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Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions. (English) Zbl 0646.60008

The authors provide a proof of the SLLN in separable Banach spaces in the general framework of pairwise independent random elements and consider almost sure convergence to 0, as \(n\to \infty\), of weighted sums, \(\sum_{k\geq 1}a_{nk}X_ k\), of random elements in a separable Banach space, where \(\{a_{nk}\); n,k\(\geq 1\}\) is a Toeplitz sequence of real numbers.
The basic results in the paper are proved by using an extension due to Blackwell and Dubins of the Skorokhod representation theorem applied to weakly equivalent sequences of probability measures.
Reviewer: V.Sakalauskas

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60B05 Probability measures on topological spaces
60F15 Strong limit theorems
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References:

[1] Billingsley, P., (Convergence of Probability Measures (1968), Wiley: Wiley New York) · Zbl 0172.21201
[2] Blackwell, D.; Dubins, L. E., An Extension of skorohod’s almost sure representation theorem, (Proc. Amer. Math. Soc., 89 (1983)), 691-692 · Zbl 0542.60005
[3] Bozorgnia, A.; Bhaskara Rao, M., Limit theorems for weighted sums of random elements in separable Banach spaces, J. Multivar. Anal., 9, 428-433 (1979) · Zbl 0415.60027
[4] Chung, K. L., (A Course in Probability Theory (1974), Academic Press: Academic Press Orlando, FL)
[5] Csörgo, S.; Tandori, K.; Totik, V., On the strong law of large numbers for pairwise independent random variables, Acta Math. Hungar., 42, 3-4, 319-330 (1983) · Zbl 0534.60028
[6] Cuesta, J. A.; Matran, C., Strong Laws of large numbers in abstract spaces via Skorohod’s representation theorem, Sankhyā, Ser. A, 48, 98-103 (1986)
[7] Daffer, P. Z.; Taylor, R. L., Tightness and strong laws of large numbers in Banach spaces, Bull. Inst. Math. Acad. Sinica, 10, 251-263 (1982) · Zbl 0496.60006
[8] Etemadi, N., An elementary proof of the strong law of large numbers, Z. Wahrsch. Verw. Gebiete, 55, 119-122 (1981) · Zbl 0438.60027
[9] Hoffman-Jørgensen, J.; Pisier, G., The law of large numbers and the central limit theorem in Banach spaces, Ann. Probab., 4, 587-599 (1976) · Zbl 0368.60022
[10] Jamison, B.; Orey, S.; Pruitt, W., Convergence of weighted averages of independent random variables, Z. Wahrsch. Verw. Gebiete, 4, 40-44 (1965) · Zbl 0141.16404
[11] Loeve, M., (Probability Theory (1963), Van Nostrand: Van Nostrand Princeton, NJ) · Zbl 0108.14202
[12] Marle, C. M., Mesures et probabilités (1974), Hermann: Hermann Paris · Zbl 0306.28001
[13] Padgett, W. J.; Taylor, R. L., Convergence of weighted sums of random elements in Banach spaces and Frechet spaces, Bull. Inst. Math. Acad. Sinica, 2, 389-400 (1974) · Zbl 0301.60004
[14] Parthasarathy, K. R., (Probability Measures on Metric Spaces (1967), Academic Press: Academic Press Orlando, FL) · Zbl 0153.19101
[15] Pollard, D., (Convergence of Stochastic Processes (1984), Springer-Verlag: Springer-Verlag Berlin/New York/Heidelberg), Springer Series in Statistics · Zbl 0544.60045
[16] Rohatgi, V. K., Convergence of weighted sums of independent random variables, (Proc. Cambridge Philos. Soc., 69 (1971)), 305-307 · Zbl 0209.20004
[17] Skorohod, A. V., Limit theorems for stochastic processes, Theory Probab. Appl., 1, 261-290 (1956)
[18] Taylor, R. L., (Stochastic Convergence of Weighted Sum of Random Elements in Linear Spaces, Vol. 672 (1978), Springer-Verlag: Springer-Verlag Berlin/New York/Heidelberg), Lecture Notes in Mathematics
[19] Taylor, R. L.; Wei, D., Laws of large numbers for tight random elements in normed linear spaces, Ann. Probab., 7, 150-155 (1979) · Zbl 0395.60005
[20] Taylor, R. L.; Padgett, W. J., Stochastic convergence of weighted sums in normed spaces, J. Multivar. Anal., 5, 434-450 (1975) · Zbl 0339.60004
[21] Wang, X. C.; Bhaskara Rao, M., A note on convergence of weighted sums of random variables, Internat. J. Math. Sti., 8, 805-812 (1985) · Zbl 0583.60021
[22] Wei, D.; Taylor, R. L., Convergence of weighted sums of tight random elements, J. Multivar. Anal., 8, 282-294 (1978) · Zbl 0376.60006
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