On Mixture Representations for the Generalized Linnik Distribution and Their Applications in Limit Theorems

Korolev, V. Yu.; Gorshenin, Andrey; Zeifman, Alexander

doi:10.1007/s10958-020-04755-8

Cited by 8 publications

(14 citation statements)

References 41 publications

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“…V. Linnik [41]. Being scale mixtures of normal distributions, see, e.g., [35], Linnik distributions can serve as the one-dimensional distributions of a special subordinated Wiener process used to model the evolution of stock prices and financial indexes. Also, generalized Linnik distributions are good candidates for modelling financial data which exhibit high kurtosis and heavy tails [44].…”

Section: Inversion Inequalitymentioning

confidence: 99%

Wasserstein convergence in Bayesian deconvolution models

Rousseau¹,

Scricciolo²

2021

Preprint

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We study the reknown deconvolution problem of recovering a distribution function from independent replicates (signal) additively contaminated with random errors (noise), whose distribution is known. We investigate whether a Bayesian nonparametric approach for modelling the latent distribution of the signal can yield inferences with asymptotic frequentist validity under the L 1 -Wasserstein metric. When the error density is ordinary smooth, we develop two inversion inequalities relating either the L 1 or the L 1 -Wasserstein distance between two mixture densities (of the observations) to the L 1 -Wasserstein distance between the corresponding distributions of the signal. This smoothing inequality improves on those in the literature. We apply this general result to a Bayesian approach bayes on a Dirichlet process mixture of normal distributions as a prior on the mixing distribution (or distribution of the signal), with a Laplace or Linnik noise. In particular we construct an adaptive approximation of the density of the observations by the convolution of a Laplace (or Linnik) with a well chosen mixture of normal densities and show that the posterior concentrates at the minimax rate up to a logarithmic factor. The same prior law is shown to also adapt to the Sobolev regularity level of the mixing density, thus leading to a new Bayesian estimation method, relative to the Wasserstein distance, for distributions with smooth densities.

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Section: Inversion Inequalitymentioning

confidence: 99%

Wasserstein convergence in Bayesian deconvolution models

Rousseau¹,

Scricciolo²

2021

Preprint

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“…Here α ∈ (0, 2] is the characteristic exponent, θ ∈ [−1, 1] is the skewness parameter (for simplicity we consider the "standard" case with unit scale coefficient at t). Any random variable with characteristic function (5) will be denoted S(α, θ) and the characteristic function (5) itself will be written as f α,θ (t). For definiteness, S(1, 1) = 1.…”

Section: Univariate Stable Distributionsmentioning

confidence: 99%

“…From (5) it follows that the characteristic function of a symmetric (θ = 0) strictly stable distribution has the form f α,0 (t) = e −|t| α , t ∈ R.…”

Section: Univariate Stable Distributionsmentioning

confidence: 99%

“…Namely, we consider mixture representations for the multivariate generalized Mittag-Leffler and multivariate generalized Linnik distributions. We continue the research we started in [2][3][4][5]. In most papers (see, e.g., [6][7][8][9][10][11][12][13][14][15][16][17][18]), the properties of the (multivariate) generalized Mittag-Leffler and Linnik distributions were deduced by analytical methods from the properties of the corresponding probability densities and/or characteristic functions.…”

Section: Introductionmentioning

confidence: 99%

“…Methodically, the present paper is very close to the work of L. Devroye [20] where many examples of mixture representations of popular probability distributions were discussed from the simulation point of view. The presented material substantially relies on the results of [2,5,15].…”

Section: Introductionmentioning

confidence: 99%

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Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems

2020

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In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag–Leffler distributions are considered in detail. Their relations with multivariate ‘ordinary’ Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag–Leffler and generalized Mittag–Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution.

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Power-law Lévy processes, power-law vector random fields, and some extensions

Ma¹

2022

Proc. Amer. Math. Soc.

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This paper introduces a power-law subordinator and a power-law Lévy process whose Laplace transform and characteristic function are simply made up of power functions or the ratio of power functions, respectively, and proposes a power-law vector random field whose finite-dimensional characteristic functions consist merely of a power function or the ratio of two power functions. They may or may not have first-order moment, and contain Linnik, variance Gamma, and Laplace Lévy processes (vector random fields) as special cases. For a second-order power-law vector random field, it is fully characterized by its mean vector function and its covariance matrix function, just like a Gaussian vector random field. An important feature of the power-law Lévy processes (random fields) is that they can be used as the building blocks to construct other Lévy processes (random fields), such as hyperbolic secant, cosine ratio, and sine ratio Lévy processes (random fields).

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On Mixture Representations for the Generalized Linnik Distribution and Their Applications in Limit Theorems

Cited by 8 publications

References 41 publications

Wasserstein convergence in Bayesian deconvolution models

Wasserstein convergence in Bayesian deconvolution models

Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems

Power-law Lévy processes, power-law vector random fields, and some extensions

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