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The Arcsine Law

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  • J. Hoffmann-Jørgensen

    (University of Aarhus)

Abstract

Let N n denote the number of positive sums in the first n trials in a random walk (S i) and let L n denote the first time we obtain the maximum in S 0,..., S n. Then the classical equivalence principle states that N n and L n have the same distribution and the classical arcsine law gives necessary and sufficient condition for (1/n) L n or (1/n) N n to converge in law to the arcsine distribution. The objective of this note is to provide a simple and elementary proof of the arcsine law for a general class of integer valued random variables (T n) and to provide a simple an elementary proof of the equivalence principle for a general class of integer valued random vectors (N n, L n).

Suggested Citation

  • J. Hoffmann-Jørgensen, 1999. "The Arcsine Law," Journal of Theoretical Probability, Springer, vol. 12(1), pages 131-145, January.
    • Handle: RePEc:spr:jotpro:v:12:y:1999:i:1:d:10.1023_a:1021748727681
    DOI: 10.1023/A:1021748727681
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    Keywords

    Random walk; arcsine law;

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