On the computation of multidimensional integrals by the Monte-Carlo method

It is shown that if $W(x)$ is an arbitrary non-negative function in $R^n$ then the Markov process with the transition density
$$ P(x'\to x)=\int\rho(x'\to x'')\sigma(x''\to x)\,dx'' $$
where $\rho(x'\to x)$ is an arbitraty transition density and
$$ \sigma(x'\to x)=\rho(x\to x')W(x)\Big/\int\rho(x\to x')W(x)\,dx $$
has the asymptotic probability density proportional to $W(x)$.
Using this fact, a method for computation of multidimensional integrals is proposed.