scholarly journals Limit Theorems for Local Cumulative Shock Models with Cluster Shock Structure

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Jianming Bai ◽  
Yun Chen ◽  
Chun Yuan ◽  
Xiaoling Yin
Keyword(s):  
Financial Risk ◽  
Infinite Divisibility ◽  
Insurance Company ◽  
Infinitely Divisible ◽  
Stable Law ◽  
Scale Parameters ◽  
Shock Process ◽  
Shock Models ◽  
Cluster Pattern ◽  
Primary Shock

This paper considers a more general shock model with insurance and financial risk background, in which the system is subject to two types of shocks called primary shocks and secondary shocks. Each primary shock causes a series of secondary shocks according to some cluster pattern. In reliability applications, a primary shock can represent an issue of insurance policies of an insurer company, and the secondary shocks then denote the relevant insurance claims generated by the policy. We focus on the local cumulative shock process where only a certain number of the most recent primary and secondary shocks are accumulated. This process is a very new topic in the available literature which is more flexible and realistic in modeling some more complex reliability situations such as bankrupt behavior of an insurance company. Based on the theory of infinite divisibility and stable distributions, we establish a central limit theorem for the local cumulative shock process and obtain the conditions for the process to converge to an infinitely divisible distribution or to anα-stable law. Also, by choosing the proper scale parameters, the process converges to a normal distribution.

scholarly journals The effect of random scale changes on limits of infinitesimal systems

1978 ◽  
Vol 1 (3) ◽  
pp. 339-372
Author(s):  
Patrick L. Brockett
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Divisible Distribution ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Stable Law ◽  
Scale Parameters ◽  
Exact Relationship ◽  
Random Scale

SupposeS={{Xnj,   j=1,2,…,kn}}is an infinitesimal system of random variables whose centered sums converge in law to a (necessarily infinitely divisible) distribution with Levy representation determined by the triple(γ,σ2,M). If{Yj,   j=1,2,…}are independent indentically distributed random variables independent ofS, then the systemS′={{YjXnj,j=1,2,…,kn}}is obtained by randomizing the scale parameters inSaccording to the distribution ofY1. We give sufficient conditions on the distribution ofYin terms of an index of convergence ofS, to insure that centered sums fromS′be convergent. If such sums converge to a distribution determined by(γ′,(σ′)2,Λ), then the exact relationship between(γ,σ2,M)and(γ′,(σ′)2,Λ)is established. Also investigated is when limit distributions fromSandS′are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.

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 Distributions with complete monotone derivative and geometric infinite divisibility

1990 ◽  
Vol 22 (03) ◽  
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.

Max-infinite divisibility

1977 ◽  
Vol 14 (02) ◽  
pp. 309-319 ◽  
Author(s):  
A. A. Balkema ◽  
S. I. Resnick
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Infinite Divisibility ◽  
Necessary And Sufficient Conditions ◽  
Infinitely Divisible ◽  
Extremal Processes ◽  
Necessary And Sufficient

Necessary and sufficient conditions are given for a distribution function in ℝ2 to be max-infinitely divisible. The d.f. F is max i.d. if F t is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.

scholarly journals Two-faced families of non-commutative random variables having bi-free infinitely divisible distributions

2016 ◽  
Vol 27 (04) ◽  
pp. 1650037 ◽  
Author(s):  
Mingchu Gao
Keyword(s):  
Limit Theorem ◽  
Fock Space ◽  
Random Variables ◽  
Infinite Divisibility ◽  
Triangular Array ◽  
Divisible Distribution ◽  
Infinitely Divisible Distributions ◽  
Space Operator ◽  
Infinitely Divisible ◽  
Operator Model

We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. Using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.

scholarly journals On characterizations and infinite divisibility of recently introduced distributions

2016 ◽  
Vol 53 (4) ◽  
pp. 467-511 ◽  
Author(s):  
G. G. Hamedani
Keyword(s):  
Weak Convergence ◽  
Order Statistic ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Simple Relationship ◽  
Infinitely Divisible

We present here characterizations of the most recently introduced continuous univariate distributions based on: (i) a simple relationship between two truncated moments; (ii) truncated moments of certain functions of the 1th order statistic; (iii) truncated moments of certain functions of the nth order statistic; (iv) truncated moment of certain function of the random variable. We like to mention that the characterization (i) which is expressed in terms of the ratio of truncated moments is stable in the sense of weak convergence. We will also point out that some of these distributions are infinitely divisible via Bondesson’s 1979 classifications.

Random compact convex sets which are infinitely divisible with respect to Minkowski addition

1979 ◽  
Vol 11 (4) ◽  
pp. 834-850 ◽  
Author(s):  
Shigeru Mase
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Infinite Divisibility ◽  
Spectral Measures ◽  
Infinitely Divisible ◽  
Minkowski Addition ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Canonical Representations ◽  
Laplace Transformations

Random closed sets (in Matheron's sense) which are a.s. compact convex and contain the origin are considered. The totality of such random closed sets are closed under the Minkowski addition and we can define the concept of infinite divisibility with respect to Minkowski addition of random compact convex sets. Using a generalized notion of Laplace transformations we get Lévy-type canonical representations of infinitely divisible random compact convex sets. Isotropic and stable cases are also considered. Finally we get several mean formulas of Minkowski functionals of infinitely divisible random compact convex sets in terms of their Lévy spectral measures.

scholarly journals A Small Remark on Hilbert’s Finitist View of Divisibility and Kanovich-Okada-Scedrov’s Logical Analysis of Real-Time Systems

2020 ◽  
pp. 39-47
Author(s):  
Mitsuhiro Okada
Keyword(s):  
Real Time ◽  
Infinite Divisibility ◽  
Early History ◽  
Logical Analysis ◽  
Real Time Systems ◽  
Infinitely Divisible ◽  
Real Time System ◽  
Natural Event ◽  
History Of ◽  
Time Systems

Abstract Hilbert remarked in the introductory part of his most famous finitism address (1925  [1]) that “[t]he infinite divisibility of a continuum is an operation that is present only in our thought”, which means that no natural event or matter is infinitely divisible in reality. We recall that Scedrov’s group including the author started logical analysis of real time systems with the principle similar to Hilbert’s no-infinite divisibility claim, in  [2]. The author would like to note some early history of the group’s work on logical analysis of real time system as well as some remark related to Hilbert’s claim of no-infinite divisibility.

scholarly journals ETHICS, SOCIAL RESPONSIBILITY AND CORRUPTION AS RISK FACTORS

10.26458/1711 ◽  
2017 ◽  
Vol 17 (1) ◽  
pp. 17
Author(s):  
Alexandru GRIBINCEA
Keyword(s):  
Risk Factors ◽  
Social Responsibility ◽  
Financial Risk ◽  
Financial Structure ◽  
Insurance Company ◽  
Financial Return ◽  
Net Profit ◽  
The Cost

 The financial risk characterises the variability of net profit, subject to the financial structure of the insurance. The capital of the insurance company has two elements (the equity and the borrowed one) that differ fundamentally in the cost they generate. If the company uses loans, it will bear systematically the related financial expenses, too. Through its size and cost, indebtedness leads to the variation and changes the size of financial risk. Resorting to the debt is justified through the high remuneration of equity in relation to borrowed capital, thus increasing the financial return.  

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