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.

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.

Comparison of density topologies on the real line

2018 ◽  
Vol 68 (1) ◽  
pp. 173-180
Author(s):  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
The Real ◽  
Closed Intervals ◽  
The Family ◽  
Necessary And Sufficient

Abstract In this paper will be considered density-like points and density-like topology in the family of Lebesgue measurable subsets of real numbers connected with a sequence 𝓙= {Jn}n∈ℕ of closed intervals tending to zero. The main result concerns necessary and sufficient condition for inclusion between that defined topologies.

scholarly journals The Topological Entropy of Cyclic Permutation Maps and Some Chaotic Properties on Their MPE sets

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Risong Li ◽  
Tianxiu Lu
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Real Line ◽  
Cyclic Permutation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Compact Subinterval ◽  
Dimensional Dynamical System ◽  
Necessary And Sufficient

In this paper, we study some chaotic properties of s-dimensional dynamical system of the form Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1, where ak∈Hk for any k∈1,2,…,s, s≥2 is an integer, and Hk is a compact subinterval of the real line ℝ=−∞,+∞ for any k∈1,2,…,s. Particularly, a necessary and sufficient condition for a cyclic permutation map Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1 to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is obtained. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is explored. Also, we proved that the topological entropy hΨ of such a cyclic permutation map is the same as the topological entropy of each of the following maps: gj∘gj−1∘⋯∘g1l∘gs∘gs−1∘⋯∘gj+1, if j=1,…,s−1and gs∘gs−1∘⋯∘g1, and that Ψ is sensitive if and only if at least one of the coordinates maps of Ψs is sensitive.

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.

The Inverse Eigenvalue Problem for Symmetric Doubly Arrow Matrices

2013 ◽  
Vol 444-445 ◽  
pp. 625-627
Author(s):  
Kan Ming Wang ◽  
Zhi Bing Liu ◽  
Xu Yun Fei
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

In this paper we present a special kind of real symmetric matrices: the real symmetric doubly arrow matrices. That is, matrices which look like two arrow matrices, forward and backward, with heads against each other at the station, . We study a kind of inverse eigenvalue problem and give a necessary and sufficient condition for the existence of such matrices.

On generalized inverses of matrices

1966 ◽  
Vol 62 (4) ◽  
pp. 673-677 ◽  
Author(s):  
M. H. Pearl
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Maximal Rank ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Finite Dimensional ◽  
The Matrix ◽  
Generalized Inverses Of Matrices ◽  
Necessary And Sufficient

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.

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| > 0, E(X) = 0 and τ = inf {n ≥ 1: Sn ≥ 0}. By Wald's equation, E(τ) =∞. If E(X 2) <∞, then by a theorem of Burkholder and Gundy (1970), E(τ 1/2) =∞. In this paper, we prove that if E((X– ) 2 ) <∞, 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 ) <∞, p > 1, is Σn–1–1p E|Sn| <∞.

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.

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| <∞.

Characterizations of r-potent matrices

1984 ◽  
Vol 96 (2) ◽  
pp. 213-222 ◽  
Author(s):  
Joseph P. McCloskey
Keyword(s):  
Quadratic Form ◽  
Random Vector ◽  
Symmetric Matrix ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Real Quadratic

A matrix A is said to be tripotent whenever A3 = A. The study of tripotent matrices is of statistical interest since if the n × 1 real random vector X follows an N(0, I) distribution and A is a symmetric matrix then the real quadratic form X′AX is distributed as the difference of two independently distributed X2 variates if and only if A3 = A. In fact, a necessary and sufficient condition that A is tripotent is that there exist two idempotent matrices B and C such that A = B – C, and BC = 0. Using properties of diagonalizable matrices, we will prove several algebraic characterizations of r-potent matrices that extend the known results for tripotent matrices. Our first result will be to obtain an analogous decomposition for an arbitrary r-potent matrix.

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