I am a computer scientist interested in analyzing stochastic processes specified as probabilistic programs. In my research, I recently encountered an idea that looks just like a martingale, but is somehow more general. I wonder if these have been already studied and if so, can someone provide me pointers.
Let $X_n$ be a vector-valued random variable. Our "martingale-like" condition looks like this:
$\mathbb{E}( X_n \ |\ X_{n-1},\ldots,X_0 ) = A X_{n-1}$
Here $A$ is a $n \times n$ matrix. If $A$ were invertible, then $(A^{-1})^n X_n$ is a martingale. But in general all we can say about $A$ is that it is a "non-negative" matrix (all entries are non-negative).
Thanks,
Sriram