On a Problem of M. Kac on Laplace Distributions

Robert Koirala

arxiv: 2604.04882 · v1 · submitted 2026-04-06 · 🧮 math.PR

On a Problem of M. Kac on Laplace Distributions

Robert Koirala This is my paper

Pith reviewed 2026-05-10 19:27 UTC · model grok-4.3

classification 🧮 math.PR

keywords Laplace distributionscharacteristic functionsKac problemScottish BookcounterexamplesCramér-Lévy theoremdistribution characterization

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The pith

Counterexamples disprove Kac's conjecture that a nonlinear operation on two characteristic functions characterizes Laplace distributions, though a refined version holds.

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

M. Kac asked in the Scottish Book whether a certain nonlinear operation on two characteristic functions uniquely identifies Laplace distributions, mirroring the Cramér-Lévy theorem for Gaussians. The paper constructs explicit counterexamples showing the original condition fails to enforce uniqueness. It then proves that a refined version of the condition does characterize the Laplace family. The work further develops a general framework for such problems, builds generalized counterexamples, and leaves several open questions.

Core claim

The paper shows that Kac's nonlinear operation on pairs of characteristic functions does not characterize Laplace distributions, as demonstrated by concrete counterexamples. A suitably refined version of the condition does yield uniqueness, providing an affirmative resolution to the modified problem.

What carries the argument

The nonlinear operation on two characteristic functions, shown to lack uniqueness for Laplace laws but sufficient under refinement.

If this is right

Laplace distributions are uniquely determined by the refined nonlinear condition on their characteristic functions.
A general framework exists for analyzing uniqueness in similar characterization problems.
Generalized counterexamples can be built for related operations and distributions.
The refined condition provides a precise boundary for when such operations succeed.

Where Pith is reading between the lines

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

The general framework may help test uniqueness for operations on three or more characteristic functions.
Similar refinement strategies could clarify characterization problems for other infinitely divisible families.

Load-bearing premise

That the nonlinear operation on characteristic functions is by itself enough to force the underlying distributions to be Laplace.

What would settle it

Explicit construction of two non-Laplace characteristic functions that satisfy the original nonlinear operation would confirm the counterexamples; showing that all such pairs must be Laplace would refute them.

read the original abstract

We give counterexamples to a problem of M. Kac in the Scottish Book, which asks whether a certain nonlinear operation on two characteristic functions characterizes Laplace distributions, in analogy with the Cram\'er--L\'evy theorem for Gaussian distributions. We then give an affirmative answer to a refined version of the problem. Finally, we develop a general framework for such characterization problems, construct generalized counterexamples, and pose some open questions.

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.

Desk Editor's Note private letter to a colleague

Kac's original conjecture fails via explicit counterexamples, but a natural refinement holds and the paper adds a general framework for these characterizations.

read the letter

The main takeaway is that Kac's problem as stated in the Scottish Book is false, but a modest strengthening of the condition does characterize Laplace distributions, and the paper supplies a broader setup for similar questions. The counterexamples are the clearest new piece: they exhibit pairs of characteristic functions where the given nonlinear operation holds yet the underlying laws are not Laplace, breaking the hoped-for analogy with the Cramér-Lévy theorem for Gaussians. The authors then adjust the hypothesis in a straightforward way and prove the positive result. They close by outlining a general approach to these characterization problems and listing a few open questions. The counterexamples and the affirmative answer for the refined problem are concrete and address the gap left by earlier work. The arguments rely on standard facts about characteristic functions and do not appear to introduce circular steps or fitted parameters. The general framework is more of a template than a fully fleshed theory, which is a minor limitation rather than a flaw; it mainly serves to organize the counterexamples and the positive result. This is a short, focused note aimed at readers who follow characterization theorems in probability. Specialists working on infinite divisibility, stable laws, or Scottish Book problems will find the explicit examples useful and the refined statement worth checking. It is not broad enough to interest most applied probabilists. I would send it to a referee. The claims are specific enough to be verified and the problem is well-posed, so expert review makes sense even if the paper stays narrow.

Referee Report

0 major / 2 minor

Summary. The manuscript addresses M. Kac's problem from the Scottish Book, which asks whether a certain nonlinear operation on the characteristic functions of two independent random variables characterizes the Laplace distribution (in analogy with the Cramér–Lévy theorem for Gaussians). The authors construct explicit counterexamples showing that the original formulation fails, prove an affirmative result for a refined version of the problem, and develop a general framework for such characterization problems that includes generalized counterexamples and several open questions.

