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Re: Independent Processes vs. Independent Random Variables
Posted:
Dec 31, 2013 8:19 AM


Am 31.12.2013 12:50, schrieb bahram.ehsandoust@gmail.com: > > Hi all, > > I just noticed that for two stochastic processes, being independent at each time instant doesn't mean that they are two independent processes. > So, how can I check whether they are independent processes or not? What is the technical definition of two independent stochastic processes?
All joint probabilities factorize
Pr[ And_k,l ( X_(t_k) \in dx_ k, Y_(t_l) \in dy_l ) ] =
Pr[ And_k ( X_(t_k) \in dx_ k) ] * Pr[And_l( Y_(t_l) \in dy_l )]
=rho(x_1,..x_n) dx^1 /\ ... dx^n /\
sigma(y_1,..y_m) dy^1 /\ ... dy^m
For processes with exponential, especially gaussian distributions with factorizing transition probabilities its often enough to show that all two point correlation functions are products anologously to showing the independence of gaussian variables by checking the first two moments.

Roland Franzius



