An Extreme-Value Analysis of the LIL for Brownian Motion

Preprint, Other literature type English OPEN
Khoshnevisan, Davar; Levin, David; Shi, Zhan;

We use excursion theory and the ergodic theorem to present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. A simplified version of our method also proves, in a paragraph, the classical theorem of Darling and Erdős (195... View more
  • References (11)
    11 references, page 1 of 2

    Davis, Philip J. 1965. Gamma Function and Related Functions, Handbook of Mathematical Functions (M. Abramowitz and I. Stegun, eds.), Dover, New York, pp. 253-293.

    Dobric, Vladimir and Lisa Marano. 2003. Rates of convergence for L´evy's modulus of continuity and Hinchin's law of the iterated logarithm, High Dimensional Probability III (J. Hoffmann-Jørgensen, M. Marcus, and J. Wellner, eds.), Progress in Probability, vol. 55, Birkha¨user, Basel, pp. 105-109.

    Einmahl, Uwe. 1987. Strong invariance principles for partial sums of independent random vectors, Ann. Probab. 15, 1419-1440.

    Erdo˝s, Paul. 1942. On the law of the iterated logarithm, Ann. Math. 43, 419-436.

    Horva´th, Lajos and Davar Khoshnevisan. 1995. Weight functions and pathwise local central limit theorems, Stoch. Proc. Their Appl. 59, 105-123.

    Itoˆ, Kyosi. 1970. Poisson point processes attached to Markov processes, Proc. Sixth. Berkeley Symp. Math. Statis. Probab., vol. 3, University of California, Berkeley, pp. 225-239.

    Khintchine, A. Ya. 1933. Asymptotische Gesetz der Wahrscheinlichkeitsrechnung, Springer, Berlin.

    Motoo, Minoru. 1959. Proof of the law of iterated logarithm through diffusion equation, Ann. Inst. Stat. Math. 10, 21-28.

    Resnick, Sidney I. 1987. Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York.

    Revuz, Daniel and Marc Yor. 1999. Continuous Martingales and Brownian Motion, Third Edition, Springer, Berlin.

  • Metrics
Share - Bookmark