Let $X_{1},X_{2},....$ be a sequence of independent and identically distributed random variables . The odd ordered random variables follows the distribution:
$P(X_{2k-1}=1)=P(X_{2k-1}=-1)=\frac{1}{2}$
And all Even ordered random variables follows:
$f(x)=\frac{1}{\sqrt(2)}e^{-\sqrt(2)|x|},$ where $-\infty < x < \infty$. It is required to obtain the probability $\lim_{n\rightarrow \infty}P\left [ \frac{X_{1}+X_{2}+...+X_{2n}}{\sqrt(2n)}\geq1 \right ]$.
My approach
Since, the mean of both odd and even ordered random variables is zero, it is only required to obtain the second order moment for variance. The second order moment of odd ordered random variables turned out to be $1$ and similarly solving the Gamma integral for the even ordered moment, the second order moment is $1$ for than also. Hence, the variance of sum of $2n$ random variables is $2n$. The expression inside the Probability turned out to be standard Normal variate. So, the required probability will be $1-\Phi(1)$.
Did I do everything correctly?. My manual is not giving me any answer. Any suggestions will be very beneficial for me. Thanks