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Ballot theorems revisited

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  • Konstantopoulos, Takis

Abstract

Ballot theorems yield a simple relation for the probability that a non-decreasing process with cyclically interchangeable increments and piecewise constant paths stay below a linear function for a certain period of time. A particular instance of the problem is the classical ballot theorem that computes the chance that, in a ballot, votes of type A are always ahead of votes of type B during counting. Traditionally, one uses a combinatorial or analytic lemma to prove such theorems. In this paper we make the simple observation that all ballot theorems can be proved by direct probabilistic arguments if the stationarity of the processes involved is properly taken into account. Our proofs use a queueing theoretic construction and thus yield a physical interpretation to ballot theorems. Furthermore, they reveal the necessity of the assumptions. The auxiliary combinatorial or analytic lemmas are finally derived as a consequence.

Suggested Citation

  • Konstantopoulos, Takis, 1995. "Ballot theorems revisited," Statistics & Probability Letters, Elsevier, vol. 24(4), pages 331-338, September.
    • Handle: RePEc:eee:stapro:v:24:y:1995:i:4:p:331-338
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    Cited by:

    1. Schweinsberg, Jason, 2001. "Applications of the continuous-time ballot theorem to Brownian motion and related processes," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 151-176, September.
    2. Claude Lefèvre & Stéphane Loisel, 2008. "On Finite-Time Ruin Probabilities for Classical Risk Models," Post-Print hal-00168958, HAL.
    3. Yuen, Fei Lung & Lee, Wing Yan & Fung, Derrick W.H., 2020. "A cyclic approach on classical ruin model," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 104-110.

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