Ordinary importance sampling has finite variance and a CLT under a weighted second moment

Given: The integrable ordinary importance sampling setup from Ordinary importance sampling is unbiased, plus ${\mathbb{E}}_Q[Y_1^2]<\mathrm{\infty }$. For the CLT conclusion, take an IID sequence $(X_i{)}_{i\ge 1}$ with law $Q$ and assume AC.

Ordinary importance sampling: The estimate is the sample average of the weighted values $Y_i=f(X_i)\tilde{p}(X_i)\mathrm{/}(Z_{p}q(X_i))$, with value $0$ when $q(X_i)=0$; the algorithm also establishes that this piecewise weighted-integrand map is measurable.

Ordinary importance sampling is unbiased: Under its support and integrability hypotheses, ${\mathbb{E}}_Q[{\widehat{\mu }}_{\mathrm{I}\mathrm{S}}]=\theta =\int fd\pi$.

Measurable coordinatewise functions preserve independence: Applying measurable coordinatewise maps to an independent family preserves independence.

Identical distribution and IID families: IID means a common law and mutual independence.

Variance adds for every finite pairwise-independent family: The variance of a finite sum of pairwise-independent square-integrable real variables is the sum of their variances. Variance and covariance identities for random variables: Covariance is bilinear on finite linear combinations, so $\mathrm{Var}(aY)=a^2\mathrm{Var}(Y)$ and $\mathrm{Var}(Y)=\mathbb{E}[Y^2]-(\mathbb{E}Y{)}^2$.

Change of variables for expectation: For a nonnegative measurable function of a random element with law $Q$, expectation equals its integral against $Q$.

Moments, variance, and covariance on a probability space: For square-integrable real $Y$, $\mathrm{Var}(Y)=\mathbb{E}[(Y-\mathbb{E}Y{)}^2]$.

Lindeberg–Lévy limit for the IID Monte Carlo mean: Under AC, a real IID sample with positive finite variance satisfies $\sqrt{n}({\bar{Y}}_n-\mathbb{E}{Y}_1)\Rightarrow N(0,\mathrm{Var}(Y_1))$.

A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere: A nonnegative measurable function has integral zero exactly when it vanishes almost everywhere.

Standard normal and normal laws: $N(m,0)$ is the Dirac law ${\delta }_m$.

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function; it is assumed only for the central limit conclusion.

Integrating against a density agrees with integrating the product: for a nonnegative measurable density $r$ and nonnegative measurable $h$, $\int hd(rd\mu )=\int hrd\mu$, including infinite values.