The Annals of Probability

Ballot Theorems and Sojourn Laws for Stationary Processes

Olav Kallenberg

Full-text: Open access

Abstract

The ballot theorem and the uniform law for sojourn times, both results known for cyclically stationary sequences and processes on a bounded index set, are here extended to infinite, stationary sequences and to stationary processes on $\mathbb{R}_+$. Our extensions contain all previously known versions as special cases.

Article information

Source
Ann. Probab., Volume 27, Number 4 (1999), 2011-2019.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022874825

Digital Object Identifier
doi:10.1214/aop/1022874825

Mathematical Reviews number (MathSciNet)
MR1742898

Zentralblatt MATH identifier
0961.60045

Subjects
Primary: 60G10: Stationary processes
Secondary: 60G57: Random measures

Keywords
Increasing sequences and processes maximum identities and inequalities sojourn and maximum times mean occupation measure

Citation

Kallenberg, Olav. Ballot Theorems and Sojourn Laws for Stationary Processes. Ann. Probab. 27 (1999), no. 4, 2011--2019. doi:10.1214/aop/1022874825. https://projecteuclid.org/euclid.aop/1022874825


Export citation

References

  • Andr´e, D. (1887). Solution directe du probl eme r´esolu par M. Bertrand. C.R. Acad. Sci. Paris 105 436-437.
  • Barbier, ´E. (1887). G´en´eralisation du probl eme r´esolu par M. J. Bertrand. C.R. Acad. Sci. Paris 105 407, 440.
  • Bertrand, J. (1887). Solution d'un probl eme. C.R. Acad. Sci. Paris 105 369.
  • Chow, Y.S. and Teicher, H. (1997). Probability Theory: Independence, Interchangeability, Martingales, 3rd ed. Springer, New York.
  • Doob, J.L. (1994). Measure Theory. Springer, New York.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications 1, 3rd ed. Wiley, New York.
  • Fitzsimmons, P. J. and Getoor, R. K. (1995). Occupation time distributions for L´evybridges and excursions. Stochastic Processes Appl. 58 73-89.
  • Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
  • Knight, F. B. (1996). The uniform law for exchangeable and L´evyprocess bridges. Hommage a P.A. Meyer et J. Neveu. Ast´erisque 236 171-188.
  • L´evy, P. (1939). Sur certains processus stochastiques homog enes. Compositio Math. 7 283-339.
  • Sparre-Andersen, E. (1953, 1954). On the fluctuations of sums of random variables I, II. Math. Scand. 1 263-285; 2 193-194, 195-223.
  • Tak´acs, L. (1967). Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.
  • Tucker, H.G. (1959). A generalization of the Glivenko-Cantelli theorem. Ann. Math. Statist. 30 828-830.