Subexponentiality and infinite divisibility

1979 ◽  
Vol 49 (3) ◽  
pp. 335-347 ◽  
Author(s):  
Paul Embrechts ◽  
Charles M. Goldie ◽  
No�l Veraverbeke
Keyword(s):  
Infinite Divisibility

Creation, Time, and Eternity

2018 ◽  
Author(s):  
T. M. Rudavsky
Keyword(s):  
Religious Belief ◽  
Infinite Divisibility ◽  
Time Function ◽  
Islamic Philosophy ◽  
The World ◽  
Acceptable Model ◽  
The Many

Of the many philosophical perplexities facing medieval Jewish thinkers, perhaps none has challenged religious belief as much as God’s creation of the world. No Jewish philosopher denied the importance of creation, that the world had a beginning (bereshit). But like their Christian and Muslim counterparts, Jewish thinkers did not always agree upon what qualifies as an acceptable model of creation. Chapter 6 is devoted to attempts of Jewish philosophers to reconcile the biblical view of creation with Greek and Islamic philosophy. By understanding the notion of creation and how an eternal, timeless creator created a temporal universe, we may begin to understand how the notions of eternity, emanation, and the infinite divisibility of time function within the context of Jewish philosophical theories of creation.

scholarly journals On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1568
Author(s):  
Shaul K. Bar-Lev
Keyword(s):  
Power Function ◽  
Exponential Family ◽  
Probability Distributions ◽  
Infinite Divisibility ◽  
Variance Function ◽  
Exponential Families ◽  
Natural Exponential Family ◽  
Natural Exponential Families ◽  
Convolution Properties ◽  
The Relationship

Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .

scholarly journals On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1571
Author(s):  
Irina Shevtsova ◽  
Mikhail Tselishchev
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Law Of Large Numbers ◽  
Negative Binomial ◽  
Infinite Divisibility ◽  
Random Sums ◽  
Generalized Gamma ◽  
Second Moments ◽  
Large Numbers ◽  
Generalized Negative Binomial Distribution

We investigate the proximity in terms of zeta-structured metrics of generalized negative binomial random sums to generalized gamma distribution with the corresponding parameters, extending thus the zeta-structured estimates of the rate of convergence in the Rényi theorem. In particular, we derive upper bounds for the Kantorovich and the Kolmogorov metrics in the law of large numbers for negative binomial random sums of i.i.d. random variables with nonzero first moments and finite second moments. Our method is based on the representation of the generalized negative binomial distribution with the shape and exponent power parameters no greater than one as a mixed geometric law and the infinite divisibility of the negative binomial distribution.

Locations

1979 ◽  
Vol 9 (2) ◽  
pp. 323-333 ◽  
Author(s):  
William J. Edgar
Keyword(s):  
Infinite Divisibility ◽  
Infinite Regress ◽  
Reductio Ad Absurdum ◽  
Ontological Analysis

Zeno's challenge to the usual mathematical characterization of extension is still with us. Butchvarov, considering the limits of ontological analysis, writes, “I shall not explore [the decision to accept the infinite regress in which the pursuit of the analytical ideal is involved], beyond noting that the infinite divisibility of space is the reductio ad absurdum of any attempt to understand space in terms of its ultimate, simple parts.” Grünbaum states this problem, commonly known as the Measure Paradox, concisely, “[How can one conceive] of an extended continuum as an aggregate of unextended elements ?”

scholarly journals Distributions with complete monotone derivative and geometric infinite divisibility

1990 ◽  
Vol 22 (3) ◽  
pp. 751-754 ◽  
Author(s):  
R. N. Pillai ◽  
E. Sandhya
Keyword(s):  
Infinite Divisibility ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Proper Subclass ◽  
Geometric Infinite Divisibility

It is shown that a distribution with complete monotone derivative is geometrically infinitely divisible and that the class of distributions with complete monotone derivative is a proper subclass of the class of geometrically infinitely divisible distributions.

scholarly journals Infinite Divisibility of Gaussian Squares with Non-zero Means

2008 ◽  
Vol 13 (0) ◽  
pp. 364-376
Author(s):  
Michael Marcus ◽  
Jay Rosen
Keyword(s):  
Infinite Divisibility

scholarly journals Some recent results in infinite divisibility

1973 ◽  
Vol 5 (1) ◽  
pp. 25-27
Author(s):  
F. W. Steutel
Keyword(s):  
Infinite Divisibility
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