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

ISBN-10: 0387963596

ISBN-13: 9780387963594

ISBN-10: 1461564204

ISBN-13: 9781461564201

Mik16s Cs6rgO and David M. Mason initiated their collaboration at the issues of this publication whereas attending the CBMS-NSF nearby Confer ence at Texas A & M college in 1981. Independently of them, Sandor Cs6rgO and Lajos Horv~th have began their paintings in this topic at Szeged college. the belief of writing a monograph jointly was once born whilst the 4 folks met within the convention on restrict Theorems in likelihood and information, Veszpr~m 1982. This collaboration led to No. 2 of Technical document sequence of the Laboratory for study in statistics and likelihood of Carleton collage and college of Ottawa, 1983. Afterwards David M. Mason has made up our minds to withdraw from this venture. The authors desire to thank him for his contributions. particularly, he has known as our realization to the opposite martingale estate of the empirical technique including the linked Birnbaum-Marshall inequality (cf.,the proofs of Lemmas 2.4 and 3.2) and to the Hardy inequality (cf. the facts of half (iv) of Theorem 4.1). those and several similar feedback helped us push down the two second situation to EX < 00 in all our vulnerable approximation theorems.

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**Example text**

L = 2. for which M. s. The latter result in turn easily implies (see top of p. 151 of M. CsorgO and Revesz (1981» the following. 9. s. s. almost surely do not cover the largest and smallest uniform order statistics Un : n and Ul : n , and in our strong approximation proofs we shall frequently need to know the order of these variables. This is the content of the following well- known estimates. 10. s. s. s. +00 Here the limsup statements follow from the more general upper-upper class result of Robbins and Siegmund (1972), while the liminf statements follow from the more general lower-lower class result of Geffroy (1958/59) • 3.

18) is known to be equivalent to the more familiar KolmogorovPetrovskii-Erd~s-Feller criterion (see in It6-McKean (1965), page 33) that 1/2 q(t) 2(t) exp( E: q-----)dt < t 3/2 2t oJ 00 for all E: > O. 21) t+O = 00 Shorack and Wellner (1982) arrive at this statement by a different (necessarily incorrect) argument. We now give a counterexample to this statement. Set EXAMPLE. = an 4 exp(-exp n ), n = 1 Ii ' 1,2, ... , and define the quantities 1 an (log log - ) an 1 q2(t) R,2 (t) ran) R,2(b n )+ n = 1,2, .

I) . 9), we obtain for the third term that A(3) (£ ) a:s. 20) with an arbitrarily small/» O. 11). We shall break up the fourth term into two terms and first we estimate the first term on the right hand side. 21), and the first term here is almost surely O(~~n n- l / 4 (loglog n)1/4(log n)1/2). s. 25) states that u 2 (y) sup n a:s. O(loglog n). ::. -)') y a:s. o(n-~ ~~n log log n) . 28) sup IA (y) I a:s. 29) 1), 6 3 -2"(log n) 2- 6 ). ::. 17), and using the last inequality when estimating A(6) (E ) we obtain n 2n An(6) (E 2n ) < CIS sup n-l n l>< 2 0 < 8 ~ 1 and Un(y) y l-E2n~y~-n- Now the cases 1!

### An Asymptotic Theory for Empirical Reliability and Concentration Processes by Miklós Csörgő, Sándor Csörgő, Lajos Horváth (auth.)

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