Strong laws of large numbers in certain linear spaces
Annales de l'Institut Fourier, Tome 24 (1974) no. 2, pp. 205-223.

Dans cet article, on explore la convergence presque sûre des séries de variables aléatoires prenant leurs valeurs dans l’espace métrique linéaire et les lois fortes des grands nombres pour suites de vecteurs aléatoires. Dans la partie 2, on considère le cas de l’espace de Banach où les résultats dépendent de la géométrie de la boule unité. Dans la partie 3, on étudie les vecteurs aléatoires dans un espace possédant une norme non-nécessairement homogène ; la partie 4 est consacrée aux suites de vecteurs indépendants et équidistribués.

In this paper we are concerned with the norm almost sure convergence of series of random vectors taking values in some linear metric spaces and strong laws of large numbers for sequences of such random vectors. Section 2 treats the Banach space case where the results depend upon the geometry of the unit cell. Section 3 deals with spaces equipped with a non-necessarily homogeneous F-norm and in Section 4 we restrict our attention to sequences of identically distributed random vectors.

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     title = {Strong laws of large numbers in certain linear spaces},
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Woyczynski, Wojbor A. Strong laws of large numbers in certain linear spaces. Annales de l'Institut Fourier, Tome 24 (1974) no. 2, pp. 205-223. doi : 10.5802/aif.514. https://aif.centre-mersenne.org/articles/10.5802/aif.514/

[1] A. Beck, A convexity condition in Banach spaces and the strog law of large numbers, Proc. Amer. Math. Soc., 13 (1962), 329-334. | MR | Zbl

[2] A. Beck and D. P. Giesy, P-uniform convergence and a vector-valued strong laws of large numbers, Trans. Amer. Math. Soc., 147 (1970), 541-559. | MR | Zbl

[3] N. Dunford and J. T. Schwartz, Linear operators, Vol. I, Wiley-Interscience, New York, 1958. | MR | Zbl

[4] R. Fortet and E. Mourier, Les fonctions aléatoires comme éléments aléatoires dans les espaces de Banach, Studia Math., 15 (1955), 62-79. | EuDML | MR | Zbl

[5] J. Hoffmann-Jorgensen, Sums of independent Banach space valued random variables, Aarhus Universitet, Matematisk Institut, Preprint Series, 1972/1973, No 15, 1-89. | Zbl

[6] J.-P. Kahane, Some random series of functions, Heath, Lexington, 1968. | MR | Zbl

[7] V. V. Petrov, Sums of independent random variables, Nauka, Moscow, 1972 (in Russian). | MR | Zbl

[8] V. V. Petrov, On the rate of growth of dependent random variables, Teor. Probability Appl., 18 (1973), 358-361. | MR | Zbl

[9] P. Révész, The laws of large numbers, Academic Press, New York, 1968. | MR | Zbl

[10] S. Rolewicz, Metric linear spaces, PWN, Warsaw, 1972. | MR | Zbl

[11] W. A. Woyczyński, Random series and laws of large numbers in some Banach spaces, Theor. Probability Appl., 18 (1973), 361-367. | MR | Zbl

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