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August, 1976 An Example Concerning CLT and LIL in Banach Space
Naresh C. Jain
Ann. Probab. 4(4): 690-694 (August, 1976). DOI: 10.1214/aop/1176996040


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.


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Naresh C. Jain. "An Example Concerning CLT and LIL in Banach Space." Ann. Probab. 4 (4) 690 - 694, August, 1976.


Published: August, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0338.60009
MathSciNet: MR451325
Digital Object Identifier: 10.1214/aop/1176996040

Primary: 60B10
Secondary: 60G15

Keywords: Banach space valued random variables , central limit theorem , Law of the iterated logarithm , Sums of independent random variables

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 4 • August, 1976
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