On the fair coin tossing process

AbstractLet Ω = {1, 0} and for each integer n ≥ 1 let Ωn = Ω × Ω × … × Ω (n-tuple) and Ωnk = {(a1, a2, …, an)|(a1, a2, … , an) ϵ Ωn and Σi=1nai = k} for all k = 0,1,…,n. Let {Ym}m≥1 be a sequence of i.i.d. random variables such that P(Y1 = 0) = P(Y1 = 1) = 12. For each A in Ωn, let TA be the first occurrence time of A with respect to the stochastic process {Ym}m≥1. R. Chen and A.Zame (1979, J. Multivariate Anal. 9, 150–157) prove that if n ≥ 3, then for each element A in Ωn, there is an element B in Ωn such that the probability that TB is less than TA is greater than 12. This result is sharpened as follows: (I) for n ≥ 4 and 1 ≤ k ≤ n − 1, each element A in Ωnk, there is an element B also in Ωnk such that the probability that TB is less than TA is greater than 12; (II) for n ≥ 4 and 1 ≤ k ≤ n − 1, each element A = (a1, a2,…,an) in Ωnk, there is an element C also in Ωnk such that the probability that TA is less than TC is greater than 12 if n ≠ 2m or n = 2m but ai = ai + 1 for some 1 ≤ i ≤ n−1. These new results provide us with a better and deeper understanding of the fair coin tossing process