scholarly journals A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory

1982 ◽  
Vol 32 (3) ◽  
pp. 412-422 ◽  
Author(s):  
Paul Embrechts ◽  
John Hawkes
Keyword(s):  
Domain Of Attraction ◽  
Divisible Distribution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Stable Law ◽  
Infinitely Divisible Laws ◽  
Ladder Epoch ◽  
Necessary And Sufficient

AbstractSuppose that (pn) is an infinitely divisible distribution on the non-negative integers having Lévy measure (vn). In this paper we derive a necessary and sufficient condition for the existence of the limit limn→∞ pn/vn. We also derive some other results on the asymptotic behaviour of the sequence (Pn) and apply some of our results to the theory of fluctuations of random walks. We obtain a necessary and sufficient condition for the first positive ladder epoch to belong to the domain of attraction of a spectrally positive stable law with index α, α ∈ (1,2).

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.

Convergence of lower records and infinite divisibility

2003 ◽  
Vol 40 (04) ◽  
pp. 865-880 ◽  
Author(s):  
Arup Bose ◽  
Sreela Gangopadhyay ◽  
Anish Sarkar ◽  
Arindam Sengupta
Keyword(s):  
Random Variable ◽  
Infinite Divisibility ◽  
Partial Sums ◽  
Necessary And Sufficient Condition ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
The Laplace Transform ◽  
Infinite Sums ◽  
Necessary And Sufficient ◽  
Lower Records

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

scholarly journals Necessary and sufficient condition for ℳ2-convergence to a Lévy process for billiards with cusps at flat points

2020 ◽  
pp. 2150024
Author(s):  
Paul Jung ◽  
Ian Melbourne ◽  
Françoise Pène ◽  
Paulo Varandas ◽  
Hong-Kun Zhang
Keyword(s):  
Recent Work ◽  
Lévy Process ◽  
Sufficient Conditions ◽  
Levy Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stable Law ◽  
Dispersing Billiards ◽  
Skorohod Topologies ◽  
Necessary And Sufficient

We consider a class of planar dispersing billiards with a cusp at a point of vanishing curvature. Convergence to a stable law and to the corresponding Lévy process in the [Formula: see text] and [Formula: see text] Skorohod topologies has been studied in recent work. Here, we show that certain sufficient conditions for [Formula: see text]-convergence are also necessary.

Convergence of lower records and infinite divisibility

2003 ◽  
Vol 40 (4) ◽  
pp. 865-880 ◽  
Author(s):  
Arup Bose ◽  
Sreela Gangopadhyay ◽  
Anish Sarkar ◽  
Arindam Sengupta
Keyword(s):  
Random Variable ◽  
Infinite Divisibility ◽  
Partial Sums ◽  
Necessary And Sufficient Condition ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
The Laplace Transform ◽  
Infinite Sums ◽  
Necessary And Sufficient ◽  
Lower Records

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

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