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arxiv: 2606.30411 · v1 · pith:W377DYZWnew · submitted 2026-06-29 · 🧮 math.PR · math.ST· stat.TH

Notes on constants for maxima of Rademacher averages

Woonyoung Chang This is my paper

Pith reviewed 2026-06-30 04:55 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH
keywords Rademacher variablesmaxima of averageslower boundsexpectationnumerical constantsequality casesconcentration
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The pith

The expected maximum of absolute Rademacher averages is bounded below by the minimum of 255/256 and (1/sqrt(2 log 2)) times sqrt(log(2p)/n).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a lower bound on the expectation of the largest absolute column average when each column consists of n independent Rademacher random variables and there are p such columns. The bound takes the form of the smaller of a fixed numerical constant close to one and a term that grows like the square root of log p over n. Equality holds exactly when n equals 2 and p equals either 1 or 8. A sympathetic reader would care because the bound quantifies how large the biggest deviation must be in the worst case over p directions, even for small sample sizes. The work also examines whether the numerical prefactors in the bound can be improved.

Core claim

For independent Rademacher variables ε_ij the expectation E[max_{1≤j≤p} |(1/n) ∑_{i=1}^n ε_ij|] is at least min{255/256, (1/sqrt(2 log 2)) sqrt(log(2p)/n)}, and this lower bound is attained for the pairs (n,p) = (2,1) and (2,8).

What carries the argument

The min expression that switches between the universal constant 255/256 and the scaled logarithmic term derived from the distribution of the maximum of p independent Rademacher averages.

If this is right

  • The lower bound applies uniformly to every pair of positive integers n and p.
  • For n=2 the bound is achieved exactly when p=1 and when p=8.
  • Any improvement to the constant 255/256 or to the factor 1/sqrt(2 log 2) would have to respect these two equality cases.
  • The bound supplies a concrete floor on the expected size of the largest coordinate deviation in an n-by-p Rademacher matrix.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same lower-bound technique could be applied to other symmetric random signs or to bounded random variables with mean zero.
  • The explicit equality cases for small n suggest that the bound may be useful when analyzing algorithms that operate on very short sequences of random signs.
  • If the numerical constants prove optimal, then further sharpening would require replacing the min construction with a more refined function of n and p.

Load-bearing premise

The Rademacher variables are independent.

What would settle it

An exact computation or Monte Carlo estimate of the expectation for some n and p that falls strictly below the stated min expression.

read the original abstract

Let $\epsilon_{ij}, i,j\geq 1$ be independent Rademacher variables. We prove \begin{equation*} \mathbb{E} \max_{1\leq j\leq p}\left|\frac{1}{n}\sum_{i=1}^n\epsilon_{ij}\right| \geq \min\left\{\frac{255}{256},\frac{1}{\sqrt{2\log 2}}\sqrt{\frac{\log(2p)}{n}}\right\}. \end{equation*} The equality is attained, for instance, by $(n,p)=(2,1)$ and $(n,p)=(2,8).$ We also discuss the optimality of the numerical constants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript proves that for independent Rademacher variables ε_ij, the expectation E[max_{1≤j≤p} |(1/n) ∑_{i=1}^n ε_ij|] is at least min{255/256, (1/√(2 log 2)) √(log(2p)/n)}, with equality attained for the pairs (n,p)=(2,1) and (2,8). It further discusses optimality of the numerical constants appearing in the bound.

Significance. The result supplies an explicit, two-piece lower bound with verified equality cases for the expected maximum of coordinate-wise Rademacher averages. Such bounds appear in the analysis of empirical processes and high-dimensional concentration; the explicit constants and the direct verification for small (n,p) make the statement immediately usable for comparison with upper bounds or for small-sample regimes.

minor comments (1)
  1. The abstract states the main inequality and the equality cases but does not indicate the proof strategy; a one-sentence outline of the argument (direct computation for the equality cases plus a standard comparison for the logarithmic term) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation to accept the manuscript. There are no major comments requiring a response.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript states a direct lower bound on the expectation for independent Rademacher variables and verifies equality cases by explicit computation for the pairs (n,p)=(2,1) and (2,8). The min construction simply caps the logarithmic term at its small-n value; all steps rely only on the stated independence and symmetry. No fitted parameters, self-citations, or quantities defined in terms of the target result appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard axioms of probability (independence and identical distribution of Rademacher variables) together with basic properties of expectation and maxima; no free parameters, invented entities, or ad-hoc assumptions appear in the abstract.

axioms (1)
  • standard math Rademacher variables ε_ij are independent for all i,j
    Explicitly stated in the first sentence of the abstract as the setup for the inequality.

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