Let $X_1,X_2,...$ be independent random variables such that $P(X_n = \pm1) = \frac {1-2^{-n}}{2}$ and $P(X_n = 2^k) = 2^{-k} $ for $k = n+1,n+2,...$
Define a new sequence of random variables by $Y_n = X_n$ if $X_n = \pm1$ and $Y_n = 0 $ otherwise. Use Lyapunov's condition to state and prove CLT for the sum of $Y_n's$ . state a similar CLT for {$X_n$}
This is what I have done so far:
$Y_n = X_nI[{X_n=\pm1}]$
$E(Y_n) = E(X_n)*P(X_n =\pm1)$ = $\frac {1-2^{-n}}{2}$
$E(Y_n^2) = E(X_n^2)*P(X_n =\pm1) = [\frac {1-2^{-n}}{2} + 2^k]*\frac {1-2^{-n}}{2}$
$V(Y_n) = E(Y_n^2)-E(Y_n)^2 = \sigma_n^2$(say)
Define $S_n = \sum_{i=1}^n Y_i$
let $s_n^2 = V(S_n)$
now to prove the required condition.
