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By M Csörgö; Barbara Szyszkowicz

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Extra resources for Asymptotic methods in probability and statistics : a volume in honor of Miklós Csörgö

Example text

1) Uk) - E S ( A k + Vk)]. k=l Since quasi-association is preserved under concatenation and ordered blocking, the random variables Ak and Uk are quasi-associated; by construction, the random variables Ak and Vk are independent. Moreover, the mapping (x, y) H f ( x + y) is an element of C~(]I~2,11~). 1) ifk=n. yields: n--1 n

For the converse, it is clear that (5) and (6) imply (27) and (28) by the same arguments as used in the proof that (3) implies (4). 1 in Kruglov and Korolev (1990), (27), (28) and (9) imply (8) and the proof of Theorem 2 is complete. 3. P r o o f of T h e o r e m 3 We may and do assume EXk - 0, k _> 1, so that Ak = 0, k _ 1. 5 of Kruglov and Korolev (1990), (22) and (13) imply max Sk) /Bn l

Sci. Hung. 11, 97-102. R@nyi, A. (1958), On mixing sequences of sets, Acta Math. Acad. Sci. 9, 215-228. R@nyi, A. (1963), On stable sequences of events, Sankhya A 25, 293-302. R@nyi, A. (1970), Probability Theory, Aka~@miai Kia~16, Budapest. R@nyi, A. and P. R@v@sz(1958), On mixing sequences of random variables, Acta Math. Acad. Sci. Hung. 9, 389-393. Rychlik, Z. (1985), Some remarks on stable sequences of random variables,, In: Sequential Methods in Statistics (Banemh Center Publications 16), PWN, Warsow, 455-463.