A mean convergence theorem and weak law for arrays of random elements in martingale type \(p\) Banach spaces. (English) Zbl 0874.60008

Adler, André; Rosalsky, Andrew; Volodin, Andrej I.

Stat. Probab. Lett. 32, No. 2, 167-174 (1997).

Summary: For weighted sums of the form \(S_n= \sum^{k_n}_{j=1} a_{nj}(V_{nj}- c_{nj})\), where \(\{a_{nj}, 1\leq j\leq k_n<\infty, n\geq 1\}\) are constants, \(\{V_{nj}, 1\leq j\leq k_n, n\geq 1\}\) are random elements in a real separable martingale type \(p\) Banach space, and \(\{c_{nj}, 1\leq j\leq k_n, n\geq1\}\) are suitable conditional expectations, a mean convergence theorem and a general weak law of large numbers are established. These results take the form \(|S_n|@>{\mathcal L}_r>> 0\) and \(S_n@>P>> 0\), respectively. No conditions are imposed on the joint distributions of the \(\{V_{nj}, 1\leq j\leq k_n, n\geq 1\}\). The mean convergence theorem is proved assuming that \(\{|V_{nj}|^r, 1\leq j\leq k_n, n\geq 1\}\) is \(\{|a_{nj}|^r\}\)-uniformly integrable whereas the weak law is proved under a Cesàro type condition which is weaker than Cesàro uniform integrability. The sharpness of the results is illustrated by an example. The current work extends that of A. Gut [ibid. 14, No. 1, 49-52 (1992; Zbl 0769.60034)] and D. H. Hong and K. S. Oh [ibid. 22, No. 1, 55-57 (1995; Zbl 0815.60023)].

MSC:

60B11	Probability theory on linear topological spaces
60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05	Central limit and other weak theorems
60F25	\(L^p\)-limit theorems

Keywords:

real separable martingale type \(p\) Banach space; weighted sums; weak law of large numbers; convergence in probability; uniformly integrable; Cesàro uniformly integrable array

Citations:

Zbl 0769.60034; Zbl 0815.60023

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