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Does the ratio of Laplace transforms of powers of a function identify the function?

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Laplace transform ratios of powers identify a nondecreasing function

desk verdict A genuine injectivity result for ratios of Laplace transforms of powers, with an auction-theory identification corollary; the core proofs are sound, but several display typos need correction. read the letter →

arxiv 1909.01884 v3 pith:TAYSMCPC submitted 2019-09-04 math.PR

Takis Konstantopoulos , Linglong Yuan This is my paper · ORCID
classification math.PR MSC 44A1026Axx91B7091B26
keywords Laplacetransformratiouniquenessorderstatisticsauctiontheoryrightanalyticfunctionsmonotoneidentificationprobleminverse
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether the ratio of Laplace transforms of two distinct powers of a function f determines f uniquely. It proves that for nonnegative, nondecreasing, cadlag, right-analytic functions of exponential order, the ratio H_{n,m}(f, λ) = L(f^n)(λ) / L(f^m)(λ) completely identifies f. For polynomials and entire functions, the same conclusion holds, with the only ambiguity being a global sign when n − m is even. The result matters because this exact ratio appears in auction theory, where it encodes the distribution of bidder-specific private values from the two highest bids.

What carries the argument

The central object is the ratio H_{n,m}(f, λ) = L(f^n)(λ) / L(f^m)(λ), an analytic quantity for large real λ when f is of exponential order. The proof's machinery is a bootstrap: Lemmas 1 and 2 compare Taylor coefficients at 0 via $\beta$-function integrals of convolutions f^n * g^m and f^m * g^n, showing all right derivatives at 0 match. Lemma 3 then takes a point a where f = g up to a, and uses monotonicity to force a sign contradiction if the first differing right derivative had opposite sign, so right analyticity lets equality step from a to a neighborhood beyond a. This pushes equality from 0 to all of [0,∞).

What would settle it

Construct two smooth, nonnegative, nondecreasing, positive-on-(0,∞) functions f and g, right analytic on [0,∞), with L($f^{2}$)L(g) = L($g^{2}$)L(f) but f ≠ g; such a pair would disprove Theorem 2. Since no such pair is known, the theorem's correctness hinges on the absence of this counterexample.

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Extended reading notes

Core claim

The central claim is Theorem 2: if f and g are nonnegative, nondecreasing, cadlag, right analytic at every point of [0,∞), of exponential order, and strictly positive for x > 0, then equality of H_{n,m}(f, ·) and H_{n,m}(g, ·) forces f = g. The proof first shows that all right derivatives of f and g at 0 coincide, using Taylor expansions and $\beta$-function integrals applied to the convolution identity f^n * g^m = f^m * g^n. It then considers the largest a such that f and g agree on [0, a); Lemma 3 shows that if they agree up to a, then their right derivatives at a also agree, so right analyticity extends the agreement past a, contradicting maximality unless a = ∞.

Load-bearing premise

The proof needs the functions to be nondecreasing; without monotonicity, the sign comparison in Lemma 3 collapses, so equality cannot be pushed from a neighborhood of zero to the whole half-line.

