Consider a stochastic vector $\mathbf{x}(t)$, that I know is ergodic and stationnary.
Are they conditions on the distribution of $\mathbf{x}(t)$, that allow me to approximate $E\left(\frac{\mathbf{x}(t)}{\mathbf{1}^T\mathbf{x}(t)}\right)\approx$ $\frac{E(\mathbf{x}(t))}{\mathbf{1}^TE(\mathbf{x}(t))}$ ? where $\mathbf{1}^T\mathbf{x}(t)$ is the sum of the elements of $\mathbf{x}(t)$.
The full question, is one of population ecology: I consider some i.i.d. environmnents (say 3), reprenseted by projection matrices $\mathbf{M}_1$,$\mathbf{M}_2$ and $\mathbf{M}_3$ which occur with probabilities $p_1$,$p_2$ and $p_3$ (that sum to 1) and an initial population as vector $\mathbf{n}(0)$.
$\mathbf{n}(t+1)=\mathbf{M}(t)\mathbf{n}(t)$, with $\mathbf{M}(t)$ having the above distribution.
$\mathbf{n}(t)$ is therefore stochastic as is the vector of relative abundances $\mathbf{w}(t)=\frac{\mathbf{n}(t)}{\mathbf{1}^T\mathbf{n}(t)}$
I know that $E(\mathbf{n}(t))=\mathbf{M}^t \mathbf{n}(0)$ with $\mathbf{M}=\sum p_i \mathbf{M}_i$ and i wish to understand how to obtain $E(\mathbf{w}(t))$. So far, I have $\mathbf{w}(t+1)=\frac{\mathbf{n}(t+1)}{\mathbf{1}^T\mathbf{n}(t+1)}=\frac{\mathbf{M}(t)\mathbf{n}(t)}{\mathbf{1}^T\mathbf{M}(t)\mathbf{n}(t)}=\frac{\mathbf{M}(t)\mathbf{w}(t)}{\mathbf{1}^T\mathbf{M}(t)\mathbf{w}(t)}$
$E(\mathbf{w}(t+1))=E_{\mathbf{M(t)}}[E(\mathbf{w}(t+1)|\mathbf{M}(t)=\mathbf{M})] $ and this is where I am blocked.
But if I can write $E(\mathbf{w}(t+1)|\mathbf{M}(t)=\mathbf{M})=E(\frac{\mathbf{M}\mathbf{w}(t)}{\mathbf{1}^T\mathbf{M}\mathbf{w}(t)})\approx\frac{\mathbf{M}E(\mathbf{w}(t))}{\mathbf{1}^T\mathbf{M}E(\mathbf{w}(t))} $ , then I can go on and I get $E(\mathbf{w}(t+1))\approx\sum_i p_i \frac{\mathbf{M}_iE(\mathbf{w}(t))}{\mathbf{1}^T\mathbf{M}_iE(\mathbf{w}(t))}$ and then equaling $E(\mathbf{w}(t+1))$ and $E(\mathbf{w}(t))$ this provides an equation in $E(\mathbf{w}(t))$ that I can solve. I did a few simulations ant it seems to be not too far off, by chance maybe?
Clearly I cannot apply this approximation to $\mathbf{n}(t)$ itself as it would give $E\mathbf{w}(t)=\mathbf{w}$ with the latter being the right eigenvector associated with the dominant eigenvalue of $\mathbf{M}$, which is not what we have: the expectation of $\mathbf{w}(t)$ as t tends toward infinity is not equal to $\mathbf{w}$. But $\mathbf{n}(t)$ is not stationnary.
Any help would be much much appreciated !