A CLT for the L^{2} modulus of continuity of Brownian local time - Mathematics > ProbabilityReport as inadecuate




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Abstract: Let $\{L^{x} {t} ; x,t\in R^{1}\times R^{1} {+}\}$ denote the local time ofBrownian motion and \ \alpha {t}:=\int {-\infty}^{\infty} L^{x} {t}^{2} dx .\ Let $\eta=N0,1$ be independent of $\alpha {t}$. For each fixed $t$ \{\int {-\infty}^{\infty} L^{x+h} {t}- L^{x} {t}^{2} dx- 4ht\over h^{3-2}}\stackrel{\mathcal{L}}{\to}{64 \over 3}^{1-2}\sqrt{\alpha {t}} \eta, \ as$h ar 0$. Equivalently \ {\int {-\infty}^{\infty} L^{x+1} {t}-L^{x} {t}^{2} dx- 4t\over t^{3-4}} \stackrel{\mathcal{L}}{\to}{64 \over 3}^{1-2}\sqrt{\alpha {1}} \eta, \ as $t ar\infty$.



Author: Xia Chen, Wenbo Li, Michael B. Marcus, Jay Rosen

Source: https://arxiv.org/







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