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Topic: Independent Processes vs. Independent Random Variables
Replies: 1   Last Post: Dec 31, 2013 8:19 AM

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Roland Franzius

Posts: 376
Registered: 12/7/04
Re: Independent Processes vs. Independent Random Variables
Posted: Dec 31, 2013 8:19 AM
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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





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