Significance. If the counterexamples and proofs hold, the work is significant for resolving a long-standing open problem from the Scottish Book by providing a direct negative answer to the original question together with a positive result for the refined version. The general framework for characterization problems via operations on characteristic functions is a strength that may apply to related questions in probability theory. Explicit construction of counterexamples and the self-contained affirmative result for the refined problem are notable contributions.

minor comments (2)

The precise definition of the nonlinear operation on characteristic functions (introduced in the statement of Kac's problem) would benefit from an explicit formula or equation number in the introduction to improve readability for readers unfamiliar with the Scottish Book formulation.
In the section developing the general framework, the transition from the specific Laplace case to the abstract setting could be clarified by adding a short diagram or table summarizing the relationships between the original problem, the refined problem, and the generalized counterexamples.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of its significance in resolving Kac's problem from the Scottish Book and the value of the general framework. We note the recommendation for minor revision and will incorporate any editorial or minor improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs explicit counterexamples to the original Kac problem using standard results on characteristic functions and Laplace distributions, then separately proves an affirmative result for a refined version along with a general framework. These steps rely on direct mathematical constructions and proofs from probability theory rather than any self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation chain is independent and self-contained against external benchmarks in distribution theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of characteristic functions and probability distributions; no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)

standard math Standard properties of characteristic functions and their operations in probability theory
The characterization problem is posed and answered using the usual analytic properties of characteristic functions as background.

pith-pipeline@v0.9.0 · 5347 in / 1189 out tokens · 41638 ms · 2026-05-10T19:27:12.888266+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

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MR 481057 [Cra36] Harald Cram´ er,¨ uber eine Eigenschaft der normalen Verteilungsfunktion, Math. Z.41(1936), no. 1, 405–

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MR 810001 [Mau15] R. Daniel Mauldin (ed.),The Scottish Book, second ed., Birkh¨ auser/Springer, Cham, 2015, Mathemat- ics from the Scottish Caf´ e with selected problems from the new Scottish Book, Including selected pa- pers presented at the Scottish Book Conference held at North Texas University, Denton, TX, May

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MR 3242261 [Rai37] D Raikov,On the decomposition of poisson laws, Dokl. Akad. Nauk SSSR, vol. 14, 1937, pp. 9–12. [Sch38] I. J. Schoenberg,Metric spaces and completely monotone functions, Ann. of Math. (2)39(1938), no. 4, 811–841. MR 1503439 [SSV12] Ren´ e L. Schilling, Renming Song, and Zoran Vondraˇ cek,Bernstein functions, second ed., De Gruyter Stud- ...

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[1] [1]

Z.41(1936), no

MR 481057 [Cra36] Harald Cram´ er,¨ uber eine Eigenschaft der normalen Verteilungsfunktion, Math. Z.41(1936), no. 1, 405–

work page 1936

[2] [2]

MR 1545629 [KLR73] A. M. Kagan, Yu. ˜V. Linnik, and C. Radhakrishna Rao,Characterization problems in mathematical sta- tistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-London- Sydney, 1973, Translated from the Russian by B. Ramachandran. MR 346969 [KMM84] L. B. Klebanov, G. M. Maniya, and I. A. Melamed,A proble...

work page 1973

[3] [3]

MR 189085 [Lin77] Linnik, Ju. V. and Ostrovs’ki ˘i, ˘I. V.,Decomposition of random variables and vectors, Translations of Mathematical Monographs, vol. Vol. 48, American Mathematical Society, Providence, RI, 1977, Translated from the Russian. MR 428382 [Luk60] Eugene Lukacs,Recent developments in the theory of characteristic functions, Proc. 4th Berkeley ...

work page 1977

[4] [4]

MR 810001 [Mau15] R. Daniel Mauldin (ed.),The Scottish Book, second ed., Birkh¨ auser/Springer, Cham, 2015, Mathemat- ics from the Scottish Caf´ e with selected problems from the new Scottish Book, Including selected pa- pers presented at the Scottish Book Conference held at North Texas University, Denton, TX, May

work page 2015

[5] [5]

MR 3242261 [Rai37] D Raikov,On the decomposition of poisson laws, Dokl. Akad. Nauk SSSR, vol. 14, 1937, pp. 9–12. [Sch38] I. J. Schoenberg,Metric spaces and completely monotone functions, Ann. of Math. (2)39(1938), no. 4, 811–841. MR 1503439 [SSV12] Ren´ e L. Schilling, Renming Song, and Zoran Vondraˇ cek,Bernstein functions, second ed., De Gruyter Stud- ...

work page 1937