Let $X_1,X_2,\dots$ be independent random variables such that $P(X_j=+a_j)=P(X_j=-a_j)=1/2$. Is it the case that for all sequences {$a_j: j\geq 1$}, where $a_j>0$ and $\sum_{j=1}^{\infty } a_j^{2}=\infty$, $\frac{X_1+\cdots+X_n}{\sqrt{a_1^2+\cdots+a_n^2}}$ converges to a standard normal distribution?
Convergence of Distribution
2
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probability
probability-theory
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0No. $ $ $ $ $ $ – 2011-11-14
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2What do you know, what did you try ? – 2011-11-14
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0You are definitely gonna need to put additional conditions on $\{a_i\}$ because if $a_i=2^i$ then $X_1+...+X_n$ will never be zero, so the limit must be zero for the value $0$, which means it cannot be normal. – 2011-11-14