\). Suppose that the sequence \(\{X_{k,n}, k \in Z, n \in N\}\) is stationary in rows and set \(S_ n = \sum^ n_{k = 1} X_{k,n}\), \(n \in N\). The author obtains necessary and sufficient conditions for \((S_ n)\), properly normalized, to converge to a generalized Poisson distribution, when \(\{X_{k,n}, k \in Z\}\) is an \(m\)-dependent or mixing \((\alpha\)-, \(\rho\)-, \(\varphi\)-mixing) sequence. For \((X_ n)\) a strictly stationary sequence of random vectors, under some mixing conditions, the author obtains a functional limit theorem for the partial sum process. Point process theory is a major tool used here. This paper extends some of the results by A. Jakubowski and the author [ibid. 29, No. 2, 219-251 (1989; Zbl 0687.60025)].

60F05 Central limit and other weak theorems
60G10 Stationary stochastic processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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