Let { X , X n , n ≥ 1 } be a sequence of i.i.d. random variables which is in the domain of attraction of the normal law with zero mean and possibly infinite variance, Q ( n ) = R ( n ) / S ( n ) be the rescaled range statistic, where R ( n ) = max 1 ≤ k ≤ n { ∑ j = 1 k ( X j ? X ˉ n ) } ? min 1 ≤ k ≤ n { ∑ j = 1 k ( X j ? X ˉ n ) } , S 2 ( n ) = ∑ j = 1 n ( X j ? X ˉ n ) 2 / n and X ˉ n = ∑ j = 1 n X j / n . Then two precise asymptotics related to probability convergence for Q ( n ) statistic are established under some mild conditions in this paper. Moreover, the precise asymptotics related to almost surely convergence for Q ( n ) statistic is also considered under some mild conditions. MSC:60F15, 60G50.
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