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Journal articles on the topic 'Distributions stables'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Zydor, Michał. "La Variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes unitaires." Canadian Journal of Mathematics 68, no. 6 (December 1, 2016): 1382–435. http://dx.doi.org/10.4153/cjm-2015-054-9.

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RésuméNous établissons une variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes unitaires. Notre formule s’obtient par intégration d'un noyau tronqué á la Arthur. Elle posséde un côté géométrique qui est une somme de distributions Jo indexée par les classes d'éléments de l'ébre de Lie de U(n + 1) stables par U(n)-conjugaison ainsi qu'un “côté spectral” formé des transformées de Fourier des distributions précédentes. On démontre que les distributions Jo sont invariantes et ne dépendent que du choix de lamesure deHaar sur U(n)(𝔸). Pour des classes o semi-simples réguliéres, Jo est une intégrale orbitale relative de Jacquet-Rallis. Pour les classes o dites relativement semi-simples régulières, on exprime Jo en terme des intégrales orbitales relatives régularisées á l'aide des fonctions zêta.

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2

Guevara, Cira, and Thiago R. Sousa. "ESTIMADOR SIMPLE Y FUERTEMENTE CONSISTENTE DE DISTRIBUCIONES ESTABLES." Selecciones Matemáticas 3, no. 1 (June 30, 2016): 25–31. http://dx.doi.org/10.17268/sel.mat.2016.01.04.

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3

Nguyen, T. T. "Conditional Distributions and Characterizations of Multivariate Stable Distribution." Journal of Multivariate Analysis 53, no. 2 (May 1995): 181–93. http://dx.doi.org/10.1006/jmva.1995.1031.

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4

Panton, Don B. "Cumulative distribution function values for symmetric standardized stable distributions." Communications in Statistics - Simulation and Computation 21, no. 2 (1992): 485–92. http://dx.doi.org/10.1080/03610919208813030.

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5

Ejsmont, Wiktor. "A Characterization of Symmetric Stable Distributions." Journal of Function Spaces 2016 (2016): 1–3. http://dx.doi.org/10.1155/2016/8384767.

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Characterization problems in probability are studied here. Using the characteristic function of an additive convolution we generalize some known characterizations of the normal distribution to stable distributions. More precisely, if a distribution of a linear form depends only on the sum of powers of the certain parameters, then we obtain symmetric stable distributions.

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6

Саенко, В. В., and V. V. Saenko. "Дробно-устойчивая статистика экспрессии генов в экспериментальных данных секвенирования нового поколения." Mathematical Biology and Bioinformatics 11, no. 2 (November 25, 2016): 278–87. http://dx.doi.org/10.17537/2016.11.278.

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As has been shown, in the author's published articles, that the application of class of the fractional-stable laws to the genes expression results obtained by DNA-microarys leads to poor agreement between experimental and theoretical distributions. This difference can be explained by the imperfection of the technology of the gene expression determination. In this article the distributions of the gene expression obtained by Next Generation Sequence technology are investigated. In this technology the determination technique of the gene expression differs from the DNA-microarrays technology. This results to more qualitative results of an approximation. In particular, it is established that the probability density function of the gene expression has a form of shift-scale mixture of probability laws, where one of the components of the mixture is the fractional-stable distribution.

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7

Ravi, S., and TS Mavitha. "THE INTERPLAY BETWEEN I-MAX, I-MIN, P-MAX AND P-MIN STABLE DISTRIBUTIONS." Mathematical Journal of Interdisciplinary Sciences 4, no. 1 (September 1, 2015): 49–53. http://dx.doi.org/10.15415/mjis.2015.41006.

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8

Khokhlov, Yury, Victor Korolev, and Alexander Zeifman. "Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems." Mathematics 8, no. 5 (May 8, 2020): 749. http://dx.doi.org/10.3390/math8050749.

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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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9

Song, Guan, An Na, Liu Jinhua, Zong Ning, He Yongtao, Shi Peili, Zhang Jinjing, and He Nianpeng. "Warming impacts on carbon, nitrogen and phosphorus distribution in soil water-stable aggregates." Plant, Soil and Environment 64, No. 2 (February 6, 2018): 64–69. http://dx.doi.org/10.17221/715/2017-pse.

