"Path" of a Random Walk

Question

Suppose that $X_1,X_2,\ldots$ are i.i.d. with $\mathbb{P}(X_i = 1) = p$ and $\mathbb{P}(X_i = -1) = 1-p$, where $p$ is a constant less than 1/2. Let $Y_k = \sum_{i=1}^k X_i$. We can show that $Y_k$ is a random walk, or we can regard it as a Markov chain. Now consider $S_n = \sum_{k=1}^n 2^{Y_k}$. Can we prove the following inequality? $$ \mathbb{P}\{\limsup_n S_n < \infty\}>0. $$

Answer 1

$(u_k)^{1/k}=2^{Y_k/k}\to 2^{2p-1}<1$ almost surely by the strong law of large numbers. Therefore $\sum_{k=1}^{\infty}u_k<\infty$ almost surely.