Majorizing-measure bounds in the realized square-function metric

Abstract

The predictable square function measures the size of a martingale. Applied to parameter differences, it also defines a random pseudometric. We explain how to perform a majorizing-measure argument directly in this realized geometry, without conditioning the terminal field to be Gaussian and without first replacing the metric by a random multiple of a deterministic one. For a finite family of predictable Gaussian sums in a $(2,D)$-smooth Banach space, and a fixed probability measure $μ$ on the parameter set, we prove \[ \norm{\max_j\osc_T f_j}_{L^p} \le CD\norm{\G_μ(d)+\sqrt p\,Δ_d}_{L^p},\qquad p\ge1, \] where $Δ_d$ is the diameter of the parameter space in the terminal square-function metric and $\G_μ(d)$ is its ball-mass integral. The proof combines a localized exponential inequality, pathwise averaging over random balls, and a stopping-time argument. Atomic measures recover logarithmically weighted maximal inequalities; Haar measure connects the result with homogeneous entropy and continuity estimates. We also give extensions, examples, and a precise account of the restrictions on choosing the averaging measure.