# joint PDF of two stochastic processes. Is factorability linked to independence?

I am processing the timeseries of two stochastic processes $X$ and $Y$. Based on other considerations, let us assume that they are stationary.

I estimate the joint probability density function $P_{X,Y}(x,y)$ of the two processes with a histogram, and observe that it is to a good approximation factorable, i.e. I can write

$P_{X,Y}(x,y)\approx P_X(x)\,P_Y(y)$

where $P_X$ is the marginal probability density function of the process $X$.

Now, if $\{X,Y\}$ were random variables, I could conclude that the two variables are independent (link). I am dealing instead with two stochastic processes.

I have two questions:

• Can I say something about the independence of the two stochastic processes $X,Y$ if their joint PDF is factorable? If not, can I introduce a weaker definition of independence that they respect?
• are there other time-series processing techniques to test for the independence of the two timeseries?
• 1) What is pdf (or joint pdf) for stochastic processes? Do you mean marginal distributions? 2) This may be an answer to your question. – zhoraster Sep 25 '17 at 13:45
• @zhoraster, I think that this is essentially the same question as math.stackexchange.com/questions/2614944/… – user512365 Jan 21 '18 at 18:51