A CLT for the third integrated moment of Brownian local time increments

2009/07/15 by Jay Rosen, Rosen, Jay
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F05 #60J55 #60J65 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60J55 #msc:60J65

paper · pdf · doi:10.48550/arxiv.0907.2693

openalex publication_date 2009/07/15 · arxiv created 2009/10/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \L^x_t ; (x,t)∈ R¹× R¹₊\ denote the local time of Brownian motion. Our main result is to show that for each fixed t ∫ (L^x+h_t- L^x_t)³ dx-12h∫ (L^x+h_t - L^x_t)L^x_t dx-24h²t\over h² \stackrelL\Longrightarrow√(192)(∫ (L^x_t)³dx)^1/2η as h→ 0, where η is a normal random variable with mean zero and variance one that is independent of L^x_t. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments

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