The Annals of Probability

A CLT for the L2 modulus of continuity of Brownian local time

Xia Chen, Wenbo V. Li, Michael B. Marcus, and Jay Rosen

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Abstract

Let {Ltx; (x, t)∈R1×R+1} denote the local time of Brownian motion, and

αt:=−∞(Ltx)2dx.

Let η=N(0, 1) be independent of αt. For each fixed t,

\[\frac{\int_{-\infty}^{\infty}(L_{t}^{x+h}-L_{t}^{x})^{2}\,dx-4ht}{h^{3/2}}\stackrel{\mathcaligr{L}}{\rightarrow}\biggl(\frac{64}{3}\biggr)^{1/2}\sqrt{\alpha_{t}}\eta \]

as h→0. Equivalently,

\[\frac{\int_{-\infty}^{\infty}(L^{x+1}_{t}-L^{x}_{t})^{2}\,dx-4t}{t^{3/4}}\stackrel{\mathcaligr{L}}{\rightarrow}\biggl(\frac{64}{3}\biggr)^{1/2}\sqrt{\alpha_{1}}\eta \]

as t→∞.

Article information

Source
Ann. Probab., Volume 38, Number 1 (2010), 396-438.

Dates
First available in Project Euclid: 25 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1264434003

Digital Object Identifier
doi:10.1214/09-AOP486

Mathematical Reviews number (MathSciNet)
MR2599604

Subjects
Primary: 60J55: Local time and additive functionals 60F05: Central limit and other weak theorems 60G17: Sample path properties

Keywords
CLT Brownian local times modulus of continuity

Citation

Chen, Xia; Li, Wenbo V.; Marcus, Michael B.; Rosen, Jay. A CLT for the L 2 modulus of continuity of Brownian local time. Ann. Probab. 38 (2010), no. 1, 396--438. doi:10.1214/09-AOP486. https://projecteuclid.org/euclid.aop/1264434003


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References

  • [1] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • [2] Chen, X. (2008). Limit laws for the energy of a charged polymer. Ann. Inst. H. Poincaré Probab. Statist. 44 638–672.
  • [3] Chen, X. and Khoshnevisan, D. (2009). From charged polymers to random walk in random scenery. In Proceedings of the Third Erich L. Lehmann Symposium. IMS Lecture Notes, Monograph Series 57 237–251.
  • [4] Chen, X., Li, W. V. and Rosen, J. (2005). Large deviations for local times of stable processes and stable random walks in 1 dimension. Electron. J. Probab. 10 577–608 (electronic).
  • [5] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
  • [6] van der Hofstad, R. and König, W. (2001). A survey of one-dimensional random polymers. J. Statist. Phys. 103 915–944.
  • [7] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [8] Kesten, H. and Spitzer, F. (1979). A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 5–25.
  • [9] Marcus, M. B. and Rosen, J. (2006). Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics 100. Cambridge Univ. Press, Cambridge.
  • [10] Marcus, M. B. and Rosen, J. (2008). Lp moduli of continuity of Gaussian processes and local times of symmetric Lévy processes. Ann. Probab. 36 594–622.
  • [11] Marcus, M. B. and Rosen, J. (2008). CLT for Lp moduli of continuity of Gaussian processes. Stochastic Process. Appl. 118 1107–1135.
  • [12] Rosen, J. (2005). Derivatives of self-intersection local times. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 263–281. Springer, Berlin.
  • [13] Stanley, R. P. (1997). Enumerative Combinatorics. Vol. 1. Cambridge Studies in Advanced Mathematics 49. Cambridge Univ. Press, Cambridge.
  • [14] Yor, M. (1983). Le drap brownien comme limite en loi des temps locaux linéaires. In Seminar on Probability, XVII. Lecture Notes in Math. 986 89–105. Springer, Berlin.
  • [15] Weinryb, S. and Yor, M. (1988). Le mouvement brownien de Lévy indexé par R3 comme limite centrale de temps locaux d’intersection. In Séminaire de Probabilités, XXII. Lecture Notes in Math. 1321 225–248. Springer, Berlin.