×

On strong versions of the central limit theorem. (English) Zbl 0661.60031

Let \(X_ 1,X_ 2,...\), be i.i.d. r.v.’s with E \(X_ 1=0\), E \(X^ 2_ 1=1\), and let \(S_ n=X_ 1+...+X_ n\), \(n\geq 1\). The main results of the paper are as follows: \[ i)\quad P[\lim_{N}\sup_{x}| N^{- 1}\sum^{N}_{n=1}\mathbf{1}_{(-\infty,x)}(S_ n/\sqrt{n})-\Phi (x)| =0]=0, \] where \(\Phi\) is the standard normal distribution function.
ii) Let a(.) be a bounded and continuous function at each point except a finite number of points on the real line. If E \(| X_ 1|^ 3<\infty\), then \[ P[\lim_{N}(\log N)^{-1}/\sum^{N}_{n=1}n^{- 1}a(S_ n/\sqrt{n})=\int_{R}a(y)d\Phi (y)]=1. \] In particular, for \(a=\mathbf{1}_{(-\infty,x)}\) the limit above equals \(\Phi\) (x) and the convergence in ii) is uniform w.r.t. \(x\in R\). A similar result for arithmetic means of appropriate subsequences is also obtained.
Remark: It seems that for the proof of ii) ess sup of a(.) on each subinterval of a finite partition of R should be used. The presented example shows that ii) does not hold if a(.) is discontinuous at infinitely many points.
Reviewer: A.M.Zapala

MSC:

60F05 Central limit and other weak theorems
60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A course in probability theory, Academic Press New York 1974
[2] Holewijn, Z. Wahrsch. Verw. Gebiete 14 pp 89– (1969)
[3] Robbins, Proc. Amer. Math. Soc. 4 pp 786– (1953)
[4] , and , Analytic methods of probability theory, Akademie-Verlag Berlin 1985 · Zbl 0583.60013
[5] Schatte, Th. Rel. Fields 77 pp 167– (1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.