The Annals of Probability

An Example Concerning CLT and LIL in Banach Space

Naresh C. Jain

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Abstract

Let $E$ be a separable Banach space with norm $\|\bullet\|$. Let $\{X_n\}$ be a sequence of $E$-valued independent, identically distributed random variables, and $S_n = X_1 + \cdots + X_n$. If $\{n^{-\frac{1}{2}}S_n\}$ converges in the sense of weak convergence of the corresponding measures in $E$, and $E$ is the real line, then it is well known that $\mathscr{E}\lbrack X_1 \rbrack = 0$ and $\mathscr{E}\lbrack\|X_1\|^2\rbrack < \infty$; consequently, the Hartman-Wintner law of the iterated logarithm also holds. We give an example here, with $E = C\lbrack 0, 1\rbrack$, such that the above convergence does not imply $\mathscr{E}\lbrack \|X_1\|^2 \rbrack < \infty$, nor does it imply the law of the iterated logarithm.

Article information

Source
Ann. Probab., Volume 4, Number 4 (1976), 690-694.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996040

Digital Object Identifier
doi:10.1214/aop/1176996040

Mathematical Reviews number (MathSciNet)
MR451325

Zentralblatt MATH identifier
0338.60009

JSTOR
links.jstor.org

Subjects
Primary: 60B10: Convergence of probability measures
Secondary: 60G15: Gaussian processes

Keywords
Banach space valued random variables sums of independent random variables central limit theorem law of the iterated logarithm

Citation

Jain, Naresh C. An Example Concerning CLT and LIL in Banach Space. Ann. Probab. 4 (1976), no. 4, 690--694. doi:10.1214/aop/1176996040. https://projecteuclid.org/euclid.aop/1176996040


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