An extreme value theory for long head runs

Abstract

For an infinite sequence of independent coin tosses with P(Heads)=p∈(0,1), the longest run of consecutive heads in the first n tosses is a natural object of study. We show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum of n(1-p) i.i.d. exponential random variables. These results are extended to the case of the longest head run interrupted by k tails. The mean length of this run is shown to be log(n)+klog(n)+(k+1)log(1-p)-log(k!)+k+γ/λ-1/2+ r1(n)+ o(1) where log=log1/p, γ=0.577 ... is the Euler-Mascheroni constant, λ=ln(1/p), and r1(n) is small. The variance is π2/6λ2+1/12 +r2(n)+ o(1), where r2(n) is again small. Upper and lower class results for these run lengths are also obtained and extensions discussed. © 1986 Springer-Verlag.