On a moment determinacy conjecture of Bertoin and Yor
Reviewed by Pith2026-07-02 17:33 UTCgrok-4.3pith:2Q433F3Qopen to challenge →
The pith
The absence of positive jumps is necessary for moment-determinacy of the reciprocal exponential functional of a Lévy process.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Bertoin and Yor proved that X_ξ is moment-determinate when ξ has no positive jumps and conjectured the condition is necessary. This paper proves the conjecture by showing that positive jumps imply moment-indeterminacy. It achieves this by establishing a lower bound near zero for the law of I_ξ through the effect of sufficiently many positive jumps near the origin. The first selected jump time acts as a smooth coordinate to produce an absolutely continuous subcomponent of the law of I_ξ. After the change of variables, the subdensity of X_ξ satisfies the Krein moment indeterminacy criterion.
What carries the argument
The first selected jump time serving as a one-dimensional smooth coordinate to extract an absolutely continuous subcomponent from the law of I_ξ, which after inversion to X_ξ yields a subdensity to which the Krein criterion applies.
If this is right
- X_ξ is moment-indeterminate whenever the Lévy process ξ has positive jumps.
- The condition of no positive jumps is both necessary and sufficient for moment-determinacy of X_ξ.
- The law of I_ξ has a positive lower bound near zero when positive jumps are allowed.
- The Krein criterion detects indeterminacy in a subdensity obtained via change of variables from the jump timing coordinate.
Where Pith is reading between the lines
- Similar coordinate-based decompositions could be used to analyze moment problems for other path-dependent functionals of processes with jumps.
- The result highlights the role of jump structure in determining uniqueness in moment problems for exponential functionals.
- Extensions might consider processes with killing or different moment conditions to see if the necessity holds more generally.
Load-bearing premise
The Lévy process must be unkilled, drift to +∞, and have positive exponential moments of all orders so that I_ξ exists and the Krein criterion can be applied after the variable change.
What would settle it
Finding a Lévy process with positive jumps that satisfies the assumptions yet has X_ξ uniquely determined by its moments would show the necessity claim is false.
Figures
read the original abstract
Let $\xi$ be an unkilled real-valued L\'evy process which drifts to $+\infty$ and has positive exponential moments of all orders, and define $I_\xi=\int_0^\infty e^{-\xi_t},dt$, and its reciprocal $X_\xi=1/I_\xi$. Bertoin and Yor proved that $X_\xi$ is moment-determinate when $\xi$ has no positive jumps, and conjectured that this condition is also necessary. We prove the latter. The proof is based on a lower bound near zero for the law of $I_\xi$. We show that a group of sufficiently many positive jumps near the origin puts $I_\xi$ on a suitable small scale. The first selected jump time is used as a one-dimensional smooth coordinate, yielding an absolutely continuous subcomponent of the law of $I_\xi$. After the change of variables, the resulting subdensity of $X_\xi$ satisfies a Krein moment indeterminacy criterion.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves the necessity part of the Bertoin-Yor conjecture: for an unkilled Lévy process ξ drifting to +∞ with positive exponential moments of all orders, the random variable X_ξ = 1/I_ξ (with I_ξ = ∫_0^∞ e^{-ξ_t} dt) is moment-determinate if and only if ξ has no positive jumps. Sufficiency was already known; necessity is shown by constructing a lower bound near zero on the law of I_ξ via sufficiently many positive jumps, using the first selected jump time as a one-dimensional smooth coordinate to produce an absolutely continuous sub-component of the law of I_ξ, performing the change of variables to X_ξ, and verifying that the resulting sub-density satisfies a Krein moment-indeterminacy criterion.
Significance. Resolving the conjecture supplies a complete if-and-only-if characterization of moment determinacy for this exponential functional of Lévy processes. The explicit construction of an absolutely continuous sub-density via jump selection, followed by direct application of the Krein criterion, is a concrete and potentially reusable technique for related moment problems in stochastic processes.
minor comments (2)
- [Abstract] The abstract states that the sub-density 'satisfies a Krein moment indeterminacy criterion' but does not name the precise form (e.g., the integral condition on the density or the reference to Krein’s theorem); a one-sentence clarification would help readers.
- Notation for the selected jump times and the resulting sub-density should be introduced with a short display equation in the first paragraph of the proof section to improve readability.
Simulated Author's Rebuttal
We thank the referee for their positive summary of our work, the recognition of its significance in resolving the Bertoin-Yor conjecture, and the recommendation to accept the manuscript.
Circularity Check
No significant circularity; derivation uses external Krein criterion on constructed sub-density
full rationale
The paper proves necessity of the no-positive-jumps condition by constructing an absolutely continuous subcomponent of the law of I_ξ via selection of positive jumps and the first jump time as a coordinate, followed by change of variables to obtain a subdensity of X_ξ that satisfies the Krein indeterminacy criterion. This chain relies on standard Lévy process properties and an external moment-indeterminacy criterion rather than any self-definition, fitted input renamed as prediction, or load-bearing self-citation. The cited Bertoin-Yor result is the conjecture being resolved, not a self-referential premise. No step reduces the target claim to an input by construction.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Lévy process ξ is unkilled, drifts to +∞, and has positive exponential moments of all orders
- standard math Krein moment indeterminacy criterion applies to the constructed sub-density of X_ξ
Reference graph
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discussion (0)
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