This book, addressed to researchers and students interested in
interacting particle systems, is a study on the application of a
law of large numbers and stability criteria to understand the
asymptotic behavior of particle systems. The aim is to investigate
the limiting behavior of a stochastic interacting particle system
both as the size of the population N grows to infinity, being the
time t fixed, and t grows to infinity, being N fixed. The limiting
behavior as the size N grows to infinity is achieved as a law of
large numbers for the empirical process associated with the
interacting particle system, while the long time behavior is
characterized in terms of the convergence of the particle
distribution to an invariant distribution. By applying the same
criterion for the convergence to the invariant measure to the
continuum time version of the Minority Game, an upper bound for the
asymptotic behavior of the waiting time for reaching the stationary
state is obtained.
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