The Annals of Probability

Ballot Theorems and Sojourn Laws for Stationary Processes

Olav Kallenberg

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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

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

First available in Project Euclid: 31 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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  • 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.