Limit Laws for Extreme Order Statistics Form Strong Mixing Processes
Limit Laws for Extreme Order Statistics Form Strong Mixing Processes
Roy E Welsch
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N — n n n n This method of constructing a strong-mixing sequence is due to Newell [6]. The limit law (2. 3) is not of the same form as H(x, y) and we conclude that by weakening the independence assumption a larger class of limit laws is possible. In the next section we prove a result which limits the size of this class. 10. 3. Possible Limit Laws . Theorem 1 . Let ^ 1> be a stationary strong-mixing sequence. If there exists a sequence of constants /a > 0, b :n > iN so that ^ n n — ' ^^"n - ^n^ ..."*" ^n' ^n - ^n^ "^ ^n^ ^^^ ^ limiting distribution, H(x, y), with G(x), the limiting distribution of P{M ^ 1) into k blocks of length m-q separated by blocks of length q. The fact that the blocks of length m-q are "nearly independent" because of the strong-mixing property is then used to obtain a functional equation for H(x, y).
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