A universal result in almost sure central limit theory

István Berkes and Endre Csáki

Stochastic Processes and their Applications, 2001, vol. 94, issue 1, 105-134

Abstract: The discovery of the almost sure central limit theorem (Brosamler, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561-574; Schatte, Math. Nachr. 137 (1988) 249-256) revealed a new phenomenon in classical central limit theory and has led to an extensive literature in the past decade. In particular, a.s. central limit theorems and various related 'logarithmic' limit theorems have been obtained for several classes of independent and dependent random variables. In this paper we extend this theory and show that not only the central limit theorem, but every weak limit theorem for independent random variables, subject to minor technical conditions, has an analogous almost sure version. For many classical limit theorems this involves logarithmic averaging, as in the case of the CLT, but we need radically different averaging processes for 'more sensitive' limit theorems. Several examples of such a.s. limit theorems are discussed.

Keywords: Almost; sure; central; limit; theorem; Logarithmic; averages; Summation; methods (search for similar items in EconPapers)
Date: 2001
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (28)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304-4149(01)00078-3
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:94:y:2001:i:1:p:105-134

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().