## Yy o

Therefore, as n ^tt, n n log (n(Ui,n — Ui-1,n))} = y \^i,n — 1 — log Wi^ + Op(log n).

By substracting n7 to the left- and right-hand side of the above equality, we obtain (36), as sought.□

Introduce now the following notation and facts. For each n > 1 and 0 < t < 1, denote, respectively by

the uniform empirical and quantile functions based upon U1,...,Un. Here and elsewhere, #A stands for the cardinality of A. The corresponding empirical and quantile processes are given, respectively, by an(t)= n1'2{Un(t) — t} and pn (t) = n1/2{Vn(t) — t}. (43)

Fact 1 below is due to Kiefer [Kie67] (see, e.g., Deheuvels and Mason [DM90]). Fact 1 For each specified 0 < to < 1, we have, almost surely, limsupn1/4(loglogn)-3/4\an(to) + pn(to)\ = 25/43-3/4{to(1 — to)}1/4. (44)

The next fact, stated below, is Lemma 3.1 of Deheuvels and Derzko [DD03] (see also Pyke [Pyk65], and Proposition 8.2.1 in Shorack and Wellner [SW86]).

Fact 2 Let Sn+1 denote a random variable, independent of U1,..., Un, and following a r(n + 1) distribution, with density sn h(s) = — e s for s > 0, h(s) =0 for s < 0. n!

Then, the random variables

are independent, exponentially distributed with unit mean, and such that n+l

In view of the above notation, letting {0inn : 1 < i < n +1} be as in (46), Fact 3 below follows from Theorems 3.1-3.2 of Deheuvels and Derzko [DD03].

Fact 3 We have, as n ^ <x>, ! j sup I £ \ - log (n(Ui,n - Ui-I,n)) - Y \