I am stuck with the following problem.
For any $k\geq 1$, let $N_k$ be a standard Gaussian, and $X_k:=\sum_{j=1}^kN_j$.
I would need to prove the following:
$$\lim_{k\rightarrow \infty} \frac{1}{k}\sum_{i=1}^k\mathbb{I}\{X_{2^i}\geq 0\}=1/2$$
or in plain english, the proportion of times $X$ is greater than 0 (along the sequence $2^i$) goes to 1/2 almost surely.
I feel like I should use somewhere a kind of symmetry of the "random walk" somewhere, but I do not really know how...
Thank you very much in advance for trying this problem!