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Chaining, tree and measure for some canonical processes

  • Xuanang Hu
  • Hanchao Wang
  • Xinglong Wu
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Abstract

We prove two deterministic results for families of distances arising in the study of canonical processes. The first derives an admissible partition scheme from a growth condition. The second gives a representation in terms of parameterized separation trees and compares it with the corresponding majorizing-measure quantities. The main point is that the proofs do not depend on the distribution of the underlying process: once the initial distance and the family of distances are given, no random variables, independence, tail functions, or moment estimates are used. For canonical processes with regular log-concave tails, the assumptions of the abstract results follow from the usual regularity conditions. One direction of the separation-tree estimate also applies to Bernoulli processes without these additional assumptions, and we prove the reverse estimate for bounded convex unconditional index sets. We also give a version of the growth argument for points which, for finite index sets, leads to a recursive construction of admissible partitions and parameterized separation trees.

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Cite

BibTeXRIS
BibTeX
@misc{hu2026chaining,
  title = {{Chaining, tree and measure for some canonical processes}},
  author = {Xuanang Hu and Hanchao Wang and Xinglong Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01709},
  eprint = {2610.01709},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01709},
}
RIS
TY  - UNPB
TI  - Chaining, tree and measure for some canonical processes
AU  - Xuanang Hu
AU  - Hanchao Wang
AU  - Xinglong Wu
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01709
UR  - https://arxiv.org/abs/2610.01709
AB  - We prove two deterministic results for families of distances arising in the study of canonical processes. The first derives an admissible partition scheme from a growth condition. The second gives a representation in terms of parameterized separation trees and compares it with the corresponding majorizing-measure quantities. The main point is that the proofs do not depend on the distribution of the underlying process: once the initial distance and the family of distances are given, no random variables, independence, tail functions, or moment estimates are used. For canonical processes with regular log-concave tails, the assumptions of the abstract results follow from the usual regularity conditions. One direction of the separation-tree estimate also applies to Bernoulli processes without these additional assumptions, and we prove the reverse estimate for bounded convex unconditional index sets. We also give a version of the growth argument for points which, for finite index sets, leads to a recursive construction of admissible partitions and parameterized separation trees.
ER  -