The statement s(t)=A1s(t-1) implies that s(t) is autoregressive of order 1. That is, it is a Markov process. There is no ground to say that s1 and s2 are independent unless you have assumed that they are independent.

I agree , s1 and s2 if measured then should be coming from sources that are isolated , else you can't say they are .

Also , causal systems for them to be independent , you can't consider or assume past may be the future ...

For example Assume:

A=(0.9 0 0 ; 0 0.5 0; 1 0 -0.5)

Can we have any considration about s1 s2 s3?

Are s1 s2 s3 independent?

No Mohammad , unless you assume that they are by virtue of the process of generation.

Now, if you say these #s are generated by a system that is of a particular type, then you can go ahead and tell they are or test by using covariance .

Or say they are random data (assumption) and will see if I can use distribution to check for closeness...