Moments of distributions attracted to operator-stable laws

Hudson, William N.; Veeh, Jerry Alan; Weiner, Daniel C.

doi:10.1016/0047-259x(88)90097-8

“…The following result generalizes Theorem 3 of [21] which considers attraction to operator-stable laws, i.e. (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.…”

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

confidence: 84%

“…The following result generalizes Theorem 3 of [21] which considers attraction to operator-stable laws, i.e. (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.…”

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

confidence: 84%

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

Becker-Kern

¹

2007

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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.

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“…Before proving the Theorem, we remark that these existence and convergence results extend Theorem 3 in Hudson, Veeh, and Weiner (1987) from absolute moments (in the general domain of attraction)…”

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

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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.

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“…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

Proc. Amer. Math. Soc.

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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.

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Moments of distributions attracted to operator-stable laws

Cited by 21 publications

References 12 publications

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

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

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

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