How to verify the following for standard Brownian motion? $$P( \bigcup_{0 \leq t \leq 1} |B(t)| < b) = \sum_{k=-\infty}^{\infty} (-1)^k P( (2k-1)b < B(1) < (2k+1)b) $$ My initial thoughts are that the LHS is the same as $P(\sup_{0\leq t \leq 1}|B(t)| < b)$ and I would use that $P(|B(t)| < b) = 2P(B(t) < b) -1$ in conjunction with the reflection principle, but I am not sure about how this links with the RHS?
2026-04-03 17:11:41.1775236301
Resnick Exercise 6.12 - Standard Brownian Motion
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