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A five-year (2010–2015) field experiment was conducted to investigate warming impacts on organic carbon (OC), total nitrogen (TN) and total phosphorus (TP) contents and their ratios in bulk soil and soil water-stable aggregates in an alpine meadow of the Tibetan Plateau. Compared with unwarmed control, warming had no significant effects on OC, TN and TP contents and their ratios in bulk soil. The contents of OC, TN and TP associated with macroaggregates and microaggregates decreased, whereas those associated with silt + clay fractions significantly increased. The C:N and C:P ratios in macro- and microaggregates and silt + clay fractions decreased, with significant differences for C:P ratio in microaggregates and C:N and C:P ratios in silt + clay fractions. The results indicated that C, N and P were protected chemically in silt- and clay-size fractions under warming, which offset the loss of C, N and P protected physically by macro- and microaggregates. Both physically and chemically protected C decomposition proceeded relatively more rapidly or accumulated relatively more slowly than did N and P. Our results suggest that C, N and P distributions within soil aggregate size fractions influence their net changes in bulk soil under future climate change scenarios.

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10

Nagaev, A. V., and S. M. Shkol’nik. "Some Properties of Symmetric Stable Distributions Close to the Normal Distribution." Theory of Probability & Its Applications 33, no. 1 (January 1989): 139–44. http://dx.doi.org/10.1137/1133014.

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11

Semelbauer, Marek, Barbara Mangová, Marek Barta, and Milan Kozánek. "The Factors Influencing Seasonal Dynamics and Spatial Distribution of Stable Fly Stomoxys calcitrans (Diptera, Muscidae) within Stables." Insects 9, no. 4 (October 16, 2018): 142. http://dx.doi.org/10.3390/insects9040142.

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The biology of the stable fly is fairly well known, but factors influencing the distribution of adult stable flies within stables are still inadequately investigated. The four experimental stables were located in south western Slovakia. Within each stable, five sticky traps were localized along the stable, and the flies were weekly counted during the flight season of years 2015–2017. Seasonal activity and stable fly abundance in relation to temperature, rainfall, light conditions, relative air humidity, and cows per stable were evaluated. The seasonal activity of the stable fly shows one large peak at the end of summer and a second smaller peak just before the end of the flight season. The spatial distribution of stable flies was unique for each stable. All of the environmental variables had significant and mostly positive effect on stable fly abundance. The strongest and most positive effect on stable fly counts was temperature and rainfall five weeks prior to collecting session. Within the stable, cow number, air humidity, and light conditions are the strongest candidates to influence their distribution.

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12

Klebanov, L. B., A. V. Kakosyan, S. T. Rachev, and G. Temnov. "On a Class of Distributions Stable Under Random Summation." Journal of Applied Probability 49, no. 02 (June 2012): 303–18. http://dx.doi.org/10.1017/s0021900200009104.

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We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

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13

Klebanov, L. B., A. V. Kakosyan, S. T. Rachev, and G. Temnov. "On a Class of Distributions Stable Under Random Summation." Journal of Applied Probability 49, no. 2 (June 2012): 303–18. http://dx.doi.org/10.1239/jap/1339878788.

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We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

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14

Zhang, Xiuzhi, Ping Zhu, Chang Peng, Hongjun Gao, Qiang Li, Jinjing Zhang, and Qaing Gao. "Phosphorus distribution and availability within soil water-stable aggregates as affected by long-term fertilisation." Plant, Soil and Environment 66, No. 11 (November 2, 2020): 552–58. http://dx.doi.org/10.17221/394/2020-pse.

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A field experiment lasting 37 years was conducted to evaluate the applications of different rates of pig manure and mineral fertilisers alone or in combination impacts on total phosphorus (P<sub>total</sub>) and Olsen phosphorus (P<sub>Olsen</sub>) contents and phosphorus activity coefficient (PAC, percentage of P<sub>Olsen</sub> to P<sub>total</sub>) within soil water-stable aggregates (WSA) in a Mollisol of Northeast China. The contents of P<sub>total</sub> and P<sub>Olsen</sub> associated with different size classes of WSA significantly (P < 0.05) increased with an increasing rate of applied P. The application of manure alone or combined with mineral fertilisers significantly increased PAC value associated with different size classes of WSA. There were positive correlations between P<sub>total</sub> and P<sub>Olsen</sub> contents with soil organic carbon (SOC) content within soil WSA. As SOC content increased 1 g/kg, P<sub>total</sub> and P<sub>Olsen</sub> contents increased 0.06–0.10 g/kg and 7.69–22.2 mg/kg, respectively, and the increase was larger in smaller size classes of WSA. The results suggested that a high manure rate combined with mineral fertilisers is more beneficial for increasing soil phosphorus content and availability. SOC is a vital factor controlling phosphorus content and availability within soil WSA.

