A strong law of large numbers for generalized almost sure central limit theorems
Authors:
Rita Giuliano Antonini and Luca Pratelli
Journal:
Theor. Probability and Math. Statist. 70 (2005), 1-9
MSC (2000):
Primary 60F15; Secondary 60F05
DOI:
https://doi.org/10.1090/S0094-9000-05-00626-5
Published electronically:
August 5, 2005
MathSciNet review:
2109818
Full-text PDF Free Access
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Additional Information
Abstract: We prove a Strong Law of Large Numbers in which the variables are assumed to be asymptotically negligible and a generalized Almost Sure Central Limit Theorem is given. As an application we obtain a result about the so-called intersective ASCLT.
References
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References
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- I. Berkes and H. Dehling, Some limit theorems in log density, Ann. Prob. 21 (1993), 1640–1670. MR 1235433 (94h:60026)
- I. Berkes and E. Csáki, A universal result in almost sure central limit theory, Stoch. Proc. Appl. 94 (2001), 105–134. MR 1835848 (2002j:60033)
- G. A. Brosamler, An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), 561–574. MR 0957261 (89i:60045)
- M. Csörgö and L. Horváth, Invariance principle for logarithmic averages, Math. Proc. Cambridge Philos. Soc. 112 (1992), 195–205. MR 1162944 (93e:60057)
- M. Dudziński, A note on the almost sure central limit theorem for some dependent random variables, Statist. Probab. Letters 61 (2003), 31–40. MR 1950451 (2003j:60038)
- R. Giuliano Antonini, On the Rosenblatt coefficient for normalized sums of real random variables, Rend. Acc. Naz. XL Mem. Mat. Appl. (5) 24 (2000), 111–120. MR 1827009 (2002c:60073)
- R. Giuliano Antonini and M. Weber, The intersective ASCLT, Stochastic Anal. Appl. 22 (2004), no. 4, 1009–1025. MR 2062956 (2005h:60088)
- I. A. Ibragimov and M. Lifshits, On the convergence of generalized moments in almost sure central limit theorem, Statist. Probab. Letters 40 (1998), 343–351. MR 1664544 (99m:60032)
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- M. Lacey and W. Philipp, A note on the almost sure central limit theorem, Statist. Probab. Letters 9 (1990), 201–205. MR 1045184 (91e:60100)
- D. Nualart, The Malliavin Calculus and Related Topics, Springer, 1995. MR 1344217 (96k:60130)
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- M. Peligrad and Q. Shao, A note on the almost sure central limit theorem for weakly dependent random variables, Statist. & Probab. Letters 22 (1995), 131–136. MR 1327738 (96b:60057)
- P. Schatte, On strong versions of the almost sure central limit theorem, Math. Nachr. 137 (1988), 249–256. MR 0968997 (89i:60070)
- M. Weber, Entropie Métrique et Convergence Presque Partout, Coll. “Travaux en Cours”, vol. 58, Herman, Paris, 1998. MR 1663938 (2000b:60084)
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Additional Information
Rita Giuliano Antonini
Affiliation:
Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56100 Pisa, Italy
Email:
giuliano@dm.unipi.it
Luca Pratelli
Affiliation:
Accademia Navale di Livorno, Viale Italia 72, 57127 Livorno, Italy
Email:
pratel@mail.dm.unipi.it
Received by editor(s):
July 14, 2003
Published electronically:
August 5, 2005
Article copyright:
© Copyright 2005
American Mathematical Society