Mixing in Stochastic Processes
Let be a sequence of random variables. Such a sequence is naturally endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a sigma algebra, the Borel sigma algebra; it is the smallest (coarsest) sigma algebra that contains the topology.
Define a function, called the strong mixing coefficient, as
In this definition, P is the probability measure on the sigma algebra. The symbol, with denotes a subalgebra of the sigma algebra; it is the set of cylinder sets that are specified between times a and b. Given specific, fixed values, etc., of the random variable, at times, etc., then it may be thought of as the sigma-algebra generated by
The process is strong mixing if as .
One way to describe this is that strong mixing implies that for any two possible states of the system (realizations of the random variable), when given a sufficient amount of time between the two states, the occurrence of the states is independent.
Read more about this topic: Mixing (mathematics)
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