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 themes of this publication 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 all started 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 chance and information, Veszpr~m 1982. This collaboration led to No. 2 of Technical document sequence of the Laboratory for study in information and chance of Carleton college and college of Ottawa, 1983. Afterwards David M. Mason has determined to withdraw from this venture. The authors desire to thank him for his contributions. specifically, he has known as our realization to the opposite martingale estate of the empirical method 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 comparable comments helped us push down the two second situation to EX < 00 in all our vulnerable approximation theorems.

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**Example text**

On the other hand, EX 2 < 00 , and and not even that this is why we assumed the latter condition separately. 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.

10, we get A(l) a=s. o(na-~(log n)l-a). 12) and the definition of O~~~£ln I () I un y £In' we have a:s. :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.

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 -.. • 2 1 exp {-c ~}ds > exp{s s J t 00 [0,1/2J. 5) the left q2(t)/t -.. 00 as O. 24) converges to even if the reciprocal of q is square-integrable. 6 can be said about it. IMPROVED EXAMPLE. o 1/2 J such that o = 00 (1/q2(t))dt < = 00 and ~O For constructing the required h(t) on o . lim inf t h*(·) there exists an O'Reilly weight inf{h*(s) q, set first : 0 < s < t} 2- h*(t), 0 < t < 1/2.

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

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