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15

Petrenz, Philipp, and Bonnie Webber. "Stable Classification of Text Genres." Computational Linguistics 37, no. 2 (June 2011): 385–93. http://dx.doi.org/10.1162/coli_a_00052.

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Every text has at least one topic and at least one genre. Evidence for a text's topic and genre comes, in part, from its lexical and syntactic features—features used in both Automatic Topic Classification and Automatic Genre Classification (AGC). Because an ideal AGC system should be stable in the face of changes in topic distribution, we assess five previously published AGC methods with respect to both performance on the same topic–genre distribution on which they were trained and stability of that performance across changes in topic–genre distribution. Our experiments lead us to conclude that (1) stability in the face of changing topical distributions should be added to the evaluation critera for new approaches to AGC, and (2) Part-of-Speech features should be considered individually when developing a high-performing, stable AGC system for a particular, possibly changing corpus.

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16

Achcar, Jorge A., and Sílvia R. C. Lopes. "Linear and Non-Linear Regression Models Assuming a Stable Distribution." Revista Colombiana de Estadística 39, no. 1 (January 18, 2016): 109–28. http://dx.doi.org/10.15446/rce.v39n1.55144.

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<p>In this paper, we present some computational aspects for a Bayesian analysis involving stable distributions. It is well known that, in general, there is no closed form for the probability density function of a stable distribution. However, the use of a latent or auxiliary random variable facilitates obtaining any posterior distribution when related to stable distributions. To show the usefulness of the computational aspects, the methodology is applied to linear and non-linear regression models. Posterior summaries of interest are obtained using the OpenBUGS software.</p>

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17

Pitman, E. J. G., and Jim Pitman. "A direct approach to the stable distributions." Advances in Applied Probability 48, A (July 2016): 261–82. http://dx.doi.org/10.1017/apr.2016.55.

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AbstractThe explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying the theory of regular variation, without appeal to the general Lévy‒Khintchine integral representation of infinitely divisible distributions.

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18

So, Jacky C. "The Distribution of Financial Ratios—A Note." Journal of Accounting, Auditing & Finance 9, no. 2 (April 1994): 215–23. http://dx.doi.org/10.1177/0148558x9400900205.

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Three competitive distributions are offered by the literature to explain the non-normality and skewness of the cross-sectional distribution of financial ratios: the mixture of normal distributions, the lognormal distribution, and the gamma distribution. Using a new technique, this paper shows that the lognormal distribution and the gamma distribution are not supported by the empirical evidence. Although these two distributions indeed capture skewness, they do not portray the correct shape of the distributions. The non-normal stable Paretian distribution seems to be good candidate to describe the distribution of financial ratios.

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19

Ahsanullah, Mohammad, and V. B. Nevzorov. "On Some Characterizations of the Levy Distribution." Stochastics and Quality Control 34, no. 1 (June 1, 2019): 53–57. http://dx.doi.org/10.1515/eqc-2018-0031.

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Abstract If the distribution of the linear combination of two independent and identically distributed random variables from a distribution belongs to the same distribution, then we call that distribution a stable distribution. The Levy distribution is a member of the family of stable distributions. In this paper, we will present some basic distributional properties and characterizations of the Levy distribution.

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20

Bielinskyi, Andrii, Serhiy Semerikov, Viktoria Solovieva, and Vladimir Soloviev. "Levy’s stable distribution for stock crash detecting." SHS Web of Conferences 65 (2019): 06006. http://dx.doi.org/10.1051/shsconf/20196506006.

