The arcsine law
The classical equivalence principle states that the number of positive sums in the first \(n\) trials in a random walk and the first time the maximal partial sum in the first \(n\) trials have the same distribution, and the classical arcsine law [see \textit{W. Feller}, ``An introduction to probability theory and its applications. Vol. II'' (1966; Zbl 0138.10207)] gives a necessary and sufficient condition for the above two random variables after dividing by \(n\) to converge in law to the same arcsine distribution. The classical proofs usually apply advanced methods from complex analysis and Fourier analysis together with some sort of Tauber theorems and tricky and ingenious combinatorial methods. With the help of a simple and elementary extension of the Cesàro lemma the author provides a proof of the equivalence principle for a general class of integer-valued random vectors of the two concerned random variables. The author also provides examples of arcsine sequences which arise naturally in Markov chains, last visiting times in Markov chains, ladder points and occurence numbers, and recurrence times in \textit{J. F. C. Kingman}'s ``Regenerative phenomena'' (1972; Zbl 0236.60040)].
scientific article; zbMATH DE number 1275650