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PreprintCC BY-NC-ND 4.0arXiv

Majorizing-measure bounds in the realized square-function metric

  • Witold M. Bednorz
  • Rafał M. Łochowski
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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 normmaxjoscT fjLp ≤ CDnormGμ(d)+√p ΔdLp, p≥1, where Δd is the diameter of the parameter space in the terminal square-function metric and Gμ(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.

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Cite

BibTeXRIS
BibTeX
@misc{bednorz2026majorizingmeasure,
  title = {{Majorizing-measure bounds in the realized square-function metric}},
  author = {Witold M. Bednorz and Rafał M. Łochowski},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38425},
  eprint = {2609.38425},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38425},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Majorizing-measure bounds in the realized square-function metric
AU  - Witold M. Bednorz
AU  - Rafał M. Łochowski
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.38425
UR  - https://arxiv.org/abs/2609.38425
AB  - 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 normmax_josc_T f_j_L^p ≤ CDnormG_μ(d)+√p Δ_d_L^p, p≥1, 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.
ER  -