Local times of deterministic paths and self-similar processes with stationary increments as normalized numbers of interval crossings

Purba Das; Rafa{\l} {\L}ochowski; Witold Bednorz

Local times of deterministic paths and self-similar processes with stationary increments as normalized numbers of interval crossings

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 2312.05570 v2 pith:Z4IDE7W3 submitted 2023-12-09 math.PR math.FA

Witold Bednorz , Purba Das , Rafa{l} {L}ochowski This is my paper · ORCID

classification math.PRmath.FA

keywords crossingsintervallocalnormalizednumbersprocessesresulttimes

verification ladder T0 review T1 audit T2 compute T3 formal

Signed reviews

No signed human review yet.

0 comments

read the original abstract

We prove a general result on a relationship between a limit of normalized numbers of interval crossings by a c\`adl\`ag path and an occupation measure associated with this path. Using this result we define local times of fractional Brownian motions (classically defined as densities of relevant occupation measure) as weak limits of properly normalized numbers of interval crossings. We also discuss a similar result for c\`adl\`ag semimartingales, in particular for alpha-stable processes, and for Rosenblatt processes, and provide natural examples of deterministic paths which possess quadratic or higher order variation but no local times.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)
math.CA 2025-01 conditional novelty 6.0 of 10

For grids (p/3^n)Z+(r/3^n) with p rational and r irrational, the Peano curve's horizontal coordinate x has quadratic variation 3^{floor(-log_3 p)}p(1-(3/4)3^{floor(-log_3 p)}p)t.

Pith tools