Bernoulli

  • Bernoulli
  • Volume 6, Number 6 (2000), 951-975.

Favourite sites, favourite values and jump sizes for random walk and Brownian motion

Endre Csáki,, Pál Révész, and Zhan Shi

Full-text: Open access

Abstract

We determine: (a) the joint almost sure asymptotic behaviour of the most visited site and the maximum local time of a one-dimensional simple random walk or Brownian motion; (b) the maximal jump size of the most visited site. In so doing, we solve two open problems of Erdös and Révész.

Article information

Source
Bernoulli, Volume 6, Number 6 (2000), 951-975.

Dates
First available in Project Euclid: 5 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1081194154

Mathematical Reviews number (MathSciNet)
MR1809729

Zentralblatt MATH identifier
0974.60033

Keywords
Brownian motion favourite site local time random walk

Citation

Csáki,, Endre; Révész, Pál; Shi, Zhan. Favourite sites, favourite values and jump sizes for random walk and Brownian motion. Bernoulli 6 (2000), no. 6, 951--975. https://projecteuclid.org/euclid.bj/1081194154


Export citation

References

  • [1] Abramowitz, M. and Stegun, I.A. (1965) Handbook of Mathematical Functions. New York: Dover.
  • [2] Bass, R.F. and Griffin, P.S. (1985) The most visited site of Brownian motion and simple random walk. Z. Wahrscheinlichkeitstheorie Verw. Geb., 70, 417-436.
  • [3] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge: Cambridge University Press.
  • [4] Csáki, E. (1989) An integral test for the supremum of Wiener local time. Probab. Theory Related Fields, 83, 207-217.
  • [5] Csáki, E. and Shi, Z. (1998) Large favourite sites of simple random walk and the Wiener process. Electron. J. Probab., 3(14).
  • [6] Eisenbaum, N. (1989) Temps locaux, excursions et lieu le plus visité par un mouvement brownien linéaire. Doctoral thesis, Université Paris VII.
  • [7] Eisenbaum, N. (1997) On the most visited sites by a symmetric stable process. Probab. Theory Related Fields, 107, 527-535. Abstract can also be found in the ISI/STMA publication
  • [8] Erdös, P. and Révész, P. (1984) On the favourite points of a random walk. In B. Sendov (ed.), Mathematical Structures - Computational Mathematics - Mathematical Modelling, Vol. 2, pp. 152-157. Sofia: Bulgarian Academy of Sciences.
  • [9] Erdös, P. and Révész, P. (1987) Problems and results on random walks. In P. Bauer, F. Konecny and W. Wertz (eds), Mathematical Statistics and Probability Theory, Vol. B, pp. 59-65. Dordrecht: Reidel.
  • [10] Fristedt, B. (1974) Sample functions of stochastic processes with stationary independent increments. In P.E. Ney and S.C. Port (eds), Advances in Probability and Related Topics, Vol. 3, pp. 241-396. New York: Dekker.
  • [11] Kesten, H. (1965) An iterated logarithm law for the local time. Duke Math. J., 32, 447-456.
  • [12] Knight, F.B. (1963) Random walks and the sojourn density process of Brownian motion. Trans. Amer. Math. Soc., 109, 56-86.
  • [13] Leuridan, C. (1997) Le point d'un fermé le plus visité par le mouvement brownien. Ann. Probab., 25, 953-996.
  • [14] Marcus, M.B. and Shepp, L.A. (1972) Sample behavior of Gaussian processes. In L. LeCam, J. Neyman and E.L. Scott (eds), Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, pp. 423-441. Berkeley: University of California Press.
  • [15] Pitman, J.W. and Yor, M. (1982) A decomposition of Bessel bridges. Z. Wahrscheinlichkeitstheorie Verw. Geb., 59, 425-457.
  • [16] Ray, D. (1963) Sojourn times of a diffusion process. Illinois J. Math., 7, 615-630.
  • [17] Révész, P. (1990) Random Walk in Random and Non-random Environments. Singapore: World Scientific.
  • [18] Revuz, D. and Yor, M. (1999) Continuous Martingales and Brownian Motion, 3rd edn. Berlin: Springer-Verlag.
  • [19] Tóth, B. and Werner, W. (1997) Tied favourite edges for simple random walk. Combin. Probab. Comput., 6, 359-369.
  • [20] Widder, D.V. (1941) The Laplace Transform. Princeton, NJ: Princeton University Press.