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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.