Given an integration problem $I=\int{f(x)dx}$, we can construct an ordinary Monte Carlo estimator as

$E[I]=\sum\limits_i\frac{f(x_i)}{p(x_i)}$

where the samples $x_i$ are usually i.i.d. and drawn from the distribution $p$.

Is it possible to use a short autocorrelated sequence $y_j$ inside an i.i.d. sequence $x_i$? For example, we could sample $x_i$ from $p$; and then generate multiple samples $\{y^i_0..y^i_j\}$ from some conditional distribution $q(y|x_i)$ with $j\ll i$. If we then construct a Monte Carlo estimator as

$E'[I]=\sum\limits_{i,j}\frac{f(y_j)}{q(y_j|x_i)p(x_i)}$

with joint probability $q(y_j|x_i)p(x_i)$, would it converge to the correct value in this case? Should I additionally account for some correlation/normalization terms in it?