The Annals of Probability

Local asymptotic classes for the successive primitives of Brownian motion

Aimé Lachal

Full-text: Open access


Let $(B(t))_{t \geq 0}$ be the linear Brownian motion starting at 0, and set $X_n (t) =(1/n!)\int_0^t(t-s)^s dB(s)$. Watanabe stated a law of the iterated logarithm for the process $(X_1(t))_{t \geq 0}$, among other things. This paper proposes an elementary proof of this fact, which can be extended to the general case $n\geq 1$. Next, we study the local asymptotic classes (upper and lower) of the $(n +1)$ -dimensional process $U_n =(B, X_1,\ldots,X_n)$ near zero and infinity, and the results obtained are extended to the case where $B$ is the $d$-dimensional Brownian motion.

Article information

Ann. Probab., Volume 25, Number 4 (1997), 1712-1734.

First available in Project Euclid: 7 June 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65] 60F15: Strong theorems 60G15: Gaussian processes
Secondary: 60G17: Sample path properties 60J25

Law of the iterated logarithm local asymptotic classes integral tests


Lachal, Aimé. Local asymptotic classes for the successive primitives of Brownian motion. Ann. Probab. 25 (1997), no. 4, 1712--1734. doi:10.1214/aop/1023481108.

Export citation


  • [1] Albin, J. M. P. (1992). On the general law of iterated logarithm with application to selfsimilar processes and to Gaussian processes in Rn and Hilbert space. Stochastic Proc. Appl. 41 1-31.
  • [2] Chaleryat-Maurel, M. and Elie, L. (1981). Diffusions gaussiennes. Ast´erisque 84-85 255-279.
  • [3] Dvoretsky, A. and Erd os, P. (1951). Some problems on random walk in space. Proc. Second Berkeley Symp. Math. Statist. Probab. 353-367. Univ. California Press, Berkeley.
  • [4] Hendricks, W. J. (1970). Lower envelopes near zero and infinity for processes with stable components.Wahrsch. Verw. Gebiete 16 261-278.
  • [5] It o, K. and McKean, H. P. (1965). Diffusion Processes and Their Sample Paths. Springer, New York.
  • [6] Khoshnevisan, D. and Shi,(1996). Some asymptotic results for the integral of Brownian motion. Preprint.
  • [7] Khoshnevisan, D. and Shi,(1996). Hitting estimates for Gaussian random fields. Preprint.
  • [8] Kolokoltsov, V. M. (1996). A note on the long time asymptotics of the Brownian motion with application to the theory of quantum measurement. Potential Analysis. To appear.
  • [9] K ono, N. (1975). Asymptotic behaviour of sample functions of Gaussian random fields. J. Math. Kyoto Univ. 15 671-707.
  • [10] K ono, N. (1975). Sur la minoration asymptotique et le caract ere transitoire des fonctions al´eatoires gaussiennes a valeurs dans Rd.Wahrsch. Verw. Gebiete 33 95-112.
  • [11] Lachal, A. (1995). Etude analytique et probabiliste d'une classe de fonctionnelles rattach´ees a la primitive du mouvement brownien. Th ese d'habilitation a diriger des recherches, Univ. Lyon 1.
  • [12] Lachal, A. (1997). Regular points for the successive primitives of Brownian motion. J. Kyoto Math. Univ. To appear.
  • [13] L´evy, P. (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villards, Paris.
  • [14] Marcus, M. B. and Shepp, L. A. (1972). Sample behaviour of Gaussian processes. Proc. Sixth Berkeley Math. Symp. Statist. Probab. 2 423-441. Univ. California Press, Berkeley.
  • [15] McKean, H. P., Jr. (1963). A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 227-235.
  • [16] Port, S. C. and Stone, C. J. (1978). Brownian Motion and Classical Potential Theory. Academic Press, New York.
  • [17] Qualls, C. and Watanabe, H. (1971). An asymptotic 0-1 behaviour of Gaussian processes. Ann. Math. Statist. 42 2029-2035.
  • [18] Shepp, L. A. (1966). Radon-Nikodym derivatives of Gaussian measures. Ann. Math. Statist. 37 321-354.
  • [19] Takeuchi, J. (1964). A local asymptotic law for the transient stable process. Proc. Japan Acad. Ser. A Math. Sci. 40 141-144.
  • [20] Takeuchi, J. (1964). On the sample paths of the symmetric stable processes in spaces. J. Math. Soc. Japan 16 109-127.
  • [21] Taylor, J. (1967). Sample paths of a transient stable process. J. Math. Mech. 16 1229-1246.
  • [22] Wahba, G. (1978). Improper priors, spline smoothing and the problem of guarding against model error in regression. J. Roy. Statist. Soc. Ser. B 40 364-372.
  • [23] Wahba, G. (1983). Bayesian "confidence intervals" for the cross-validated smoothing spline. J. Roy. Statist. Soc. Ser. B 45 133-150.
  • [24] Watanabe, H. (1970). An asymptotic property of Gaussian processes. I. Trans. Amer. Math. Soc. 148 233-248.
  • [25] Weber, M. (1978). Classes sup´erieures de processus gaussiens.Wahrsch. Verw. Gebiete 42 113-128.
  • [26] Weber, M. (1980). Analyse asymptotique des processus gaussiens stationnaires. Ann. Inst. H. Poincar´e 16 117-176.