## The Annals of Probability

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

### An Example Concerning CLT and LIL in Banach Space

#### 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