On the expected ℓ∞-norm of high-dimensional martingales

Abstract

Motivated by a problem from theoretical machine learning, we show asymptotically optimal bounds on $E\left({\left(X_{\tau }\right)}_{\infty }\right)/E\left(\sqrt{\tau }\right)$, where ${\left(X_t\right)}_{t\ge 0}$ is a continuous stochastic process in $R^n$ with ${\left(X_{t,i}\right)}_{t\ge 0}$ being a Brownian motion for each $i\in \{1,\dots ,n\}$ and $\tau$ being a stopping time such that $E\left(\sqrt{\tau }\right)<\infty$. We further extend this result to the setting where the entries of ${\left(X_t\right)}_{t\ge 0}$ have smooth quadratic variation. Finally, we show a similar result for discrete-time processes using analogous techniques, together with a discrete version of Itô’s formula.