Suppose I have a two sided stationary sequence of random variables $\ldots,X\_{-1},X\_0,X\_1,\ldots$ such that all finite dimensional joint densities $f(x\_1,\ldots,x\_n)$, $n\in\mathbb{N}$ exist. I want to ensure the following:



> 
> Let $A$ and $B$ be events such that $P(\ldots,X\_{-1},X\_0\in A)>0$ and $P(X\_1,X\_2,\ldots\in B)>0$, then
>  $$
> P(\ldots,X\_{-1},X\_0\in A \qquad\text{and}\qquad X\_1,X\_2,\ldots\in B)>0.
> $$
> 
> 
> 


Is it enough to assume that the finite dimensional joint densities all have full support?



Stationary sequence and nonzero probabilities
No, this is not enough. Take $Y\_n$ i.i.d. Gaussians, $r$ an independent Bernoulli taking values in $\{-1,1\}$, and set $X\_n = r + Y\_n$.
Then the events $A = \{X\,:\,\limsup\_{n \to \infty} {1\over n}\sum\_{k=1}^n X\_k = 1\}$
and $B = \{X\,:\,\limsup\_{n \to \infty} {1\over n}\sum\_{k=1}^n X\_{-k} = -1\}$ both have probability ${1\over 2}$, but $A \cap B$ has probability $0$.


**Edit**
In the case when $X$ is a stationary Gaussian sequence, these kind of problems are very well studied. A necessary condition which guarantees that finite-dimensional densities have full support and that the sequence is strongly mixing (and a fortiori ergodic) is that its spectral measure has a density $f$ with respect to Lebesgue measure such that $f$ is almost everywhere strictly positive and $\log f$ is integrable. The property you want to ensure however is equivalent (in the Gaussian case) to the property of the sequence being *off-white* (in the sense of Tsirelson) which holds if and only if $\log f$ belongs to the Sobolev space $H^{1/2}$. This shows that imposing ergodicity or mixing isn't sufficient either. See for example the old books by Dym & McKean and by Ibragimov & Rozanov for statements along these lines.


