Characteristic functions of random variables attracted to $1$-stable laws

Aaronson, Jon; Denker, Manfred

doi:10.1214/aop/1022855426

Cited by 42 publications

(107 citation statements)

References 6 publications

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“…One can take A n = nE(Z) if Z is integrable, and A n = 0 if Z ∈ D 3 with p < 1 (if p = 1 but Z is not integrable, the value of A n is more complicated to express, see [AD98]). Moreover, the random variables in D are the only ones to satisfy such a limit theorem: if a random variable Z is such that the sequence n−1 k=0 Z k satisfies a nondegenerate limit theorem, then Z ∈ D.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

Gouëzel

2010

Isr. J. Math.

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Abstract. We investigate limit theorems for Birkhoff sums of locally Hölder functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For L 2 observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.

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Section: Introduction and Resultsmentioning

confidence: 99%

Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

Gouëzel

2010

Isr. J. Math.

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“…It follows from the characterization of domains of attraction of multidimensional symmetric stable distributions in terms of characteristic functions (see Corollaries 1 and 2 in [1]) that the covariance function C must have the form (34), (35). Inserting this in (39) for some t with t = 1, we obtain which yields (37).…”

mentioning

confidence: 99%

Stationary max-stable fields associated to negative definite functions

Kabluchko¹,

Schlather²,

Haan³

2009

Ann. Probab.

337

561

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Let Wi, i ∈ N, be independent copies of a zero-mean Gaussian process {W (t), t ∈ R d } with stationary increments and variance σ 2 (t). Independently of Wi, letδU i be a Poisson point process on the real line with intensity e −y dy. We show that the law of the random family of functions {Vi(·), i ∈ N}, where Vi(t) = Ui + Wi(t) − σ 2 (t)/2, is translation invariant. In particular, the process η(t) =Vi(t) is a stationary max-stable process with standard Gumbel margins. The process η arises as a limit of a suitably normalized and rescaled pointwise maximum of n i.i.d. stationary Gaussian processes as n → ∞ if and only if W is a (nonisotropic) fractional Brownian motion on R d . Under suitable conditions on W , the process η has a mixed moving maxima representation.

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“…By studying P t on an appropriate function space, motivated by [HH1,HH2,GP], we prove that P t is a quasi-compact operator with a simple maximal eigenvalue λ t . We then apply a result from [AD1,AD2] to study the asymptotic expansion of λ t , allowing us to relate λ t to the sums in question. §2 Statement of results Let (X, d) be a locally compact (but not necessarily compact) second countable metric space.…”

Section: §1 Introductionmentioning

confidence: 99%

Distributional and Local Limit Laws for a Class of Iterated Maps That Contract on Average

Santos

Walkden

2013

Stoch. Dyn.

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We consider iterated function schemes that contract on average with place-dependent probabilities. We are interested in generalisations of the central limit theorem, particularly to observations with infinite variance. By studying the spectral properties of an associated one-parameter family of transfer operators acting on an appropriate function space, we prove both a distributional and local limit law with convergence to a stable distribution.

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Characteristic functions of random variables attracted to $1$-stable laws

Abstract: The domain of attraction of a 1-stable law on R d is characterized by the expansions of the characteristic functions of its elements.

Cited by 42 publications

References 6 publications

Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

Stationary max-stable fields associated to negative definite functions

Distributional and Local Limit Laws for a Class of Iterated Maps That Contract on Average

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