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In this paper we study the possibility of construction indicators-precursors relying on one of the most power-law tailed distributions - Levy’s stable distribution. Here, we apply Levy’s parameters for 29 stock indices for the period from 1 March 2000 to 28 March 2019 daily values and show their effectiveness as indicators of crisis states on the example of Dow Jones Industrial Average index for the period from 2 January 1920 to 2019. In spite of popularity of the Gaussian distribution in financial modeling, we demonstrated that Levy’s stable distribution is more suitable due to its theoretical reasons and analysis results. And finally, we conclude that stability α and skewness β parameters of Levy’s stable distribution which demonstrate characteristic behavior for crash and critical states, can serve as an indicator-precursors of unstable states.

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21

WERON, RAFAŁ. "LEVY-STABLE DISTRIBUTIONS REVISITED: TAIL INDEX > 2 DOES NOT EXCLUDE THE LEVY-STABLE REGIME." International Journal of Modern Physics C 12, no. 02 (February 2001): 209–23. http://dx.doi.org/10.1142/s0129183101001614.

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Power-law tail behavior and the summation scheme of Levy-stable distributions is the basis for their frequent use as models when fat tails above a Gaussian distribution are observed. However, recent studies suggest that financial asset returns exhibit tail exponents well above the Levy-stable regime (0 < α ≤ 2). In this paper, we illustrate that widely used tail index estimates (log–log linear regression and Hill) can give exponents well above the asymptotic limit for α close to 2, resulting in overestimation of the tail exponent in finite samples. The reported value of the tail exponent α around 3 may very well indicate a Levy-stable distribution with α ≈ 1.8.

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22

Garay, József, and Zoltán Varga. "Evolutionarily Stable Allele Distributions." Journal of Theoretical Biology 191, no. 2 (March 1998): 163–72. http://dx.doi.org/10.1006/jtbi.1997.0577.

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23

Wang, Cheng, and Zhihao Ma. "Operator Semi-Stable Distributions." Stochastic Analysis and Applications 23, no. 4 (July 2005): 659–64. http://dx.doi.org/10.1081/sap-200050124.

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24

Sato, Ken-iti. "Strictly operator-stable distributions." Journal of Multivariate Analysis 22, no. 2 (August 1987): 278–95. http://dx.doi.org/10.1016/0047-259x(87)90091-1.

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25

Pillai, R. N. "Generalized geometric stable distributions." Journal of Soviet Mathematics 52, no. 2 (November 1990): 2974–77. http://dx.doi.org/10.1007/bf01103754.

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26

Bouzar, Nadjib. "Discrete semi-stable distributions." Annals of the Institute of Statistical Mathematics 56, no. 3 (September 2004): 497–510. http://dx.doi.org/10.1007/bf02530538.

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27

Khokhlov, Yury, and Victor Korolev. "On a Multivariate Analog of the Zolotarev Problem." Mathematics 9, no. 15 (July 22, 2021): 1728. http://dx.doi.org/10.3390/math9151728.

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A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random index having a special multivariate geometric distribution are considered. Anisotropic-geometric stable distributions are introduced. It is demonstrated that these distributions are coordinate-wise scale mixtures of elliptically contoured stable distributions with the Marshall–Olkin mixing distributions. The corresponding “anisotropic” analogs of multivariate Laplace, Linnik and Mittag–Leffler distributions are introduced. Some relations between these distributions are presented.

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28

Kagan, Y. Y. "Distribution of incremental static stress caused by earthquakes." Nonlinear Processes in Geophysics 1, no. 2/3 (September 30, 1994): 172–81. http://dx.doi.org/10.5194/npg-1-172-1994.

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Abstract. Theoretical calculations, simulations and measurements of rotation of earthquake focal mechanisms suggest that the stress in earthquake focal zones follows the Cauchy distribution which is one of the stable probability distributions (with the value of the exponent α equal to 1). We review the properties of the stable distributions and show that the Cauchy distribution is expected to approximate the stress caused by earthquakes occurring over geologically long intervals of a fault zone development. However, the stress caused by recent earthquakes recorded in instrumental catalogues, should follow symmetric stable distributions with the value of α significantly less than one. This is explained by a fractal distribution of earthquake hypocentres: the dimension of a hypocentre set, δ, is close to zero for short-term earthquake catalogues and asymptotically approaches 2¼ for long-time intervals. We use the Harvard catalogue of seismic moment tensor solutions to investigate the distribution of incremental static stress caused by earthquakes. The stress measured in the focal zone of each event is approximated by stable distributions. In agreement with theoretical considerations, the exponent value of the distribution approaches zero as the time span of an earthquake catalogue (ΔT) decreases. For large stress values α increases. We surmise that it is caused by the δ increase for small inter-earthquake distances due to location errors.

