Functional L.I.L. for the Upper Extremes of i.i.d. Random Variarbles from a Sub-class of Heavy Tailed Distributions

Abstract

Let $\left\{ X_n\right\}$ be a sequence of independent and identically distributed non-negative valued random variables defined over a common probability space and let the common distribution function F be continuous. Suppose that F belongs to the domain of partial attraction of a max-semi Frechet law. Define for any $r\ge 1$ and $n\ge r$, $M_{r,[nt]}$ = the rth highest among $\left\{ X_1,X_2,\ldots ,X_{[nt]}\right\}$ if $\frac{r}{n} \le t \le 1,$ and = min $\left\{ X_1,X_2,\ldots ,X_r\right\}$ if $0\le t \le \frac{r}{n}.$ We establish below, a functional law of the iterated logarithm for the sequence $(M_{r,[nt]}, 0\le t \le 1)$, properly normalized, under the $M_1$-topology.