A Counterexample to Small-time Limit Theorems for Stochastic Processes

Pietro Maria Sparago

arxiv: 2605.15931 · v1 · pith:7RFXJRX3new · submitted 2026-05-15 · 🧮 math.PR

A Counterexample to Small-time Limit Theorems for Stochastic Processes

Pietro Maria Sparago This is my paper

Pith reviewed 2026-05-19 18:56 UTC · model grok-4.3

classification 🧮 math.PR MSC 60F1760J60

keywords diffusionsItô SDEssmall-time limitsfinite-dimensional distributionsweak convergencecàdlàg processesexit timescounterexample

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The pith

A scaling of diffusions by exit from shrinking balls converges in finite-dimensional distributions but not weakly in the càdlàg topology.

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

The paper takes time-homogeneous diffusions defined by Itô stochastic differential equations and scales them by stopping at the first exit time from a ball whose radius shrinks like n to the power of minus one half, leaving time unscaled. It shows that the resulting sequence of processes converges in finite-dimensional distributions on a dense subset of the nonnegative reals. At the same time the sequence of their laws fails to converge weakly when the processes are viewed as random elements in the space of càdlàg paths. The construction therefore supplies a concrete counterexample to the expectation that small-time limit theorems for semimartingales automatically hold under this spatial-stopping scaling.

Core claim

For the class of time-homogeneous Itô diffusions under consideration, the processes stopped at the first exit from balls of radius n^{-1/2} converge in finite-dimensional distributions over a dense subset of [0, ∞), yet their laws do not converge weakly in the Skorokhod space of càdlàg functions; the law of each stopped process at its exit time is also characterised explicitly.

What carries the argument

The exit-time scaling that stops each diffusion upon first leaving a ball of radius n^{-1/2} without any accompanying time rescaling.

If this is right

The small-time functional central limit theorem for semimartingales does not extend to this exit-time scaling without additional conditions.
Finite-dimensional convergence on a dense time set is insufficient for weak convergence of càdlàg laws under the chosen scaling.
The distribution of the position at the stopping time admits a direct probabilistic description independent of the path convergence question.

Where Pith is reading between the lines

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

The gap between finite-dimensional and functional convergence here may stem from the lack of control on oscillations between the dense time points induced by the spatial stopping rule.
Analogous counterexamples could be constructed for other Markov processes whose exit times from shrinking sets interact poorly with path regularity.
Identifying the minimal regularity on the diffusion coefficients that would restore tightness under this scaling remains open.

Load-bearing premise

The processes are time-homogeneous diffusions driven by Itô SDEs and the scaling consists solely of stopping at the first exit from balls of radius shrinking as n^{-1/2}.

What would settle it

An explicit tightness estimate or modulus-of-continuity bound showing that the stopped processes are relatively compact in the Skorokhod space would establish weak convergence and thereby falsify the claimed absence of convergence.

read the original abstract

The standard small-time functional central limit theorem of semimartingales has been established in (Gerhold, S., Kleinert, M., Porkert, P., and Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications. Stochastics, 87), proving that the scaling limit law of a large class of stochastic processes in increasingly small time scales is that of a Brownian motion with a possibly nontrivial variance-covariance matrix. In this paper we focus on the time-homogeneous diffusion processes described by It\^{o} SDEs. Instead of the simple time scaling $1/n$ of (Gerhold, S., Kleinert, M., Porkert, P., and Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications. Stochastics, 87) we consider the scaled processes stopped at the first exit times from the balls of decreasing radius $n^{-1/2}$ without scaling time itself. To the best of our knowledge, this particular scaling has not been investigated in the literature. We prove that this is a nontrivial example of a sequence of processes which converges in the sense of finite-dimensional distributions over a dense subset of $[0,\infty)$, but it does not converge weakly in the sense of laws of c\`{a}dl\`{a}g processes. We also characterise the limit law of the scaled processes evaluated at their respective first exit times.

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

This paper gives a direct counterexample showing finite-dimensional convergence on a dense time set but failure of weak convergence in the càdlàg topology under exit-time scaling without time rescaling.

read the letter

This paper's key point is that certain scaled versions of time-homogeneous diffusions converge in finite-dimensional distributions on a dense time grid but do not converge weakly as càdlàg processes. The scaling here stops the process at the first exit from a ball of radius n to the minus one half, with no adjustment to the time variable. This differs from the standard small-time scaling in the literature. The new element is the focus on this exit-time scaling, which the authors say has not been studied before. They position it as a counterexample to the 2015 result by Gerhold and others, but under a different regime. They also characterize what the limit looks like when evaluated at the exit times. The paper does a good job laying out an explicit construction for the counterexample. It uses direct scaling and stopping times on Itô SDEs, which keeps things straightforward and avoids circular reasoning. The claim about convergence in finite dimensions over a dense subset while failing in the full path space is clearly stated. The soft spot is that the abstract only outlines the proof strategy for non-convergence in the càdlàg topology and the exit-time characterization. Without seeing the full details on tightness arguments or how they handle the dense time set, it's difficult to spot any gaps. The soundness depends on whether those steps hold up in the manuscript. This work is for researchers in probability theory who study limit theorems for stochastic processes, particularly those dealing with semimartingales and small-time asymptotics. A reader who wants to understand the boundaries of functional central limit theorems would get value from the distinction drawn here. I think this paper deserves a serious referee. The construction raises a valid question about when weak convergence holds, and it should be checked carefully by experts in the field.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs a family of time-homogeneous Itô diffusions and examines their spatially scaled versions, obtained by stopping at the first exit time from balls of radius n^{-1/2} with no time rescaling. It claims that these processes converge in finite-dimensional distributions over a dense subset of [0, ∞) but fail to converge weakly in the Skorohod space of càdlàg paths. The limit law at the respective exit times is also characterized.

