2
$\begingroup$

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?

  • 0
    No. $ $ $ $ $ $2011-11-14
  • 2
    What do you know, what did you try ?2011-11-14
  • 0
    You 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

1 Answers 1