= Exponential moment bound for a Gaussian random-walk maximum
{title2=$\mathbb E e^{bZ}\leq2/(2-b),\quad0<b<2$}
For <independent and identically distributed random variables> with law $N(-1,1)$, $e^{2S_n}$ is a nonnegative <martingale>, since the increment's <moment-generating function> equals one at parameter two. The <Doob maximal inequality> gives $\mathbb P(Z>t)\leq e^{-2t}$ for the all-time maximum $Z\geq0$. Integrating the tail gives the displayed bound. For $b\leq0$, $e^{bZ}\leq1$. The <strong law of large numbers> guarantees finiteness and attainment of the maximum because $S_n\to-\infty$.
Back to article page
