Monte Carlo Integration on the Real Line

Question

Suppose that we want to compute $$\int_{\mathbb{R}} f(x) dx$$ using monte carlo integration so that we can normalize $f(x)$ and make it a pdf. The examples you typically see involve integrals over a finite domain such as $[a,b]$. In such cases we can sample from the uniform distribution on $[a,b]$. In the case we integrate over $\mathbb{R}$ we need to sample from a pdf whose support is $\mathbb{R}$, say the standard normal pdf which we represent as $p(x)$. We can then compute $$\frac{1}{n} \sum_{i=1}^{n} \frac{f(x_i)}{p(x_i)}$$ as our unbiased estimate of $$\int_{\mathbb{R}}f(x)dx=\int_{\mathbb{R}}\frac{f(x)}{p(x)}p(x)dx=\mathbb{E}\left[\frac{f(x)}{p(x)}\right]$$ so long as $$(1)\;\;\;\;\int_{\mathbb{R}}\frac{f(x)^2}{p(x)}dx < \infty$$ We can also rely on estimates of the variance of the estimator as well as employ the central limit theorem to construct confidence intervals so long as $$(2)\;\;\;\;\int_{\mathbb{R}}\frac{f(x)^4}{p(x)^3}dx < \infty$$ My problem is that I struggle to construct and example where (1) and (2) hold. Can someone give examples of monte carlo integration techniques used to normalize a function over $\mathbb{R}$ which satisfy the necessary assumptions so that the estimate is reliable?

Answer 1

Extreme example: $f$ a standard normal distribution, and $p$ a Cauchy distribution (just remove a constant from $f$ to make it more similar to your context). Then,

$$\frac{f(x)^2}{p(x)} = \dfrac{\pi(1+x^2)}{(2\pi)}\exp(-x^2),$$

and

$$\frac{f(x)^4}{p(x)^3} = \dfrac{\pi^3(1+x^2)^3}{(2\pi)^2}\exp(-2x^2),$$

which are clearly integrable since they are related to the moments of normal distributions, which always exist.