Local Limit Theorems for Sample Extremes

Abstract

A local limit theorem for maxima of i.i.d. random variables is proved. Also it is shown that under the so-called von Mises' conditions the density of the normalized maximum converges to the limit density in $L_p(0 < p \leq \infty)$ provided both the original density and the limit density are in $L_p$. Finally an occupation time result is proved. The methods of proof are different from those used for the corresponding results concerning partial sums.