By Miklós Csörgő, Sándor Csörgő, Lajos Horváth (auth.)

Mik16s Cs6rgO and David M. Mason initiated their collaboration at the issues of this booklet whereas attending the CBMS-NSF local Confer ence at Texas A & M collage in 1981. Independently of them, Sandor Cs6rgO and Lajos Horv~th have started their paintings in this topic at Szeged college. the belief of writing a monograph jointly used to be born whilst the 4 folks met within the convention on restrict Theorems in likelihood and information, Veszpr~m 1982. This collaboration ended in No. 2 of Technical record sequence of the Laboratory for learn in information and chance of Carleton collage and college of Ottawa, 1983. Afterwards David M. Mason has determined to withdraw from this undertaking. The authors desire to thank him for his contributions. particularly, he has referred to as our awareness to the opposite martingale estate of the empirical procedure including the linked Birnbaum-Marshall inequality (cf.,the proofs of Lemmas 2.4 and 3.2) and to the Hardy inequality (cf. the evidence of half (iv) of Theorem 4.1). those and several comparable feedback helped us push down the two second to EX < 00 in all our susceptible approximation theorems.

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**Example text**

2) is of be course automatically satisfied, then 2/(1-6)), and hence Ex2 < 00. 5) diverges. 1, the- first term of our limit process T(u) = f u o l-u B(y)dQ(y) - f(Q(u)) B(u) is almost surely bounded, but the second term is almost surely unbounded on [O,lJ. ) is almost surely unbounded and thus no process can converge to it weakly in the space of continuous functions on [O,lJ. 2) is almost optimal. For further discussion of this condition we refer to Section 8. 3. 14) Jl = sup u(l-u) O__
__

O(n- T (l/2-1/r-o)). 3, since ily small and T < 1 can be as close to 1 ° > 0 as we wish. is arbitrar- 4. MEAN RESIDUAL LIFE PROCESSES. 15) as a consequence of the preceding section. Clearly, (l-Fn(X» -1 1 J F(x) cxn(y)dQ(y) + (l-Fn(X»-l~(Q(F(xl»nl::l(Fn(X)-F(X»' and hence its approximating Gaussian process will be Zn(x) = (l-F(x»- 1 1 J B (y)dQ(y) F(x) n -(l-F(X»-lMF(Q(F(X»)Bn(F(X» Setting . inf{t:F(t)=l}, we have the following result. 1. > 0 O

Since the left side upper bound tends to zero as s(q (t)/t) - loglog (l/t) ~ Then 2 exp{-(E 2 [0,1/2J. exp{-s ~)dS s exp {-s mEet) q is already nonincreasing on as 00 t + < t O. 24) h q(t) = t 2 h(t), 0 < t ~ 1/2. 1 in O'Reilly (1974, p. 644). For easy reference we formulate this result and give also O'Reilly's simple proof. 6. is, q E Q* If J. o Proof. 5) is satisfied, then Fixing t 0 < c < 1 = ct we have for any = using only that q (log t -.. 0 t (0,1/2J E t q2~t)} J -1s ct 1:) exp{- q c is nondecreasing on side converges to zero as -..