# Measuring the independence between the components of a stochastic process

In a context of blind source separation (e.g. you want to extract the voice of a singer from a song), many approaches consist in maximizing the independence between the components of a certain decomposition of an input signal. However, the signals are often considered only as random variables, although their temporal structure is often strong. I am interested in reading / learning / developing about an appropriate generalization of this approach where the signal is now modeled as a stochastic process (and even a diffusion process is it makes things simpler). So first I need a manageable definition and measure of the independence between the components of a stochastic process.

For a random variable $X$ on $\mathbb{R}^n$, the Kullback-Leibler divergence between the joint and the product of the marginal distributions of $X$ is often used to measure the degree of independence between the components of $X$. It is written $$I = \int p(X) log \frac{p(X)}{\prod_{a=1}^n p_a(X_a)}dX$$ where $p_a(X_a)$ is a marginal probability function. If $I=0$, the components are completely independent.

How can I generalize it to stochastic processes? On this link http://planetmath.org/IndependentStochasticProcesses.html, is given the definition of the independence between two stochastic processes. I guess this could lead to a formal generalization but I do not manage to write it. Can anyone do it?

Besides, I am afraid it would lead to unpractical computations. So I am thinking that maybe with sufficient strong conditions on the stochastic process (which will not be worse than the usual ergodic random variable approach), one could simplify this expression up to a practical one. For instance, assuming that the process is solution to a stochastic differential equation, I am dreaming of the power of Girsanov theorem to compute the fraction in the above equation...

Do you have the answers to my questions? Do you know any litterature on this topic? Any advice about the direction I should take?

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