Let $\{x_i\}_{i = 1, ...,n}$ , $\{y_i\}_{i = 1, ...,n}$ be sequences generated by a pseudo-random number generator using different seed keys, for example $ x_0$ and $y_0$. Are $\{x_i\}$ and $\{y_i\}$ independent?
If not, under what conditions can I say $\{x_i\}$ and $\{y_i\}$ are independent?
Asymptotically, if it is a good random number generator, then the answer should be yes i.e, $\forall (i\neq j,\epsilon>0) \space \exists N: \{\forall k> N,|P(x_{ik}\cap x_{jk})- P(x_{ik})P(x_{jk})|<\epsilon\}$
Practically, I usually just assume all numbers generated by two different seed sequences are independent, although it would be an interesting study to look at the plot of $(x_i,y_i)$ for different separations in the seed values and see how fast the pattern "washes out". My guess is that it will be extremely rapid, if any pattern presents at all (which it shouldn't otherwise random number generators would be a pain to use).
In general, what you would need would be to test the independence of the two sequences as a time series. This paper presents such a method.