A CLT for the L² moduli of continuity of local times of Levy processes

2009/06/25 by Michael B. Marcus, Marcus, Michael B., Jay Rosen +1
Economics, Econometrics and Finance · Mathematics · #60F05 #60G51 #60J55 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60G51 #msc:60J55

paper · pdf · doi:10.48550/arxiv.0906.4770

arxiv created 2009/06/25 · openalex publication_date 2009/06/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X=\X_t,t∈ R₊\ be a symmetric Lévy process with local time \L^x_t ; (x,t)∈ R¹× R¹₊\. When the Lévy exponent ψ(\la) is regularly varying at infinity with index 1<β≤ 2 and satisfies some additional regularity conditions && √hψ²(1/h) \lc ∫ (L^x+h₁- L^x₁)² dx- E(∫ (L^x+h₁- L^x₁)² dx)\rc\nn && 1 in \stackrelL\Longrightarrow (8c_β,1)^1/2 η(∫ (L₁^x)² dx)^1/2 \nn, as h\rar 0, where η is a normal random variable with mean zero and variance one that is independent of L^x_t, and c_β,1 is a known constant.

Related

• A CLT for the L² norm of increments of local times of Lévy processes as time goes to infinity
• Variation and Rough Path Properties of Local Times of Lévy Processes
• Local moduli of continuity for permanental processes that are zero at zero
• A CLT for the L² modulus of continuity of Brownian local time
• CLT for L^p moduli of continuity of Gaussian processes