Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, let $S_N = \sum_{n=0}^N X_n a_n$. Express the expectation and variance of $S_{\phi(j+1)} - S_{\phi(j)}$ in terms of the terms of the sequence $(a_n)_{n \in \mathbb{N}}$.