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29

SenGupta, Ashis, and Moumita Roy. "An Universal, Simple, Circular Statistics-Based Estimator of α for Symmetric Stable Family." Journal of Risk and Financial Management 12, no. 4 (November 23, 2019): 171. http://dx.doi.org/10.3390/jrfm12040171.

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The aim of this article is to obtain a simple and efficient estimator of the index parameter of symmetric stable distribution that holds universally, i.e., over the entire range of the parameter. We appeal to directional statistics on the classical result on wrapping of a distribution in obtaining the wrapped stable family of distributions. The performance of the estimator obtained is better than the existing estimators in the literature in terms of both consistency and efficiency. The estimator is applied to model some real life financial datasets. A mixture of normal and Cauchy distributions is compared with the stable family of distributions when the estimate of the parameter α lies between 1 and 2. A similar approach can be adopted when α (or its estimate) belongs to (0.5,1). In this case, one may compare with a mixture of Laplace and Cauchy distributions. A new measure of goodness of fit is proposed for the above family of distributions.

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30

Finkelstein, G. S. "Maturity Guarantees Revisited: Allowing for Extreme Stochastic Fluctuations using Stable Distributions." British Actuarial Journal 3, no. 2 (June 1, 1997): 411–82. http://dx.doi.org/10.1017/s1357321700004992.

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ABSTRACTThe paper examines the suitability of the stable family of distributions with the Maturity Guarantees Working Party's stochastic investment model (Ford et al, 1980). It then examines the effect of replacing the Gaussian assumption made by the working party with a more general stable distribution. It also explains how the appropriate stable distribution can be fitted.

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31

Semelbauer, Marek, Jozef Oboňa, Marek Barta, Barbara Mangová, and Milan Kozánek. "Spatial distribution and seasonal dynamics of non-biting moth flies (Diptera, Psychodidae) in confound conditions of a stable." Polish Journal of Entomology 89, no. 4 - Ahead of print (December 31, 2020): 190–99. http://dx.doi.org/10.5604/01.3001.0014.4974.

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Many small Diptera adopted at some level endophilic life style, including man-made buildings. Stables create a specific type of microhabitat, which provides shady and relatively humid conditions in combination with excess of organic matter. Unlike the parasites (mosquitos, biting midges, etc.), the commensal fauna of stables is poorly studied. Moth flies (Psychodidae) were collected in cow stable located in Šenkvice, SW Slovakia. Special traps (derived from Malaise traps) were installed along the stable internal wall and in three different heights. In total, we recorded 6325 moth flies belonging to 8 species. The flight period lasted from spring to autumn. Seasonal dynamics was strongly influenced by rainfall and mean week temperature, e.g. high temperature in mid-summer caused drop in moth flies captures. The moth flies clearly preferred the ground and moderately preferred the interior of stable.

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El-Hafsi, Boukhalfa. "Average Sample Number Function for Pareto Heavy Tailed Distributions." ISRN Applied Mathematics 2013 (June 10, 2013): 1–8. http://dx.doi.org/10.1155/2013/938545.

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The main purpose of this work is shortly to give the average sample number function after a sequential probability ratio test on the index parameter alpha of stable densities, which we give a mean of the number of data required to take decision in the case , we use the fact that the tails of Levy-stable distributions are asymptotically equivalent to a Pareto law for large data. Stable distributions are a rich class of probability distributions that allow skewness and heavy tails and have many intriguing mathematical properties. The lack of closed formulas for densities and distribution functions for all has been a major drawback to the use of stable distributions by practitioners, but few stable distributions have the analytical formula of their densities functions which are Gauss, Levy, and Cauchy.

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Naradh, Kimera, Knowledge Chinhamu, and Retius Chifurira. "Estimating the value-at-risk of JSE indices and South African exchange rate with Generalized Pareto and stable distributions." Investment Management and Financial Innovations 18, no. 3 (August 20, 2021): 151–65. http://dx.doi.org/10.21511/imfi.18(3).2021.14.

