The law of large numbers with exceptional sets

2001 ◽  
Vol 55 (4) ◽  
pp. 431-438
Author(s):  
István Berkes
Keyword(s):  
Law Of Large Numbers ◽  
Exceptional Sets ◽  
Large Numbers ◽  
The Law

scholarly journals THE LAW OF LARGE NUMBERS AND THE LAW OF THE ITERATED LOGARITHM FOR INFINITE DIMENSIONAL INTERACTING DIFFUSION PROCESSES

2003 ◽  
Vol 06 (03) ◽  
pp. 489-503 ◽  
Author(s):  
BYRON SCHMULAND ◽  
WEI SUN
Keyword(s):  
Markov Process ◽  
Large Scale ◽  
Law Of Large Numbers ◽  
Diffusion Processes ◽  
Gibbs Measures ◽  
Exceptional Sets ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law ◽  
Scale Properties

The classical Dirichlet form given by the intrinsic gradient on Γℝd is associated with a Markov process consisting of a countable family of interacting diffusions. By considering each diffusion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure valued diffusion. The quasi-sure analysis of Dirichlet forms is used to find exceptional sets of configurations for this Markov process. We consider large scale properties of the configuration and show that, for quite general measures, the process never hits those unusual configurations that violate the law of large numbers. Furthermore, for certain Gibbs measures, which model random particles in ℝd that interact via a potential function, we show, for d=1, 2, that the process never hits those unusual configurations that violate the law of the iterated logarithm.

scholarly journals A new proof for the generalized law of large numbers under Choquet expectation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jing Chen ◽  
Zengjing Chen
Keyword(s):  
Probability Theory ◽  
Law Of Large Numbers ◽  
Moment Condition ◽  
Large Numbers ◽  
Choquet Expectation ◽  
The Law ◽  
Linear Probability ◽  
First Moment ◽  
Independent And Identically Distributed

Abstract In this article, we employ the elementary inequalities arising from the sub-linearity of Choquet expectation to give a new proof for the generalized law of large numbers under Choquet expectations induced by 2-alternating capacities with mild assumptions. This generalizes the Linderberg–Feller methodology for linear probability theory to Choquet expectation framework and extends the law of large numbers under Choquet expectation from the strong independent and identically distributed (iid) assumptions to the convolutional independence combined with the strengthened first moment condition.

Killing the Law of Large Numbers: Mortality Risk Premiums and the Sharpe Ratio

2006 ◽  
Vol 73 (4) ◽  
pp. 673-686 ◽  
Author(s):  
M. A. Milevsky ◽  
S. D. Promislow ◽  
V. R. Young
Keyword(s):  
Mortality Risk ◽  
Law Of Large Numbers ◽  
Sharpe Ratio ◽  
Risk Premiums ◽  
Large Numbers ◽  
The Law

SYNCHRONIZATION IN RANDOMLY DRIVEN SYSTEMS

1995 ◽  
Vol 09 (16) ◽  
pp. 985-988 ◽  
Author(s):  
A.M. JAYANNAVAR
Keyword(s):  
Simple Model ◽  
Law Of Large Numbers ◽  
Driven Systems ◽  
Large Numbers ◽  
Long Time ◽  
The Law

We have solved analytically a simple model of evolution of particles driven by identical noise. We show that the trajectories of all particles collapse into a single trajectory at long time. This synchronization also leads to violation of the law of large numbers.

Do globally coupled maps really violate the law of large numbers?

1994 ◽  
Vol 72 (11) ◽  
pp. 1644-1646 ◽  
Author(s):  
Arkady S. Pikovsky ◽  
Jürgen Kurths
Keyword(s):  
Law Of Large Numbers ◽  
Large Numbers ◽  
Coupled Maps ◽  
The Law
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