Editorial extensions

If this is right

  • Any two functions in the stated class with the same ratio of Laplace transforms of powers are the same function, so the map f ↦ H_{n,m}(f, ·) is injective on that class.
  • In the symmetric auction model with unobserved common value and idiosyncratic shocks, the common distribution F of the idiosyncratic component is identified from the Laplace transforms of the highest and second-highest bids (Theorem 3).
  • For polynomials and entire functions, equality of the ratio identifies f exactly when n − m is odd; when n − m is even, the only possible ambiguity is f versus −f.
  • The derivative-matching argument provides a constructive route: the Taylor data of f at 0 are recovered from H_{n,m}, and the bootstrap then determines f on the whole half-line.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the authors' conjecture that monotonicity can be dropped is correct, the auction identification result would hold for all distribution functions, not only nondecreasing smooth ones.
  • The same ratio-injectivity phenomenon may hold for other integral transforms with a convolution structure, where ratios of powers act as enough information to determine the underlying function up to symmetry.
  • The bootstrap suggests a numerical recovery scheme for f from H_{n,m}, though the reliance on derivatives at 0 likely makes the reconstruction sensitive to small errors in the ratio.
  • When n − m is odd, an explicit inversion formula analogous to the inverse Laplace transform may exist, since all Taylor coefficients are then determined without sign ambiguity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the injectivity of the map f ↦ H_{n,m}(f,·) = \widehat{f^n}/\widehat{f^m} on suitable classes of functions on [0,∞). It proves that for distinct positive integers m,n the ratio determines f uniquely among polynomials (up to global sign when n-m is even) and among nonnegative nondecreasing càdlàg functions that are right analytic at every point and of exponential order. The proof compares the small-time asymptotics of the convolution f^n * g^m, uses beta-function computations to transfer all right derivatives at 0, and then bootstraps equality from a right neighborhood of 0 to the whole line via a comparison lemma. The paper also applies the result to an auction model, where F is the common CDF of idiosyncratic shocks and H_{N-1,N}(F,·) is shown to determine F, so that the ratio of Laplace transforms of the two highest bids identifies F.

Significance. The main uniqueness result (Theorem 2) is a genuine contribution: it provides a positive answer to a natural inverse problem for nonlinear Laplace-transform ratios under explicit and checkable hypotheses. The proof is self-contained and, modulo the display errors listed below, rigorous; it uses only beta-function asymptotics and real analytic continuation, with no parameter fitting. The paper is also honest in stating the open conjecture without monotonicity, and the polynomial/entire-function cases are cleanly separated. If the displayed errors are corrected, the auction application gives a useful identification result for a model with unobserved heterogeneity.

major comments (3)
  1. [Section 2, Lemma 2 (equation after (8))] The displayed simplification of the beta-function ratio is incorrect because the factor (km)!/(kn)! is omitted. The ratio equals n/m times (km)!/(kn)! times (ℓ-k+kn)!/(ℓ-k+km)!; after restoring this factor, the conclusion that the ratio is >1 for n>m and <1 for n<m still follows (because ℓ>k), so the argument can be repaired, but the printed equality is false.
  2. [Section 3, proof of Theorem 3 (formula for K)] The formula K(F,λ) = H_{N-1,N}(F,λ)/(N+(N-1)H_{N-1,N}(F,λ)) is wrong. Combining (14) and (15) gives K = \widehat{F^N}/(N\widehat{F^{N-1}}-(N-1)\widehat{F^N}) = 1/(N H_{N-1,N}(F,λ)-(N-1)). The conclusion of Theorem 3 survives because the corrected map H ↦ 1/(N H - (N-1)) is injective, but the displayed equality must be fixed.
  3. [Section 3, Example 2] The lognormal CDF is not right analytic at 0; all right derivatives of F at 0 vanish, so the Taylor series of F at 0 is identically zero, which cannot represent F on any right neighborhood of 0. Consequently Theorem 3, as stated with right analyticity at every point of [0,∞), does not apply to the lognormal family, and the claim 'By Theorem 3, the mapping (μ,σ) ↦ K(μ,σ,·) is injective' is unsupported. The authors should either replace the example with a distribution satisfying the theorem's assumptions (e.g., a Weibull distribution with shape parameter 2) or clarify and weaken the regularity condition at 0 and prove the needed variant.
minor comments (4)
  1. [Section 2, Lemma 1] In the definition of the constants, b := f^{(ℓ)}(0) should read b := g^{(ℓ)}(0), as b is used in (5) for the ℓ-th derivative of g.
  2. [Section 2, Lemma 1 proof] The comparison of p1 and p2 contains a garbled identity; the intended contradiction is that if p1>p2 then t^{-p2}h(t) = t^{p1-p2} t^{-p1}h(t) tends to 0, contradicting the nonzero limit C2.
  3. [Section 2, Lemma 3 proof] In the sentence after defining i0, the equality f^{(j)}(a+) = f^{(j)}(a+) should read f^{(j)}(a+) = g^{(j)}(a+).
  4. [Section 3, Example 2 and display (13)] The denominator in the displayed definition of K(μ,σ,λ) and in (13) is missing the factor λ; it should be E e^{-λε_(N-1)}, not E e^{-ε_(N-1)}.

