Nonexistence conjecture for nonnegative jump amplitudes satisfying diffusion-scaling moment conditions

Let $(J_n)_{n\in\mathbb{N}}$ be a sequence of nonnegative random variables, and let $(\lambda_n)_{n\in\mathbb{N}}$ satisfy the limiting conditions denoted by

. Suppose the moment conditions denoted by

require the mean and second moment of $J_n$ to have the stated asymptotic orders and the fourth moment to be $o(\lambda_n^{-1})$. **Nonexistence conjecture.** A sequence of nonnegative random variables $(J_n)_{n\in\mathbb{N}}$ satisfying the conditions

under

does not exist. The claim would rule out the proposed nonnegative jump-amplitude scaling for the diffusion approximation, despite the subsequent discussion of the resulting Gaussian Ornstein--Uhlenbeck limit if these conditions were fulfilled.

Progress summary

Solved

A 2020 paper proves the proposed nonnegative-jump scaling cannot produce the stated diffusion limit, while a later posted proof reaches the same conclusion but has not been independently verified.

The conjecture asserts that nonnegative jump amplitudes cannot satisfy the moment scaling required for the proposed Gaussian diffusion approximation. No proposer is identified in the retrieved sources.

Known results

Tamborrino and Lansky (2020) derive the necessary diffusion-limit conditions and show that nonnegative jumps cannot satisfy them unless negative jumps occur with probability tending to zero.

Posted attempt

A posted argument claims a complete quantitative proof via Hölder's inequality: the scaled fourth moment must have a positive lower bound, contradicting its required vanishing. The argument has not been independently verified, although its conclusion agrees with the 2020 paper.

Current status (as of August 2026): The nonexistence claim is settled by the Tamborrino--Lansky result for the stated nonnegative-jump regime; the posted Hölder proof itself remains unverified.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Massimiliano Tamborrino and Petr Lansky, “Shot noise, weak convergence and diffusion approximations”, arXiv:2005.06067 (2020).

diffusion-approximations shot-noise-processes stochastic-processes

1 week ago

The conjecture holds, with a sharp quantitative strengthening. Let $J\geq0$ and $\lambda>0$ . Hölder's inequality gives

\mathbb E[J^2] =\mathbb E[J^{2/3}J^{4/3}] \leq (\mathbb E[J])^{2/3}(\mathbb E[J^4])^{1/3},

and therefore

\boxed{\quad (\lambda\mathbb E[J^2])^3 \leq (\lambda\mathbb E[J])^2(\lambda\mathbb E[J^4]). \quad}

Under the proposed conditions, the left-hand side converges to $\sigma^6>0$ , whereas the right-hand side converges to $\mu^2\cdot0=0$ , an impossibility. The additional assumptions $\lambda_n\to\infty$ and $\mathbb E[J_n]\to0$ are unnecessary.

More precisely, if $\mu>0$ , then

\liminf_{n\to\infty}\lambda_n\mathbb E[J_n^4] \geq\frac{\sigma^6}{\mu^2}>0;

if $\mu=0$ , then

\lambda_n\mathbb E[J_n^4]\longrightarrow+\infty.

The first bound is sharp: for any $\lambda\geq\mu^2/\sigma^2$ , take

J=\frac{\sigma^2}{\mu} \quad\text{with probability}\quad \frac{\mu^2}{\lambda\sigma^2},

and $J=0$ otherwise. Then

\lambda\mathbb E[J]=\mu,\qquad \lambda\mathbb E[J^2]=\sigma^2,\qquad \lambda\mathbb E[J^4]=\frac{\sigma^6}{\mu^2}.

Thus the moment conditions are incompatible throughout their entire stated parameter range.

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Shivam Patel · 2 days ago