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South Africa’s economy has faced many downturns in the previous decade, and to curb the spread of the novel SARS-CoV-2, the lockdown brought South African financial markets to an abrupt halt. Therefore, the implementation of risk mitigation approaches is becoming a matter of urgency in volatile markets in these unprecedented times. In this study, a hybrid generalized autoregressive conditional heteroscedasticity (GARCH)-type model combined with heavy-tailed distributions, namely the Generalized Pareto Distribution (GPD) and the Nolan’s S0-parameterization stable distribution (SD), were fitted to the returns of three FTSE/JSE indices, namely All Share Index (ALSI), Banks Index and Mining Index, as well as the daily closing prices of the US dollar against the South African rand exchange rate (USD/ZAR exchange rate). VaR values were estimated and back-tested using the Kupiec likelihood ratio test. The results of this study show that for FTSE/JSE ALSI returns, the hybrid exponential GARCH (1,1) model with SD model (EGARCH(1,1)-SD) outperforms the GARCH-GPD model at the 2.5% VaR level. At VaR levels of 95% and 97.5%, the fitted GARCH (1,1)-SD model for FTSE/JSE Banks Index returns performs better than the GARCH (1,1)-GPD. The fitted GARCH (1,1)-SD model for FTSE/JSE Mining Index returns is better than the GARCH (1,1)-GPD at 5% and 97.5% VaR levels. Thus, this study suggests that the GARCH (1,1)-SD model is a good alternative to the VaR robust model for modeling financial returns. This study provides salient results for persons interested in reducing losses or obtaining a better understanding of the South African financial industry.

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Molchanov, Ilya S. "Limit theorems for convex hulls of random sets." Advances in Applied Probability 25, no. 2 (June 1993): 395–414. http://dx.doi.org/10.2307/1427659.

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Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn, and independent copies A1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

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35

Sato, Ken-Iti, and Makoto Yamazato. "Completely operator-selfdecomposable distributions and operator-stable distributions." Nagoya Mathematical Journal 97 (March 1985): 71–94. http://dx.doi.org/10.1017/s0027763000021267.

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Urbanik introduces in [16] and [17] the classes Lm and L∞ of distributions on R1 and finds relations with stable distributions. Kumar-Schreiber [6] and Thu [14] extend some of the results to distributions on Banach spaces. Sato [7] gives alternative definitions of the classes Lm and L∞ and studies their properties on Rd. Earlier Sharpe [12] began investigation of operator-stable distributions and, subsequently, Urbanik [15] considered the operator version of the class L on Rd. Jurek [3] generalizes some of Sato’s results [7] to the classes associated with one-parameter groups of linear operators in Banach spaces. Analogues of Urbanik’s classes Lm (or L∞) in the operator case are called multiply (or completely) operator-selfdecomposable. They are studied in relation with processes of Ornstein-Uhlenbeck type or with stochastic integrals based on processes with homogeneous independent increments (Wolfe [18], [19], Jurek-Vervaat [5], Jurek [2], [4], and Sato-Yamazato [9], [10]). The purpose of the present paper is to continue the preceding papers, to give explicit characterizations of completely operator-selfdecomposable distributions and operator-stable distributions on Rd, and to establish relations between the two classes. For this purpose we explore the connection of the structures of these classes with the Jordan decomposition of a basic operator Q.

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36

Molchanov, Ilya S. "Limit theorems for convex hulls of random sets." Advances in Applied Probability 25, no. 02 (June 1993): 395–414. http://dx.doi.org/10.1017/s0001867800025416.

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Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn , and independent copies A 1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

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37

Rroji, Edit, and Lorenzo Mercuri. "Mixed tempered stable distribution." Quantitative Finance 15, no. 9 (October 23, 2014): 1559–69. http://dx.doi.org/10.1080/14697688.2014.969763.

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38

Fallahgoul, Hassan A., Young S. Kim, and Frank J. Fabozzi. "Elliptical tempered stable distribution." Quantitative Finance 16, no. 7 (February 2, 2016): 1069–87. http://dx.doi.org/10.1080/14697688.2015.1111522.

