A result on the almost sure convergence for the R/S statistic
Let {X(i), i >= 1} be a sequence of non-degenerate, i.i.d. random variables, which is in the domain of attraction of the normal law with mean zero and possibly infinite variance, Q(n) = R(n)/S(n) be the rescaled range statistic, where R(n) = max(1 <= k <= n){Sigma(k)(j=1)(X(j) - (X) over bar (n))} - min(1 <= k <= n){Sigma(k)(j-1)(X(j) - (X) over bar (n))}, S(2)(n) = Sigma(n)(j-1) (X(j) - (X) over bar (n))(2)/n and (X) over barn = Sigma(n)(j-1) X(j)/n. Then a result about the almost sure convergence for Q(n) is established under a mild condition in this paper. (C) 2011 Elsevier Ltd. All rights reserved.
almost sure r/s statistic domain of attraction of the normal law brownian bridge