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Convergence of weighted sums of random variables in vector lattices. (English) Zbl 0599.60039
In the present paper the order-convergence of weighted sums \(S_ n=\sum^{n}_{k=1}a_{nk}f_ k\) is obtained under various conditions on the weights \(\{a_{nk}\}\) and random variables \(f_ k\). I recall that in many spaces (e.g. \(L^ p\)-spaces, \(1\leq p<\infty)\) this convergence is stronger than convergence in the norm. The first theorem extends a result of V. K. Rohatgi [Proc. Camb. Philos. Soc. 69, 305-307 (1971; Zbl 0209.200)] for weighted sums of random variables to vector lattices. The other main result is an order-version of a theorem of W. J. Padgett and R. L. Taylor [Proc. 1st int. Conf. Probab. Banach Spaces, Oberwolfach 1975, Lect. Notes Math. 526, 187-202 (1976; Zbl 0342.60008)].
MSC:
60F25 \(L^p\)-limit theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15 Strong limit theorems
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References:
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[2] FREMLIN D. H.: Topological Riesz Spaces and Measure Theory. Cambridge, 1974. · Zbl 0273.46035
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