= Brownian exit-skeleton clock convergence
{c}
{title2=$\sigma_{2^{-n}}^{2^{2n}}\longrightarrow\mathbb E\sigma_1\quad\text{almost surely}$}
Unit-radius <Brownian exit time> has finite <variance>, since independent unit-time increments give an exponential upper tail. The <Brownian exit-time skeleton> clock after $2^{2n}$ radius-$2^{-n}$ exits has mean $\mathbb E\sigma_1$ and <variance> $2^{-2n}\operatorname{Var}(\sigma_1)$. <Chebyshev's inequality> and the <Borel-Cantelli first lemma> yield the displayed pathwise convergence. <Independence> between different rows indexed by $n$ is unnecessary.
Back to article page
