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# Limit Laws for the Maximum of Weighted and Shifted I.I.D. Random Variables

D. J. Daley and Peter Hall
The Annals of Probability
Vol. 12, No. 2 (May, 1984), pp. 571-587
Stable URL: http://www.jstor.org/stable/2243488
Page Count: 17
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## Abstract

Gnedenko's (1943) study of the class G of limit laws for the sequence of maxima $M_n \equiv \max\{X_0, \cdots, X_{n - 1}\}$ of independent identically distributed random variables X0, X1, ⋯ is extended to limit laws for weighted sequences {wn(γ)Xn} (the simplest case {γnX n} has geometric weights $(0 \leq \gamma < 1))$ and translated sequences {Xn - vn(δ)} (the simplest case is $\{X_n - n\delta\} (\delta > 0))$. Limit laws for these simplest cases belong to the family G characterized by Gnedenko; with more general weights or translates, limit laws outside G may arise.

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