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.