Precise Asymptotics in the Baum–Katz and Davis Laws of Large Numbers

Gut, Allan; Spătaru, Aurel

doi:10.1006/jmaa.2000.6892

Cited by 115 publications

(76 citation statements)

References 14 publications

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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%

“…case) Spȃtaru [28], followed by Gut and Spȃtaru [9], obtained versions of Theorems 2.1 and 2.3-2.4, respectively, in the case of partial sums S n of i.i.d. random variables belonging to the (normal) domain of attraction of a stable law with index α ∈ (1, 2] (see, e.g., Theorems 1.3-1.5 above).…”

Section: Examples In Terms Of Weaker Assumptionsmentioning

confidence: 99%

“…Remaining values of r and p have later been taken care of in [2,28,9]. For ease of reference we state those results here.…”

Section: Introductionmentioning

confidence: 99%

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Precise asymptotics – A general approach

Gut

Steinebach

2012

Acta Math Hung

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The legendary 1947-paper by Hsu and Robbins, in which the authors introduced the concept of "complete convergence", generated a series of papers culminating in the like-wise famous Baum-Katz 1965-theorem, which provided necessary and sufficient conditions for the convergence of the series ∞ n=1 n r/p−2 P (|Sn| ≥ εn 1/p ) for suitable values of r and p, in which Sn denotes the n-th partial sum of an i.i.d. sequence. Heyde followed up the topic in his 1975-paper in that he investigated the rate at which such sums tend to infinity as ε 0 (for the case r = 2 and p = 1). The remaining cases have been taken care of later under the heading "precise asymptotics". An abundance of papers have since then appeared with various extensions and modifications of the i.i.d.-setting. The aim of the present paper is to show that the basis for the proof is essentially the same throughout, and to collect a number of examples. We close by mentioning that Klesov, in 1994, initiated work on rates in the sense that he determined the rate, as ε 0, at which the discrepancy between such sums and their "Baum-Katz limit" converges to a nontrivial quantity for Heyde's theorem. His result has recently been extended to the complete set of r-and p-values by the present authors.

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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%

Section: Examples In Terms Of Weaker Assumptionsmentioning

confidence: 99%

“…Remaining values of r and p have later been taken care of in [2,28,9]. For ease of reference we state those results here.…”

Section: Introductionmentioning

confidence: 99%

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Precise asymptotics – A general approach

Gut

Steinebach

2012

Acta Math Hung

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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%