On the moments of ladder epochs for driftless random walks

1994 ◽  
Vol 31 (A) ◽  
pp. 201-205 ◽  
Author(s):  
Yuan S. Chow
Keyword(s):  
Random Walks ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Let X, X1, X2, … be i.i.d. Sn=Σ1nXj, E|X| > 0, E(X) = 0 and τ = inf {n ≥ 1: Sn ≥ 0}. By Wald's equation, E(τ) =∞. If E(X2) <∞, then by a theorem of Burkholder and Gundy (1970), E(τ1/2) =∞. In this paper, we prove that if E((X–)2) <∞, then E(τ1/2) =∞. When X is integer-valued and X ≥ −1 a.s., a necessary and sufficient condition for E(τ1–1/p) <∞, p > 1, is Σn–1–1p E|Sn| <∞.

On the moments of ladder epochs for driftless random walks

1994 ◽  
Vol 31 (A) ◽  
pp. 201-205
Author(s):  
Yuan S. Chow
Keyword(s):  
Random Walks ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Let X, X1, X2, … be i.i.d. Sn =Σ1 n Xj , E|X| &gt; 0, E(X) = 0 and τ = inf {n ≥ 1: Sn ≥ 0}. By Wald's equation, E(τ) =∞. If E(X 2) &lt;∞, then by a theorem of Burkholder and Gundy (1970), E(τ 1/2) =∞. In this paper, we prove that if E((X– ) 2 ) &lt;∞, then E(τ 1/2) =∞. When X is integer-valued and X ≥ −1 a.s., a necessary and sufficient condition for E(τ 1–1/p ) &lt;∞, p &gt; 1, is Σn–1–1p E|Sn| &lt;∞.

Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov

1982 ◽  
Vol 14 (2) ◽  
pp. 272-294 ◽  
Author(s):  
Gérard Letac
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Spherical Symmetry ◽  
Spherical Functions ◽  
Plancherel Measure ◽  
Sufficient Condition ◽  
Gelfand Pairs ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

After an elementary description of Gelfand pairs, spherical functions and Plancherel measure, some explicit computations on the related Markov chains are performed. Random walks on polyhedra belong to this class of Markov chains; two more examples of chains on graphs are worked out, and the necessary and sufficient condition of transcience of random walks on p-adic numbers with spherical symmetry is given as an application of the techniques of the paper.

Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov

1982 ◽  
Vol 14 (02) ◽  
pp. 272-294 ◽  
Author(s):  
Gérard Letac
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Spherical Symmetry ◽  
Spherical Functions ◽  
Plancherel Measure ◽  
Sufficient Condition ◽  
Gelfand Pairs ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

After an elementary description of Gelfand pairs, spherical functions and Plancherel measure, some explicit computations on the related Markov chains are performed. Random walks on polyhedra belong to this class of Markov chains; two more examples of chains on graphs are worked out, and the necessary and sufficient condition of transcience of random walks on p-adic numbers with spherical symmetry is given as an application of the techniques of the paper.

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 &gt; 0) ds → c as t → ∞ is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs &gt;0} ds to converge in P 0 law to Fc. Moreover, P 0(Xt &gt; 0) = c for all t &gt; 0 is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs &gt;0} ds under P 0 to have law Fc for all t &gt; 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.

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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