Circularity Check

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No circularity: Theorem 2 is proved from first principles, and the auction application uses it as an internally proved lemma.

full rationale

The derivation chain is self-contained. Lemma 1 and Lemma 2 compare Taylor coefficients at 0 using the convolution identity f^n*g^m = f^m*g^n derived from equality of Laplace-transform ratios; Lemma 3 uses monotonicity to force f(a+)=g(a+), and right analyticity then extends equality from a neighborhood of 0 to all of [0,∞). No parameter is fitted, no known result is renamed, and no load-bearing claim is justified by a self-citation. Theorem 3 uses Theorem 2 as an internally proved lemma, and the displayed formula for K(F,lambda), although it contains a sign typo in the denominator, is derived by direct integration from the order-statistics distributions. The one external citation used for a structural idea, [10], is not by the present authors and does not replace a proof step. Thus no circular step is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no free parameters and no new entities. It relies on standard Laplace and real-analysis facts and on the stated regularity and monotonicity hypotheses. The main mathematical burden is the monotonicity and right-analyticity class, which the authors explicitly flag as a conjecture to be relaxed.

assumptions (7)
  • standard math Injectivity of the Laplace transform and its inversion on [0,∞) for functions of exponential order.
    Used in Section 1 to say the m=0 case is solved by inverse Laplace transform, and in Section 2 to pass from ratio equality to convolution equality f^n*g^m = f^m*g^n, equation (3).
  • standard math Analyticity of Laplace transforms of powers of exponential-order functions in a right half-plane.
    Section 1 invokes Doetsch, Theorem 6.1, for analyticity of \hat{f^n} and \hat{f^m}.
  • standard math Beta function identity B(α,β)=Γ(α)Γ(β)/Γ(α+β).
    Equation (2), used in Lemmas 1 and 2 to compute limits of convolution products near zero.
  • standard math Right analyticity implies local power series representation and unique determination by right derivatives at every point.
    Definition of right analyticity in Section 1 and the real analytic extension property from Krantz and Parks are used in Theorem 2 to bootstrap f=g from a neighborhood to all of [0,∞).
  • domain assumption The domain assumptions on f and g: nonnegative, nondecreasing, cadlag, right analytic, exponential order, positive on (0,∞).
    These are the hypotheses of Theorem 2; the authors conjecture the theorem without monotonicity, and Section 1 shows some support condition is necessary to remove translation ambiguity.
  • domain assumption Monotonicity in Lemma 3 yields g(a+δ-u) ≥ g(u) for u<δ<a.
    This inequality is the engine of the contradiction in Lemma 3; without monotonicity the proof breaks down.
  • domain assumption Nonnegativity selects the positive sign in Corollary 1 when n-m is even.
    In the proof of Theorem 2, |f^{(k)}(0)|=|g^{(k)}(0)| becomes f^{(k)}(0)=g^{(k)}(0) because f and g are nonnegative; for general sign-changing functions the even case would also allow f=-g.

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Pith. "Pith review of Does the ratio of Laplace transforms of powers of a function identify the function?." pith.science (2026). https://pith.science/paper/TAYSMCPC

@misc{pith2026190901884,
  author       = {Pith},
  title        = {Pith review of: Does the ratio of Laplace transforms of powers of a function identify the function?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TAYSMCPC}},
  note         = {Machine review of arXiv:1909.01884}
}
abstract

We study the following question: if $f$ is a nonzero measurable function on $[0,\infty)$ and $m$ and $n$ distinct nonnegative integers, does the ratio $\widehat{f^n}/\widehat{f^m}$ of the Laplace transforms of the powers $f^n$ and $f^m$ of $f$ uniquely determine $f$? The answer is yes if one of $m, n$ is zero, by the inverse Laplace transform. Under some assumptions on the smoothness of $f$ we show that the answer in the general case is also affirmative. The question arose from a problem in economics, specifically in auction theory where $f$ is the cumulative distribution function of a certain random variable. This is also discussed in the paper.

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Reference graph

Works this paper leans on

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