Pith Number

pith:7RFXJRX3

pith:2026:7RFXJRX3GWXL3ZSD4CURLQN7AR

not attested not anchored not stored refs resolved

A Counterexample to Small-time Limit Theorems for Stochastic Processes

Pietro Maria Sparago

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

arxiv:2605.15931 v1 · 2026-05-15 · math.PR

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{7RFXJRX3GWXL3ZSD4CURLQN7AR}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

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,∞), but it does not converge weakly in the sense of laws of càdlàg processes.

C2weakest assumption

The processes under consideration are time-homogeneous diffusions given by Itô SDEs, and the scaling is performed by stopping at the first exit time from balls of radius n^{-1/2} without any time rescaling (abstract, paragraph on the scaling choice).

C3one line summary

The paper exhibits a sequence of scaled diffusion processes that converge in finite-dimensional distributions over a dense time set but fail to converge weakly in the càdlàg topology.

References

8 extracted · 8 resolved · 0 Pith anchors

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

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

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

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

[5] (2013).Probability theory: a comprehensive course 2013

Receipt and verification

First computed	2026-05-20T00:01:45.563620Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

fc4b74c6fb35aebde643e0a915c1bf046688f6437fa70cc07564882a15ccd607

Aliases

arxiv: 2605.15931 · arxiv_version: 2605.15931v1 · doi: 10.48550/arxiv.2605.15931 · pith_short_12: 7RFXJRX3GWXL · pith_short_16: 7RFXJRX3GWXL3ZSD · pith_short_8: 7RFXJRX3

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/7RFXJRX3GWXL3ZSD4CURLQN7AR \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: fc4b74c6fb35aebde643e0a915c1bf046688f6437fa70cc07564882a15ccd607

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "2ce544c77d49a8629daf56445d5fe8798d1d807fbad91a7e5cf69e3f57d19bf4",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-15T13:13:50Z",
    "title_canon_sha256": "2361b96fd67303b4588de55312eb9708da174c4200c88e6f3db369cf1a38b428"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15931",
    "kind": "arxiv",
    "version": 1
  }
}