Convergence of sums of mixing triangular arrays of random vectors with stationary rows. (English) Zbl 0542.60012

This paper deals with the convergence in distribution to Gaussian, generalized Poisson and infinitely divisible laws of the row sums of triangular arrays of weakly dependent, Banach space valued random vectors with stationary rows. Under hypotheses of \(\phi\)- or \(\psi\)-mixing type together with certain dependence restrictions about contiguous r.v.’s, necessary and sufficient conditions for convergence, some of them in terms of the individual r.v.’s, are proved; in the Hilbert space case and assuming some specified mixing rates, simpler sufficient conditions for convergence are given. An invariance principle in the case of convergence to Gaussian laws is included.
After this paper was written, the author has proved that: a) the results of the article can be applied to the sequence of partial quotients of the continued fraction expansion of a random number in (0,1); b) the given sufficient conditions for convergence to Gaussian laws hold, in the \(\phi\)-mixing case, without imposing dependence restrictions on contiguous r.v.’s; c) there are necessary and sufficient conditions for the validity of the invariance principle in distribution for stationary, \(\phi\)-mixing triangular arrays, in the case of convergence to an infinitely divisible law.


60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
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