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A stochastic calculus proof of the CLT for the \(L^2\) modulus of continuity of local time. (English) Zbl 1219.60028

Donati-Martin, Catherine (ed.) et al., Séminaire de Probabilités XLIII, Poitiers, France, Juin 2009. Berlin: Springer (ISBN 978-3-642-15216-0/pbk; 978-3-642-15217-7/ebook). Lecture Notes in Mathematics 2006, 95-104 (2011).
The purpose of the paper is to give a new and shorter proof of the central limit theorem \[ \frac{\int {(L_t^{x + h} - L_t^x)}^2 dx - 4ht}{h^{3/2}}\overset {w}{\Rightarrow} c \left( \int {(L_t^x)}^2 dx \right)^{1/2}\eta \] as \(h \to 0\) for Brownian local time \(L_t^x\). Here \(\eta \) is an independent normal random variable with mean zero and variance one.
For the entire collection see [Zbl 1202.60007].

MSC:

60F05 Central limit and other weak theorems
60J55 Local time and additive functionals
60J65 Brownian motion
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[1] Chen, X.; Li, W.; Marcus, M.; Rosen, J., A CLT for the L 2 modulus of continuity of local times of Brownian motion, Ann. Probab., 38, 1, 396-438 (2010) · Zbl 1190.60071
[2] van der Hofstad, R.; Klenke, A.; Konig, W., The critical attractive random polymer in dimension one, J. Stat. Phys., 106, 3-4, 477-520 (2002) · Zbl 1001.82124
[3] Marcus, MB; Rosen, J., L^p moduli of continuity of Gaussian processes and local times of symmetric Lévy processes, Ann. Probab., 36, 594-622 (2008) · Zbl 1260.60156
[4] Marcus, M.B., Rosen, J.: Markov processes, Gaussian processes and local times. Cambridge studies in advanced mathematics, vol. 100. Cambridge University Press, Cambridge (2006) · Zbl 1129.60002
[5] Revuz, D.; Yor, M., Continuous martingales and Brownian motion (1999), Berlin: Springer, Berlin · Zbl 0917.60006
[6] Rosen, J., Joint continuity of renormalized intersection local times, Ann. Inst. Henri Poincare., 32, 671-700 (1996) · Zbl 0867.60049
[7] Rosen, J.: Continuous differentiability of renormalized intersection local times in R^1. Ann. Inst. Henri Poincare, to appear. arxiv:0910.2919 · Zbl 1210.60084
[8] Weinryb, S., Yor, M.: Le mouvement brownien de Lévy indexé par R^3 comme limite centrale des temps locaux d’intersection. In: Séminaire de Probabilités XXII. Lecture Notes in Mathematics vol. 1321, pp. 225-248. Springer, New York (1988). To appear · Zbl 0653.60074
[9] Yor, M.: Le drap brownien comme limite en lois des temps locaux linéaires. Séminaire de Probabilités XVII, Lecture Notes in Mathematics vol. 986, pp. 89-105. Springer, New York (1983) · Zbl 0514.60075
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