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- W2050646152 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 $mathscr{E}lbrack|X_1|^2rbrack < infty$; consequently, the Hartman-Wintner law of the iterated logarithm also holds. We give an example here, with $E = Clbrack 0, 1rbrack$, 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." @default.
- W2050646152 created "2016-06-24" @default.
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- W2050646152 date "1976-08-01" @default.
- W2050646152 modified "2023-10-18" @default.
- W2050646152 title "An Example Concerning CLT and LIL in Banach Space" @default.
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- W2050646152 doi "https://doi.org/10.1214/aop/1176996040" @default.
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