scholarly journals The empirical mean position of a branching Lévy process

2020 ◽  
Vol 57 (4) ◽  
pp. 1252-1259
Author(s):  
David Cheek ◽  
Seva Shneer
Keyword(s):  
Limit Theorem ◽  
Real Line ◽  
Lévy Process ◽  
Second Order ◽  
Levy Process ◽  
The Real

AbstractWe consider a supercritical branching Lévy process on the real line. Under mild moment assumptions on the number of offspring and their displacements, we prove a second-order limit theorem on the empirical mean position.

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.

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.

Dependent thinning of point processes

1980 ◽  
Vol 17 (04) ◽  
pp. 987-995 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Real Line ◽  
Point Processes ◽  
Compound Poisson Process ◽  
Second Order ◽  
Binary Variables ◽  
Compound Poisson ◽  
The Real ◽  
Markov Sequence

A point process, N, on the real line, is thinned using a k -dependent Markov sequence of binary variables, and is rescaled. Second-order properties of the thinned process are described when k = 1. For general k, convergence to a compound Poisson process is demonstrated.

A central limit theorem for Lp transportation cost on the real line with application to fairness assessment in machine learning

2019 ◽  
Vol 8 (4) ◽  
pp. 817-849
Author(s):  
Eustasio del Barrio ◽  
Paula Gordaliza ◽  
Jean-Michel Loubes
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Real Line ◽  
Asymptotic Variance ◽  
Transportation Cost ◽  
Consistent Estimate ◽  
Data Set ◽  
The Real ◽  
Empirical Distributions

Abstract We provide a central limit theorem for the Monge–Kantorovich distance between two empirical distributions with sizes $n$ and $m$, $\mathcal{W}_p(P_n,Q_m), \ p\geqslant 1,$ for observations on the real line. In the case $p>1$ our assumptions are sharp in terms of moments and smoothness. We prove results dealing with the choice of centring constants. We provide a consistent estimate of the asymptotic variance, which enables to build two sample tests and confidence intervals to certify the similarity between two distributions. These are then used to assess a new criterion of data set fairness in classification.

Peano arithmetic may not be interpretable in the monadic theory of linear orders

1997 ◽  
Vol 62 (3) ◽  
pp. 848-872 ◽  
Author(s):  
Shmuel Lifsches ◽  
Saharon Shelah
Keyword(s):  
Real Line ◽  
Order Theory ◽  
Peano Arithmetic ◽  
Second Order ◽  
Linear Orders ◽  
The Real ◽  
Monadic Theory

AbstractGurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We will show here that it is consistent that the monadic second-order theory of no chain interprets Peano Arithmetic.

Dependent thinning of point processes

1980 ◽  
Vol 17 (4) ◽  
pp. 987-995 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Real Line ◽  
Point Processes ◽  
Compound Poisson Process ◽  
Second Order ◽  
Binary Variables ◽  
Compound Poisson ◽  
The Real ◽  
Markov Sequence

A point process, N, on the real line, is thinned using a k -dependent Markov sequence of binary variables, and is rescaled. Second-order properties of the thinned process are described when k = 1. For general k, convergence to a compound Poisson process is demonstrated.

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