2000
DOI: 10.1006/jmaa.2000.6892
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Precise Asymptotics in the Baum–Katz and Davis Laws of Large Numbers

Abstract: Let X, X , X , . . . be a sequence of i.i.d. random variables such that EX s 0, let 1 2 Z be a random variable possessing a stable distribution G with exponent ␣, 1 -␣ F 2, assume the distribution of X is attracted to G, and set S s X n 1 q иии qX . We prove that n p Ž ␣ p rŽ ␣yp..Ž r r py1. r r py2 1 r p yŽ ␣ p rŽ ␣yp..Ž r r py1.

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Cited by 115 publications
(76 citation statements)
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“…For the proof of Theorem 2.1 we follow the arguments in Subsection 2.1 (see also the proof of Theorem 1 in Gut and Spȃtaru [9]). That is, in a first step we show that the asymptotic (2.1) holds if |Z n | is replaced by n 1/α |Z|, and in a second step it is verified that the discrepancy incurred by this replacement is asymptotically negligible.…”
Section: Remark 22mentioning
confidence: 99%
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“…For the proof of Theorem 2.1 we follow the arguments in Subsection 2.1 (see also the proof of Theorem 1 in Gut and Spȃtaru [9]). That is, in a first step we show that the asymptotic (2.1) holds if |Z n | is replaced by n 1/α |Z|, and in a second step it is verified that the discrepancy incurred by this replacement is asymptotically negligible.…”
Section: Remark 22mentioning
confidence: 99%
“…Next we turn our attention to the limiting case r = p, which is not covered by Theorems 2.1 and 2.2 (cf. Gut and Spȃtaru [9], Theorem 2, for case of sums of i.i.d. random variables).…”
Section: Theorem 22mentioning
confidence: 99%
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“…The first step of both methods consists in finding l under the special assumption that X is stable. The classical method, as illustrated in Heyde [6], Spȃtaru [12], Gut and Spȃtaru [3], etc., proceeds with the assumption that X is in the domain of attraction of a stable law, and uses a version of the powerful Fuk-Nagaev inequality (see Spȃtaru [12]). This inequality is applicable only if the working condition has the form E |X| r < ∞ for some r. The second approach, introduced and practised by Gut and Spȃtaru [4], and Spȃtaru [13], is suitable for cases when Fuk-Nagaev type inequalities prove useless.…”
Section: Aurel Spȃtarumentioning
confidence: 99%