On the Multiplicity of the Sample Maximum and the Longest Head Run

Abstract

Tossing a (not necessarily unbiased) coin n times let us denote the length of the longest head run by Z _n and the number of head runs of such length by M _n . Once Erdős asked about the asymptotic behavior of M _n as n → ∞, and these questi motivated the problems that will be discussed in the present paper.

In an array of a double sequence of integer valued random variables, i.i.d. within rows, let μ(n) denote the multiplicity of the maximal value in the nth row. In Section 2 the asymptotic distribution of μ(n) is computed. Though limit distribution does not exist in the ordinary sense, a.s. limit distribution does, as proved in Section 3.

In Section 4 the multiplicity M _n of the maximal run is investigated in a general model of waiting times. By applying the results of Sections 2 and 3 an asymptotic formula is derived for the distribution of M _n , together with an a.s. limit distribution theorem.

Two interesting examples are discussed in Section 5. One of them is the motivating problem of longest head run, with a generalization of allowing at most d tails in between. The other one concerns the longest flat segment (or tube, in other words) of a (discrete) random walk.

The last section contains multivariate extensions.