Convergence of moments and related functional in the general central limit theorem in banach spaces

Acosta, A. de; Giné, Evarist

doi:10.1007/bf01886874

Cited by 61 publications

(24 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this 7Î choose TV so large that n > TV implies To prove the Theorem, we modify an argument of de Acosta (1979 and; as compared to the argument in Hudson, Veeh, and Weiner (1987), the work here is simplified because "regular variation" of the operators {n~A} for normal attraction is built in; no convergence-of-types arguments are required to see, for example, that for every m, we have (mn)~AnA -* m~A as n -+ oo.…”

Section: Results and Proofsmentioning

confidence: 99%

On the Existence and Convergence of Pseudomoments for Variables in the Domain of Normal Attraction of an Operator Stable Distribution

Weiner¹

1987

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

ABSTRACT. Integrals are constructed to replace absolute moments for variables in the domain of normal attraction of an operator stable law. These integrals, called pseudomoments, improve on the geometric information contained in absolute moments. Existence and convergence to appropriate values of these integrals are shown for the variables and their affine normalized sums. Introduction.The purpose of this note is to prove the existence and convergence of certain integrals (called pseudomoments) involving weakly convergent affine normalized sums of independent, identically distributed random vectors taking values in a finite-dimensional Euclidean space, when the normalizing operators take on a special form. The existence of these pseudomoments for the common distribution of the summands, and the convergence of the pseudomoments of the affine normalized sums to the corresponding pseudomoments of the limiting distribution, are shown to occur precisely when the latter exist, in the case of a non-Gaussian limit. The case of Gaussian limits is also covered.The pseudomoments of a distribution (for definitions, see §2) are designed to reflect the geometry of the distribution more accurately and informatively than do ordinary absolute moments. Since the great utility of affine normalization is its attention to variable decay rates in the probability tails of the one-dimensional projections (marginals) of a distribution, it is clearly an advantage to construct moment-integrals showing similar attention to these marginal tails, as we will see pseudomoments do.

show abstract

Section: Results and Proofsmentioning

confidence: 99%

On the Existence and Convergence of Pseudomoments for Variables in the Domain of Normal Attraction of an Operator Stable Distribution

Weiner¹

1987

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

show abstract

“…(4.11) holds for k n = n. In fact, the Corollary to Theorem 3 on page 8 of [21] for stochastically compact sequences might enable us to conclude uniform integrability. But following the proof of Theorem 3 on the basis of the proof of Theorem 6.1 in [14] the exact knowledge of the weak accumulation points (4.14) enables us to sharpen the result. Theorem 4.7.…”

Section: By (42) We Have Convergence Of the Riemann Sumsmentioning

confidence: 96%

Almost sure limit theorems of mantissa type for semistable domains of attraction

Becker-Kern

2007

Acta Math Hung

View full text Add to dashboard Cite

A certain class of stochastic summability methods of mantissa type is introduced and its connection to almost sure limit theorems is discussed. The summability methods serve as suitable weights in almost sure limit theory, covering all relevant known examples for, e.g., normalized sums or maxima of i.i.d. random variables. In the context of semistable domains of attraction the methods lead to previously unknown versions of semistable almost sure limit theorems.

show abstract

“…Now F ∈ D 2 implies that there exists a positive non-decreasing function slowly varying at infinity such that both (9) and Since F ∈ D 2 , we can use any number of asymptotically equivalent normings n in (20) such that n = n satisfies (9). We shall only supply a proof in the case when E X 2 = .…”

Section: Proofsmentioning

confidence: 99%

The Chow and Robbins Fixed Width Confidence Interval Procedure When the Second Moment Barely Does Not Exist

Mason

2004

Sequential Analysis

View full text Add to dashboard Cite

show abstract

Convergence of moments and related functional in the general central limit theorem in banach spaces

Cited by 61 publications

References 16 publications

On the Existence and Convergence of Pseudomoments for Variables in the Domain of Normal Attraction of an Operator Stable Distribution

On the Existence and Convergence of Pseudomoments for Variables in the Domain of Normal Attraction of an Operator Stable Distribution

Almost sure limit theorems of mantissa type for semistable domains of attraction

The Chow and Robbins Fixed Width Confidence Interval Procedure When the Second Moment Barely Does Not Exist

Contact Info

Product

Resources

About