Significance. If the explicit construction and associated arguments hold, the result is significant: it supplies a concrete counterexample showing that small-time limit theorems for semimartingales are sensitive to the choice of scaling. The direct construction via stopping times (rather than time rescaling as in Gerhold et al. 2015) avoids circularity and provides a falsifiable example where f.d.d. convergence on a dense set does not imply weak convergence in càdlàg topology. This clarifies the scope of existing theorems and supplies a useful reference case for the field.

major comments (1)

The non-convergence claim in the càdlàg topology rests on the failure of tightness; the manuscript should explicitly verify that the candidate limit process violates the modulus-of-continuity condition in Skorohod space (e.g., by exhibiting a sequence of times where the oscillation exceeds any δ with positive probability uniformly in n).

minor comments (2)

Clarify the precise definition of the dense subset of [0, ∞) on which f.d.d. convergence is asserted; a short remark on why this set is dense yet insufficient for the Skorohod topology would help readers.
In the characterization of the exit-time limit law, state explicitly whether the limiting distribution depends on the starting point or is translation-invariant.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for the constructive suggestion. We address the major comment below.

read point-by-point responses

Referee: The non-convergence claim in the càdlàg topology rests on the failure of tightness; the manuscript should explicitly verify that the candidate limit process violates the modulus-of-continuity condition in Skorohod space (e.g., by exhibiting a sequence of times where the oscillation exceeds any δ with positive probability uniformly in n).

Authors: We agree that an explicit verification strengthens the presentation. Our current argument establishes non-convergence by showing that the sequence fails to be tight in the Skorohod space D([0,∞),R^d). To make this fully explicit as suggested, we will add a short paragraph (in the section on weak convergence) constructing, for arbitrary δ>0 and h>0, a sequence of times t_n such that P(sup_{s,t∈[t_n,t_n+h], |s-t|<h} |X^n_t - X^n_s| > δ) is bounded away from zero uniformly in n. This directly violates the modulus-of-continuity criterion and confirms the failure of tightness. The revision will be incorporated without changing any theorems or proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs an explicit family of time-homogeneous Itô diffusions and applies a spatial scaling via first-exit times from radius-n^{-1/2} balls (no time rescaling). It then proves finite-dimensional convergence on a dense subset of [0,∞) while showing failure of weak convergence in the Skorohod space of càdlàg paths, plus characterization of the exit-time limit law. All steps rely on standard stochastic-process arguments (SDEs, stopping times, finite-dimensional distributions, tightness criteria) rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation. The sole external reference (Gerhold et al. 2015) is by unrelated authors and is invoked only for contrast with the conventional time-scaled CLT; it supplies no uniqueness theorem or ansatz that the present derivation reduces to. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard existence/uniqueness theory for Itô SDEs and on the definition of the càdlàg Skorokhod topology; no free parameters or new entities are introduced.

axioms (2)

domain assumption Solutions to the Itô SDEs exist and are unique (standard Lipschitz or linear-growth conditions on coefficients).
Invoked when the processes are introduced as time-homogeneous diffusions described by Itô SDEs.
standard math The càdlàg space equipped with the Skorokhod topology is the appropriate setting for weak convergence of the scaled processes.
Used when stating failure of weak convergence in the sense of laws of càdlàg processes.

pith-pipeline@v0.9.0 · 5796 in / 1447 out tokens · 41003 ms · 2026-05-19T18:56:44.841466+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

We may then provide an explicit illustrative example of Theorem 1 as following

that in the literature there are (possibly nontrivial) counterexamples in functional 3 limit theorems, however no specific details are mentioned except general references. We may then provide an explicit illustrative example of Theorem 1 as following. Example 1.Letd= 1, µ= 0, σ= 1, x= 0, so thatXis a standard Brownian motion onRand ˜τ n(X) = ˜τn(W) = inf{...

work page
[2]

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[1] [1]

We may then provide an explicit illustrative example of Theorem 1 as following

that in the literature there are (possibly nontrivial) counterexamples in functional 3 limit theorems, however no specific details are mentioned except general references. We may then provide an explicit illustrative example of Theorem 1 as following. Example 1.Letd= 1, µ= 0, σ= 1, x= 0, so thatXis a standard Brownian motion onRand ˜τ n(X) = ˜τn(W) = inf{...

work page

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Gerhold, S., Kleinert, M., Porkert, P., and Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications.Stochastics, 87

work page 2015

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and Watanabe, S

Ikeda, N. and Watanabe, S. (1989).Stochastic Differential Equations and Diffusion Processes (Second Edition). North-Holland Mathematical Library

work page 1989

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and Shiryaev, A

Jacod, J. and Shiryaev, A. N. (2003).Limit theorems for stochastic processes. Springer-Verlag

work page 2003

[5] [5]

(2013).Probability theory: a comprehensive course

Klenke, A. (2013).Probability theory: a comprehensive course. Springer

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and Yor, M

Revuz, D. and Yor, M. (1999).Continuous Martingales and Brownian Motion. Springer

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Stoyanov, J. M. (2013).Counterexamples in Probability. Dover Books in Mathematics

work page 2013

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(2017).Measures, Integrals and Martingales

Schilling, R. (2017).Measures, Integrals and Martingales. Cambridge University Press. 13

work page 2017