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39

Voorn, W. J. "Characterization of the logistic and loglogistic distributions by extreme value related stability with random sample size." Journal of Applied Probability 24, no. 4 (December 1987): 838–51. http://dx.doi.org/10.2307/3214209.

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Maximum stability of a distribution with respect to a positive integer random variable N is defined by the property that the type of distribution is not changed when considering the maximum value of N independent observations. The logistic distribution is proved to be the only symmetric distribution which is maximum stable with respect to each member of a sequence of positive integer random variables assuming value 1 with probability tending to 1. If a distribution is maximum stable with respect to such a sequence and minimum stable with respect to another, then it must be logistic, loglogistic or ‘backward' loglogistic. The only possible sample size distributions in these cases are geometric.

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40

Voorn, W. J. "Characterization of the logistic and loglogistic distributions by extreme value related stability with random sample size." Journal of Applied Probability 24, no. 04 (December 1987): 838–51. http://dx.doi.org/10.1017/s0021900200116729.

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Abstract:

Maximum stability of a distribution with respect to a positive integer random variable N is defined by the property that the type of distribution is not changed when considering the maximum value of N independent observations. The logistic distribution is proved to be the only symmetric distribution which is maximum stable with respect to each member of a sequence of positive integer random variables assuming value 1 with probability tending to 1. If a distribution is maximum stable with respect to such a sequence and minimum stable with respect to another, then it must be logistic, loglogistic or ‘backward' loglogistic. The only possible sample size distributions in these cases are geometric.

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41

Osypchuk, M. M., and M. I. Portenko. "On the distribution of a rotationally invariant α-stable process at the hitting time of a given hyperplane." Reports of the National Academy of Sciences of Ukraine, no. 12 (December 15, 2018): 14–20. http://dx.doi.org/10.15407/dopovidi2018.12.014.

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42

Rodgers, G. J., and M. K. Hassan. "Stable distributions in fragmentation processes." Physica A: Statistical Mechanics and its Applications 233, no. 1-2 (November 1996): 19–30. http://dx.doi.org/10.1016/s0378-4371(96)00234-8.

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43

Küchler, Uwe, and Stefan Tappe. "Tempered stable distributions and processes." Stochastic Processes and their Applications 123, no. 12 (December 2013): 4256–93. http://dx.doi.org/10.1016/j.spa.2013.06.012.

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Tapiero, Charles S., and Pierre Vallois. "Randomness and fractional stable distributions." Physica A: Statistical Mechanics and its Applications 511 (December 2018): 54–60. http://dx.doi.org/10.1016/j.physa.2018.07.019.

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45

SenGupta, A., and S. Roy. "Classification rules for stable distributions." Mathematical and Computer Modelling 34, no. 9-11 (November 2001): 1073–93. http://dx.doi.org/10.1016/s0895-7177(01)00117-0.

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Hofert, Marius. "Sampling Exponentially Tilted Stable Distributions." ACM Transactions on Modeling and Computer Simulation 22, no. 1 (December 2011): 1–11. http://dx.doi.org/10.1145/2043635.2043638.

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47

Fan, Zhaozhi. "Parameter Estimation of Stable Distributions." Communications in Statistics: Theory and Methods 35, no. 2 (March 1, 2006): 245–55. http://dx.doi.org/10.1080/03610920500439992.

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Krystek, Anna Dorota, and Łukasz Jan Wojakowski. "Conditionally free semi-stable distributions." Infinite Dimensional Analysis, Quantum Probability and Related Topics 18, no. 02 (June 2015): 1550015. http://dx.doi.org/10.1142/s0219025715500150.

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We define a notion of semi–stability in the conditionally free probability and explain that the semi–stable measures are infinitely divisible. We also show that in the conditionally free probability stable measures are semi–stable, and that semi–stability for all r implies stability.

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Krapivsky, P. L., E. Ben-Naim, and I. Grosse. "Stable distributions in stochastic fragmentation." Journal of Physics A: Mathematical and General 37, no. 8 (February 11, 2004): 2863–80. http://dx.doi.org/10.1088/0305-4470/37/8/002.

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50

Buckle, D. J. "Bayesian Inference for Stable Distributions." Journal of the American Statistical Association 90, no. 430 (June 1995): 605–13. http://dx.doi.org/10.1080/01621459.1995.10476553.

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