M Csörgö; Barbara Szyszkowicz's Asymptotic methods in probability and statistics : a volume PDF

By M Csörgö; Barbara Szyszkowicz

ISBN-10: 0444500839

ISBN-13: 9780444500830

ISBN-10: 0585474680

ISBN-13: 9780585474687

ISBN-10: 1281047120

ISBN-13: 9781281047120

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

Example text

The l o w e r - b o u n d argument. Let 0 < ( < 1 be given and let r E ((2, 1) and q E (1, 1 / r ) . For n _ 1, let k n - [exp(nq)] and m n - k 4. Finally, let Po - 0 and, for n _ 1, let Pn - P n - 1 - k n m n . A significant feature of this last sequence is that lira P n - 1 = O. s. 3) j---*oo Let us accept, for the moment, that this claim is true and see how the remainder of the argument rests upon it. 2), lim apt-1 r = 0 These remarks, in conjunction with the result of the upper-bound argument, show that the sequence { I S p ~ _ l l / ( x / - ~ a k ~ r converges to 0 almost surely as j ~ c~.

Let t3 - {Y, Y ( a ) 9 a E Z d} be a collection of independent and identically distributed real-valued random variables. In this context, is called the random scenery. We will assume that W and tj are defined on a common probability space and generate independent a-algebras. For n >_ 0, let n ZY(S ). k--O The process {Zn " n >_ 0}, which is called random walk in random scenery, was introduced by Kesten and Spitzer (1979), who investigated its weak limits. 1 of Khoshnevisan and Lewis (1997)). In the sequel, we will assume that E(Y) - 0 and E(Y 2) - 1.

R6v6sz" A limit distribution theorem for sums of dependent random variables, Acta Math. Acad. Sci. Hungar. 10 (1959), 125-131. Z. Rychlik: Some remarks on stable sequences of random variables, In: Sequential Methods in Statistics (Banach Center Publications 16), pp. 455-463, PWN, Warsaw, 1985. C. Stone: Ratio limit theorems for random walks on groups, Trans. Amer. Math. Soc. 125 (1966), 86-100. Asymptotic Methods in Probability and Statistics B. V. All fights reserved. 13 Limit Theorems for Maximal Random Sums Piotr Kowalski a and Zdzistaw Rychlik a'b a Department of Mathematics, Maria Curie-Sktodowska University, P1.

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Asymptotic methods in probability and statistics : a volume in honor of Miklós Csörgö by M Csörgö; Barbara Szyszkowicz


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