Inner product estimator - random variable

Question

I'm curently working on the functional space $L^2(\mathbb{R}^n,B(\mathbb{R}^n),\mathbb{P}_X)$ where $\mathbb{P}_X$ is a probability measure.

If I generate randomly $N$ realizations of $x_i$ following $P_X$, it is reasonable to assume $$\int_{\mathbb{R}^n} f(x)g(x)p_X(x)dx = \frac{1}{N}\sum_{i=1}^N f(x_i)g(x_i)$$ where $p_X$ is the probability density ?

I'm not used to probabilistic and Monte-Carlo estimators...

Thanks.

Answer 1

This is the principle of Monte-Carlo methods.

You have $$\int_{\mathbb{R}^n} f(x)g(x)p_X(x)dx=E\left\lbrack f(X)g(X)\right\rbrack$$

If your realizations are i.i.d., by the law of large numbers, $\frac{1}{N}\sum_{i=1}^N f(x_i)g(x_i)$ converges almost surely to this expected value. Therefore this a way to approximate your integral. This is basic Monte-Carlo but it can be refined in many ways.