Abstract
Let (X k) be a sequence of independent random variables, defined on a probability space (Ω,.F, P) and taking their values in a real, separable, Banach space (B, || ||) (that Banach space being equipped with its Borel σ — field B ). To that sequence (X k) one associates the partial sums : S n = X 1 + … + X n and the σ- fields F n generated by X 1,…, X n. One says that (X k) satisfies the weak law of large numbers (WLLN) if ( ||S n/n|| ) converges to 0 in probability; the strong law of large numbers (SLLN) holds for (X k) if (||S n/n||) converges a.s. to 0.
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References
Chacon (R.V.), Sucheston (L.). — On convergence of vector-valued asymptotic martingales, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, t. 33, 1975, p. 55–59.
Dam (B.K.). — On the strong law of large numbers for amarts, Eötvös Sect. Math., t. 33, 1990, p. 139–142.
Edgar (G.A.), Sucheston (L.). — Amarts. A class of asymptotic martingales. A. Discrete parameter, Journal of Multivariate Analysis, t. 6, 1976, p. 193–221.
Edgar (G.A.), Sucheston (L.). — Stopping times and directed processes. — Cambridge, Cambridge University Press, 1992.
Egghe (L.). — Stopping time techniques for analysts and probabilists. — Cambridge, Cambridge University Press, 1984.
Fuk (D. Kh.), Nav (S.V.). — Probability inequalities for sums of independent random variables, Theor. Probab. Appl., t. 16, 1971, p. 643–660.
Godbole, (A.P.). — Strong laws of large numbers and laws of the iterated logarithm in Banach spaces, Ph. D. Thesis of the Michigan State University, 1984.
Gut (A.), Schmidt (K.D.). — Amarts and Set Function Processes, Lecture Notes in Math. 1042, Berlin, Springer, 1983.
Heinkel (B.). — A law of large numbers for random vectors having large norms, Probability in Banach spaces 7, Oberwolfach 1988, p. 105–125, Progress in Probability 21, Boston, Birkhäuser, 1990.
Heinkel (B.). — On the Kolmogorov quasimartingale property, preprint, 1992.
Heinkel (B.). — A probabilistic property of the space ℓ 2 m, preprint, 1992.
Heinkel (B.). — When is \(\frac{ s_n p}{n p}\) an amart?, in preparation, 1993.
Hoffmann-Jørgensen (J.). — Sums of independent Banach space valued random variables, Studia Math., t. 52, 1974, p. 159–186.
Hoffmann-Jørgensen (J.), Pisier (G.). — The law of large numbers and the central limit theorem in Banach spaces, Ann. Probab., t. 4, 1976, p. 587–599.
Kolmogorov (A.). — Sur la loi forte des grands nombres, C.R. Acad. Sci. Paris, t. 191, 1930, p. 910–912.
Krengel (U.), Sucheston (L.). — Semiamarts and finite values, Bull. Amer. Math. Soc, t. 83, 1977, p. 745–747.
Kuelbs (J.), Zinn (J.). — Some stability results for vector valued random variables, Ann. Probab., t. 7, 1979, p. 75–84.
Ledoux (M.), Talagrand (M.). — Probability in Banach spaces. — Berlin, Springer, 1991.
Peligrad (M.). — Local convergence for sums of dependent random variables and the law of large numbers, Revue Roumaine de Mathematiques Pures et Appliquées, t. 25, 1980, p. 89–98.
Petrov (V.V.). — Sums of independent random variables. — Berlin, Springer, 1975.
Rao (M.M.). — Foundations of stochastic analysis. — New York, Academic Press, 1981.
Talagrand (M.). — Some structure results for martingales in the limit and pramarts, Ann. Probab., t. 13, 1985, p. 1192–1203.
Woyczynski (W.A.). — Geometry and martingales in Banach spaces, Probability Winter School, Proceedings Karpacz 1975, p. 229–275, Lecture Notes in Math. 472, Berlin, Springer, 1975.
Woyczynski (W.A.). — Asymptotic behavior of martingales in Banach spaces II, Martingale Theory in harmonic analysis and Banach spaces, Proceedings Cleveland 1981, p. 216–225, Lecture Notes in Math. 939, Berlin, Springer, 1982.
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Heinkel, B. (1994). Some Generalized Martingales Arising from the Strong Law of Large Numbers. In: Hoffmann-Jørgensen, J., Kuelbs, J., Marcus, M.B. (eds) Probability in Banach Spaces, 9. Progress in Probability, vol 35. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0253-0_6
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DOI: https://doi.org/10.1007/978-1-4612-0253-0_6
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