A Note on Mixture Representations for the Linnik and Mittag-Leffler Distributions and Their Applications*

Korolev, V. Yu.; Zeifman, Alexander

doi:10.1007/s10958-016-3032-6

Cited by 13 publications

(24 citation statements)

References 24 publications

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“…Proof. This theorem is a direct consequence of Theorem 2 with the account of relations (17) and (18).…”

Section: Convergence Of the Distributions Of Random Sums Of Random Vementioning

confidence: 69%

“…The class of scale mixtures of normal laws is very rich and involves distributions with various character of decrease of tails. For example, this class contains Student distributions with arbitrary (not necessarily integer) number of degrees of freedom (and the Cauchy distribution included), symmetric stable distributions (see the "multiplication theorem" 3.3.1 in [15]), symmetric fractional stable distributions (see [16]), symmetrized gamma distributions with arbitrary shape and scale parameters (see [10]), and symmetrized Weibull distributions with shape parameters belonging to the interval 0, 1 ð (see [17,18]). As an example, in the next section, we will discuss the conditions for the convergence of the distributions of the statistics constructed from samples with random sizes to the multivariate Student distribution.…”

Section: Some Remarks On the Heavy-tailedness Of Scale Mixtures Of Nomentioning

confidence: 99%

See 1 more Smart Citation

From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums and Statistics with Random Sample Sizes

Korolev¹,

Zeifman²

2020

Probability, Combinatorics and Control

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This chapter contains a possible explanation of the emergence of heavy-tailed distributions observed in practice instead of the expected normal laws. The bases for this explanation are limit theorems for random sums and statistics constructed from samples with random sizes. As examples of the application of general theorems, conditions are presented for the convergence of the distributions of random sums of independent random vectors with finite covariance matrices to multivariate elliptically contoured stable and Linnik distributions. Also, conditions are presented for the convergence of the distributions of asymptotically normal (in the traditional sense) statistics to multivariate Student distributions. The joint asymptotic behavior of sample quantiles is also considered.

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“…Proof. This theorem is a direct consequence of Theorem 2 with the account of relations (17) and (18).…”

Section: Convergence Of the Distributions Of Random Sums Of Random Vementioning

confidence: 69%

Section: Some Remarks On the Heavy-tailedness Of Scale Mixtures Of Nomentioning

confidence: 99%

From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums and Statistics with Random Sample Sizes

Korolev¹,

Zeifman²

2020

Probability, Combinatorics and Control

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“…In this paper we continue the research we started in [19,20]. We study the interrelationship between the (generalized) Linnik and (generalized) Mittag-Leffler distributions.…”

Section: Introductionmentioning

confidence: 88%

“…of the r.v. Y is identifiable, then condition (20) is not only sufficient for (21), but is necessary as well. Let X 1 , X 2 , .…”

Section: Convergence Of the Distributions Of Random Sums To The Genermentioning

confidence: 99%

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

2020

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We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, exponential and stable laws and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e. g., random sums of independent random variables with finite variances and maximum random sums, in which the generalized Linnik distribution plays the role of the limit law. Thus we demonstrate that the scheme of geometric (or, in general, negative binomial) summation is far not the only asymptotic setting (even for sums of independent random variables) in which the generalized Linnik law appears as the limit distribution.

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“…For each censoring threshold h = min i m i the sample is formed according to the rule (18). For each value of the threshold on the upper graph there are…”

Section: The Statistical Analysis Of Real Datamentioning

confidence: 99%

The probability distribution of extreme precipitation

Korolev

Gorshenin

2017

Dokl. Earth Sc.

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Based on the negative binomial model for the duration of wet periods measured in days [2], an asymptotic approximation is proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Fréchet distribution with the gamma mixing distribution and coincides with the distribution of a positive power of a random variable having the Snedecor-Fisher distribution. The proof of this result is based on the representation of the negative binomial distribution as a mixed geometric (and hence, mixed Poisson) distribution [7] and limit theorems for extreme order statistics in samples with random sizes having mixed Poisson distributions [22]. Some analytic properties of the obtained limit distribution are described.In particular, it is demonstrated that under certain conditions the limit distribution is mixed exponential and hence, is infinitely divisible. It is shown that under the same conditions the limit distribution can be represented as a scale mixture of stable or Weibull or Pareto or folded normal laws. The corresponding product representations for the limit random variable can be used for its computer simulation. Several methods are proposed for the estimation of the parameters of the distribution of the maximum daily precipitation volume. The results of fitting this distribution to real data are presented illustrating high adequacy of the proposed model. The obtained mixture representations for the limit laws and the corresponding asymptotic approximations provide better insight into the nature of mixed probability ("Bayesian") models.

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A Note on Mixture Representations for the Linnik and Mittag-Leffler Distributions and Their Applications*

Cited by 13 publications

References 24 publications

From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums and Statistics with Random Sample Sizes

From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums and Statistics with Random Sample Sizes

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

The probability distribution of extreme precipitation

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