= Threshold stopping of an independent Gaussian series
{title2=$\mathbb EM_T=0\iff\sum_ka_k^2<\infty$}
Let $M_n=\sum_{k\leq n}a_kZ_k$, and stop on first reaching a fixed positive threshold. If the coefficient squares are summable, <uniform integrability> gives the stopped mean zero, with the terminal limit used on $T=\infty$. If the coefficient squares diverge, $\mathbb P(M_n\geq R)\to1/2$ for each fixed $R$. Thus $\mathbb P(\sup_nM_n=\infty)\geq1/2$. This event is unaffected by changing finitely many of the <independent> variables, so the <Kolmogorov zero-one law> makes its probability one. Hence $T<\infty$ and $M_T$ is at least the positive threshold; its <expectation>, finite or infinite, cannot be zero.
Back to article page
