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

Nicholas J.A. Harvey, Christopher Liaw and Victor S. Portella

Stochastic Processes and their Applications, 2025, vol. 183, issue C

Abstract: Motivated by a problem from theoretical machine learning, we show asymptotically optimal bounds on EXτ∞/Eτ, where (Xt)t≥0 is a continuous stochastic process in Rn with (Xt,i)t≥0 being a Brownian motion for each i∈{1,…,n} and τ being a stopping time such that Eτ<∞. We further extend this result to the setting where the entries of (Xt)t≥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.

Keywords: Expected norm; Continuous martingales; Itô’s formula; Discrete martingales; Confluent Hypergeometric Function (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spa.2025.104575

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