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 -norm and in Section 4 we restrict our attention to sequences of identically distributed random vectors.
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.
@article{AIF_1974__24_2_205_0, author = {Woyczynski, Wojbor A.}, title = {Strong laws of large numbers in certain linear spaces}, journal = {Annales de l'Institut Fourier}, pages = {205--223}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {24}, number = {2}, year = {1974}, doi = {10.5802/aif.514}, mrnumber = {53 #9318}, zbl = {0275.60039}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.514/} }
TY - JOUR AU - Woyczynski, Wojbor A. TI - Strong laws of large numbers in certain linear spaces JO - Annales de l'Institut Fourier PY - 1974 SP - 205 EP - 223 VL - 24 IS - 2 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.514/ DO - 10.5802/aif.514 LA - en ID - AIF_1974__24_2_205_0 ER -
Woyczynski, Wojbor A. Strong laws of large numbers in certain linear spaces. Annales de l'Institut Fourier, Volume 24 (1974) no. 2, pp. 205-223. doi : 10.5802/aif.514. http://archive.numdam.org/articles/10.5802/aif.514/
[1] A convexity condition in Banach spaces and the strog law of large numbers, Proc. Amer. Math. Soc., 13 (1962), 329-334. | MR | Zbl
,[2] P-uniform convergence and a vector-valued strong laws of large numbers, Trans. Amer. Math. Soc., 147 (1970), 541-559. | MR | Zbl
and ,[3] Linear operators, Vol. I, Wiley-Interscience, New York, 1958. | MR | Zbl
and ,[4] Les fonctions aléatoires comme éléments aléatoires dans les espaces de Banach, Studia Math., 15 (1955), 62-79. | EuDML | MR | Zbl
and ,[5] Sums of independent Banach space valued random variables, Aarhus Universitet, Matematisk Institut, Preprint Series, 1972/1973, No 15, 1-89. | Zbl
,[6] Some random series of functions, Heath, Lexington, 1968. | MR | Zbl
,[7] Sums of independent random variables, Nauka, Moscow, 1972 (in Russian). | MR | Zbl
,[8] On the rate of growth of dependent random variables, Teor. Probability Appl., 18 (1973), 358-361. | MR | Zbl
,[9] The laws of large numbers, Academic Press, New York, 1968. | MR | Zbl
,[10] Metric linear spaces, PWN, Warsaw, 1972. | MR | Zbl
,[11] Random series and laws of large numbers in some Banach spaces, Theor. Probability Appl., 18 (1973), 361-367. | MR | Zbl
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