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arxiv: 2502.01254 · v4 · submitted 2025-02-03 · 🧮 math.ST · stat.TH

A necessary and sufficient condition for convergence in distribution of the quantile process in L¹(0,1)

Brendan K. Beare , Tetsuya Kaji This is my paper

Pith reviewed 2026-05-23 04:38 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords quantile processconvergence in distributionL^1(0,1)quantile functionlocal absolute continuitysquare integrabilitybootstrap approximationiid sampling
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The pith

The quantile process converges in distribution in L^1(0,1) if and only if the quantile function is locally absolutely continuous and satisfies a strengthened square integrability.

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

The paper determines the precise conditions under which the quantile process from iid samples converges in distribution in the L^1 norm on (0,1). It shows that local absolute continuity of the quantile function together with a slight strengthening of square integrability is necessary and sufficient for this convergence. This also implies that the bootstrap can approximate the process under the condition. A reader cares because it identifies exactly when asymptotic results for quantiles hold in this topology.

Core claim

We establish a necessary and sufficient condition for the quantile process based on iid sampling to converge in distribution in L^1(0,1). The condition is that the quantile function is locally absolutely continuous and satisfies a slight strengthening of square integrability. If the quantile process converges in distribution then it may be approximated using the bootstrap.

What carries the argument

The quantile process in the L^1(0,1) topology, with the carrying condition being local absolute continuity and strengthened square integrability of the quantile function.

Load-bearing premise

The data are iid samples from a distribution whose quantile function is locally absolutely continuous and satisfies the strengthened square integrability.

What would settle it

Finding a counterexample distribution where the quantile function lacks local absolute continuity but the quantile process converges in L^1 distribution would disprove the necessity of the condition.

read the original abstract

We establish a necessary and sufficient condition for the quantile process based on iid sampling to converge in distribution in $L^1(0,1)$. The condition is that the quantile function is locally absolutely continuous and satisfies a slight strengthening of square integrability. If the quantile process converges in distribution then it may be approximated using the bootstrap.

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 / 3 minor

Summary. The manuscript establishes a necessary and sufficient condition for distributional convergence of the empirical quantile process (constructed from iid samples) in the Banach space L^1(0,1). The condition requires the quantile function to be locally absolutely continuous together with a mild strengthening of square-integrability; under this condition the bootstrap is shown to approximate the limiting distribution.

Significance. If correct, the result supplies a sharp characterization of weak convergence in L^1, a topology relevant for integrated functionals and certain risk measures. The necessity direction is especially useful because it identifies precisely when convergence fails. The bootstrap approximation adds immediate practical value. The paper delivers an explicit mathematical equivalence together with its proof.

minor comments (3)
  1. [Abstract, §1] Abstract and §1: the phrase “slight strengthening of square integrability” is imprecise; the exact integrability requirement (presumably something like ∫ Q²(u)/(u(1-u)) du < ∞) should be stated explicitly at the first appearance of the main theorem.
  2. [§2] §2 (notation): the definition of the quantile process and the precise topology on L^1(0,1) appear only after several paragraphs; moving the definitions to the beginning of §2 would improve readability.
  3. [Introduction] References: several classical works on quantile-process convergence (e.g., Csörgő–Horváth, Shorack) are cited only in passing; a short paragraph contrasting the L^1 result with existing sup-norm and L^p results would help situate the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of the significance of the necessary-and-sufficient characterization, and the recommendation of minor revision. Since no specific major comments were listed, we provide no point-by-point responses below.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves a necessary and sufficient condition (local absolute continuity of the quantile function plus a mild integrability strengthening) for weak convergence of the empirical quantile process in the Banach space L^1(0,1). This is a self-contained mathematical equivalence theorem in empirical process theory; the statement is an if-and-only-if characterization whose proof relies on standard external tools (e.g., properties of empirical processes and functional analysis) rather than any fitted parameters, self-referential definitions, or load-bearing self-citations that collapse the result to its own inputs. No equations or quantities are constructed by renaming or fitting within the paper itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard definitions of quantile functions, L1 convergence, and iid sampling that are taken from prior probability theory.

axioms (2)
  • standard math Quantile function is the generalized inverse of the cumulative distribution function
    Invoked implicitly when the quantile process is defined from the samples.
  • standard math L1(0,1) is equipped with the usual integral norm
    Required for the notion of convergence in distribution in that space.

pith-pipeline@v0.9.0 · 5578 in / 1131 out tokens · 32216 ms · 2026-05-23T04:38:43.624794+00:00 · methodology

discussion (0)

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Reference graph

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