Approximations of small jumps of Lévy processes with a view towards simulation

2001 ◽  
Vol 38 (02) ◽  
pp. 482-493 ◽  
Author(s):  
Søren Asmussen ◽  
Jan Rosiński
Keyword(s):  
Levy Processes ◽  
Error Rates ◽  
Levy Process ◽  
Series Expansions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compound Poisson ◽  
Absolute Value ◽  
In Series ◽  
Necessary And Sufficient

Let X = (X(t):t ≥ 0) be a Lévy process and X ∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ2(∊) = var(X ∊(1)). In simulation, X - X ∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X ∊/σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X ∊/σ(∊) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.

Approximations of small jumps of Lévy processes with a view towards simulation

2001 ◽  
Vol 38 (2) ◽  
pp. 482-493 ◽  
Author(s):  
Søren Asmussen ◽  
Jan Rosiński
Keyword(s):  
Levy Processes ◽  
Error Rates ◽  
Levy Process ◽  
Series Expansions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compound Poisson ◽  
Absolute Value ◽  
In Series ◽  
Necessary And Sufficient

Let X = (X(t):t ≥ 0) be a Lévy process and X∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ2(∊) = var(X∊(1)). In simulation, X - X∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X∊/σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X∊/σ(∊) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.

scholarly journals Spectral Properties of Unitary Cayley Graphs of Finite Commutative Rings

10.37236/2390 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Xiaogang Liu ◽  
Sanming Zhou
Keyword(s):  
Spectral Properties ◽  
Line Graph ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Absolute Value ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Unitary Cayley Graph ◽  
Necessary And Sufficient

Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set $\left\{\{a,b\}:a,b\in R, a-b\in R^\times\right\}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than $\pm r$ is at most $2\sqrt{r-1}$. In this paper we give a necessary and sufficient condition for $G_R$ to be Ramanujan, and a necessary and sufficient condition for the complement of $G_R$ to be Ramanujan. We also determine the energy of the line graph of $G_R$, and compute the spectral moments of $G_R$ and its line graph.

On the arc-sine laws for Lévy processes

1994 ◽  
Vol 31 (01) ◽  
pp. 76-89 ◽  
Author(s):  
R. K. Getoor ◽  
M. J. Sharpe
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Lévy Process ◽  
Elementary Proof ◽  
Levy Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Necessary And Sufficient ◽  
Spitzer's Theorem

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t −1 ⨍0 t P 0(X s > 0) ds → c as t → ∞ is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs >0} ds to converge in P 0 law to Fc. Moreover, P 0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs >0} ds under P 0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

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.

On the arc-sine laws for Lévy processes

1994 ◽  
Vol 31 (1) ◽  
pp. 76-89 ◽  
Author(s):  
R. K. Getoor ◽  
M. J. Sharpe
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Lévy Process ◽  
Elementary Proof ◽  
Levy Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Necessary And Sufficient ◽  
Spitzer's Theorem

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t−1 ⨍0tP0(Xs > 0) ds → c as t → ∞ is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds to converge in P0 law to Fc. Moreover, P0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds under P0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

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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