Suppose
$$x_1, x_2,...,x_N$$
are samples from a Markov Chain $P(x_t|x_{t-1})$ with all the nice properties (unique steady state $\pi$ or any other reasonable conditions you need)
Is there a nice way to decompose the product of trajectory with the expectation of individual samples? In other words, can we have some function $f$ such that $$E[\prod_{i=1}^N x_i] = f(E[x_1],...,E[x_N]).$$
The equality can be relaxed to $\leq$ and it would be even better if the function $f$ only needs the steady state expectation $E_{\pi}[x]$ as input (i.e, replace the right size with $f(E_{\pi}[x])$.
I am aware that using Cauchy-Schwarz (CS) inequality can achieve the purpose above but the square and square root in CS inequality is a nightmare for later computation. I wonder if there is a better way achieve this.
Any pointers on the related material and suggestions are greatly appreciated.