The strong law of large numbers: a weak-$l_ 2$ view
HTML articles powered by AMS MathViewer
- by Bernard Heinkel
- Proc. Amer. Math. Soc. 123 (1995), 273-280
- DOI: https://doi.org/10.1090/S0002-9939-1995-1213860-X
- PDF | Request permission
Abstract:
Let $({X_k})$ be a sequence of independent, centered, and square integrable real-valued random variables. To that sequence one associates \[ \forall n \in \mathbb {N},\quad {\xi _n} = {\left \| {({2^{ - n}}{X_k}),\;{2^n} + 1 \leq k \leq {2^{n + 1}}} \right \|_{2,\infty }}.\] When there exists $K \geq 1$ such that \[ \sum \limits _{n \geq 1} {{P^K}({\xi _n} > {c_n}) < + \infty ,} \] where $({c_n})$ is a suitable sequence of positive constants, then the strong law of large numbers holds if and only if $({X_k}/k)$ converges almost surely to 0.References
- Jean-Christian Alt, Stabilité de sommes pondérées de variables aléatoires vectorielles, J. Multivariate Anal. 29 (1989), no. 1, 137–154 (French, with English summary). MR 991061, DOI 10.1016/0047-259X(89)90081-X
- Niels T. Andersen, Evarist Giné, Mina Ossiander, and Joel Zinn, The central limit theorem and the law of iterated logarithm for empirical processes under local conditions, Probab. Theory Related Fields 77 (1988), no. 2, 271–305. MR 927241, DOI 10.1007/BF00334041
- Bernard Heinkel, Rearrangements of sequences of random variables and exponential inequalities, Probab. Math. Statist. 10 (1989), no. 2, 247–255. MR 1057932
- Bernard Heinkel, Some exponential inequalities with applications to the central limit theorem in $C[0,1]$, Probability in Banach spaces 6 (Sandbjerg, 1986) Progr. Probab., vol. 20, Birkhäuser Boston, Boston, MA, 1990, pp. 162–184. MR 1056710
- Bernard Heinkel, A law of large numbers for random vectors having large norms, Probability in Banach spaces, 7 (Oberwolfach, 1988) Progr. Probab., vol. 21, Birkhäuser Boston, Boston, MA, 1990, pp. 105–125. MR 1105554
- Jørgen Hoffmann-Jørgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159–186. MR 356155, DOI 10.4064/sm-52-2-159-186
- Tord Holmstedt, Interpolation of quasi-normed spaces, Math. Scand. 26 (1970), 177–199. MR 415352, DOI 10.7146/math.scand.a-10976 A. N. Kolmogorov, Sur la loi forte des grands nombres, C. R. Acad. Sci. Paris 191 (1930), 910-912.
- M. Ledoux and M. Talagrand, Some applications of isoperimetric methods to strong limit theorems for sums of independent random variables, Ann. Probab. 18 (1990), no. 2, 754–789. MR 1055432, DOI 10.1214/aop/1176990857
- Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR 1102015, DOI 10.1007/978-3-642-20212-4
- M. B. Marcus and G. Pisier, Characterizations of almost surely continuous $p$-stable random Fourier series and strongly stationary processes, Acta Math. 152 (1984), no. 3-4, 245–301. MR 741056, DOI 10.1007/BF02392199
- S. J. Montgomery-Smith, The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109 (1990), no. 2, 517–522. MR 1013975, DOI 10.1090/S0002-9939-1990-1013975-0
- S. V. Nagaev, Necessary and sufficient conditions for the strong law of large numbers, Teor. Verojatnost. i Primenen. 17 (1972), 609–618 (Russian, with English summary). MR 0312548
- Gilles Pisier, De nouvelles caractérisations des ensembles de Sidon, Mathematical analysis and applications, Part B, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 685–726 (French, with English summary). MR 634264
- Yu. V. Prokhorov, Some remarks on the strong law of large numbers, Teor. Veroyatnost. i Primenen. 4 (1959), 215–220 (Russian, with English summary). MR 0105740
- William F. Stout, Almost sure convergence, Probability and Mathematical Statistics, Vol. 24, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0455094
- Michel Talagrand, Isoperimetry and integrability of the sum of independent Banach-space valued random variables, Ann. Probab. 17 (1989), no. 4, 1546–1570. MR 1048946
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 273-280
- MSC: Primary 60F15; Secondary 46B45, 46N30, 60E15, 60G50
- DOI: https://doi.org/10.1090/S0002-9939-1995-1213860-X
- MathSciNet review: 1213860