Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$, and let $S_N = \sum_{n=0}^N X_n a_n$. With the events $$B_{j,m} = \left\{|S_m - S_{\phi(j)}| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \; |S_n - S_{\phi(j)}| \leqslant 2^{-j}\right\},$$ $$B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} |S_n - S_{\phi(j)}| > 2^{-j}\right\},$$ for all $j \in \mathbb{N}$, prove that the events $B_{j,m}$, for $m$ ranging over $\llbracket \phi(j)+1, \phi(j+1) \rrbracket$, are pairwise disjoint and that we have the equality of events $$B_j = \bigcup_{\phi(j) < m \leqslant \phi(j+1)} B_{j,m}.$$
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$, and let $S_N = \sum_{n=0}^N X_n a_n$. With the events
$$B_{j,m} = \left\{|S_m - S_{\phi(j)}| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \; |S_n - S_{\phi(j)}| \leqslant 2^{-j}\right\},$$
$$B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} |S_n - S_{\phi(j)}| > 2^{-j}\right\},$$
for all $j \in \mathbb{N}$, prove that the events $B_{j,m}$, for $m$ ranging over $\llbracket \phi(j)+1, \phi(j+1) \rrbracket$, are pairwise disjoint and that we have the equality of events
$$B_j = \bigcup_{\phi(j) < m \leqslant \phi(j+1)} B_{j,m}.$$