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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
##### Keywords:
order-convergence of weighted sums; vector lattices
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##### References:
 [1] POTOCKÝ R.: A strong law of large numbers for identically distributed vector lattice-valued random variables. Mathematica Slovaca, to appear. · Zbl 0599.60038 [2] FREMLIN D. H.: Topological Riesz Spaces and Measure Theory. Cambridge, 1974. · Zbl 0273.46035 [3] PADGETT W. J., TAYLOR R. L.: Almost sure convergence of weighted sums of random elements in Banach spaces. Lecture Notes in Mathematics. Vol. 526. Springer, Berlin 1976. · Zbl 0342.60008 [4] TAYLOR R. L.: Stochastic Convergence of weighted Sums of Random Elements in Linear Spaces. Springer, Berlin 1978. · Zbl 0443.60004 [5] BOZORGINA A., RAO M. BHASKARA: Limit theorems for weighted sums of random elements in separable B-spaces. J. Multivariate Anal. 9 (1979), no. 3, 428-433. · Zbl 0415.60027 [6] ROHATGI W. K.: Convergence of weighted sums of independent random variables. Proc. Cambridge Philos. Soc. 69 (1971), 305-307. · Zbl 0209.20004
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