Majorizing Measures for Canonical Processes with Log-Concave Tails
Abstract
Let Y1,…,Yn be independent symmetric random variables with log-concave tails. We give a dimension-free characterization of the expected supremum of the canonical process Xx=∑i=1n xiYi without any Δ2 or regular-growth assumption on the coordinate tails. The characterization is governed by scale-dependent intrinsic costs determined directly by the convex tail potentials. It has an information-theoretic rate-distortion formulation as well as two equivalent majorizing-measure formulations on the original index set, one multiscale and one based on a single probability measure. More strongly, the comparison holds for every prescribed law of the index. In particular, the Gaussian majorizing-measure theorem of Talagrand and the Bernoulli theorem of Bednorz-Latała are recovered from the same formula, through quadratic and truncated quadratic costs, respectively. The new geometry is necessary: once the Δ2 growth condition is removed, the usual chaining functional based on increment moments can exceed the expected supremum by an unbounded factor. The proof is based on a uniform convex-order truncation principle for symmetric exponentials and a dyadic decomposition of the tail slopes that transfers simultaneously the random process and its intrinsic cost, with universal constants.
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BibTeX
@misc{hu2026majorizing,
title = {{Majorizing Measures for Canonical Processes with Log-Concave Tails}},
author = {Xuanang Hu and Hanchao Wang and Xinglong Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13959},
eprint = {2609.13959},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13959},
}
RIS
TY - UNPB
TI - Majorizing Measures for Canonical Processes with Log-Concave Tails
AU - Xuanang Hu
AU - Hanchao Wang
AU - Xinglong Wu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13959
UR - https://arxiv.org/abs/2609.13959
AB - Let Y_1,…,Y_n be independent symmetric random variables with log-concave tails. We give a dimension-free characterization of the expected supremum of the canonical process X_x=∑_i=1^n x_iY_i without any Δ_2 or regular-growth assumption on the coordinate tails. The characterization is governed by scale-dependent intrinsic costs determined directly by the convex tail potentials. It has an information-theoretic rate-distortion formulation as well as two equivalent majorizing-measure formulations on the original index set, one multiscale and one based on a single probability measure. More strongly, the comparison holds for every prescribed law of the index. In particular, the Gaussian majorizing-measure theorem of Talagrand and the Bernoulli theorem of Bednorz-Latała are recovered from the same formula, through quadratic and truncated quadratic costs, respectively. The new geometry is necessary: once the Δ_2 growth condition is removed, the usual chaining functional based on increment moments can exceed the expected supremum by an unbounded factor. The proof is based on a uniform convex-order truncation principle for symmetric exponentials and a dyadic decomposition of the tail slopes that transfers simultaneously the random process and its intrinsic cost, with universal constants.
ER -