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Download Asymptotic methods in probability and statistics : a volume by M Csörgö; Barbara Szyszkowicz PDF

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By M Csörgö; Barbara Szyszkowicz

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If, in addition, we assume that supn>l F~(ISn/anl 3) < c~, then, as our proof will show, the second term on the right can be replaced by c~ IIf"lloo/V ~ , which gives a more appealing bound. For associated random variables, the papers of Woods (1983) and Birkel (1988a) give sufficient conditions for such moment bounds. 2, is a generalization of a result attributed to Hoeffding (see Lemma 2 of Lehmann (1966)), which states that C o v (Z, Y) - / R 2 Qx,y(s, t)dsdt, whenever the covariance is defined.

Mj)) >_0 for each integer j >_ 1 and for each coordinatewise-nondecreasing function f " I~J ~ I~, whenever this expectation is defined. Consequently, {Sn "n >_ 0} is an L 2 demimartingale; this will be an important observation in the sequel, when we develop a maximal inequality for this process. 2, respectively; in {}6 we present applications of our theorems to Kesten and Spitzer's random walk in random scenery. 2. PRELIMINARY RESULTS Let H(1) - E(X2), and, for m _>2, let 7n H(m) = E(X12) + 2 E E ( x 1 x j ) .

Guiasu, S. (1971), On the asymptotic distribution of the sequences of random variables with random indices, Ann. Math. Statist. 42, 2018-2028. Krajka, A. and Z. Rychlik (1993), Necessary and sufficient conditions for weak convergence of random sums of independent random variables, Comment. Math. Univ. Carolinae 34, 465-482. M. Yu. Korolev (1990), Limit Theorems for Random Sums, Moscow (in Russian). M. and Zhgan Bo (1996), Limit theorems for maximal random sums, Teor. Verojatnost. Primenen. 41,520-532.

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