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

Gnedenko's (1943) study of the class $\mathscr{G}$ of limit laws for the sequence of maxima $M_n \equiv \max\{X_0, \cdots, X_{n - 1}\}$ of independent identically distributed random variables $X_0, X_1, \cdots$ is extended to limit laws for weighted sequences $\{w_n(\gamma)X_n\}$ (the simplest case $\{\gamma^nX_n\}$ has geometric weights $(0 \leq \gamma < 1))$ and translated sequences $\{X_n - v_n(\delta)\}$ (the simplest case is $\{X_n - n\delta\} (\delta > 0))$. Limit laws for these simplest cases belong to the family $\mathscr{G}$ characterized by Gnedenko; with more general weights or translates, limit laws outside $\mathscr{G}$ may arise.