IID Monte Carlo integration and error — Examples

Given: AC, $\sigma >0$, and a real IID sequence $(Y_i{)}_{i\ge 1}$ with common law $N(\mu ,{\sigma }^2)$.

The Axiom of Choice: AC is the axiom that every family of nonempty sets has a choice function.

Location-scale normal laws have their stated mean and variance: Under AC, a real random variable with law $N(m,{\sigma }^2)$ for $\sigma \ge 0$ has mean $m$ and variance ${\sigma }^2$.

Unbiasedness and exact variance of the IID Monte Carlo mean: For integrable IID outputs, $\mathbb{E}[{\bar{Y}}_n]=\mu$; with a finite second moment, $\mathrm{Var}({\bar{Y}}_n)={\sigma }^2\mathrm{/}n$.

Consistency of the IID Monte Carlo sample variance: With finite second moment, $s_n^2\to {\sigma }^2$ almost surely, and for ${\sigma }^2>0$, ${\widehat{se}}_n\mathrm{/}(\sigma \mathrm{/}\sqrt{n})\to 1$ almost surely.

Lindeberg–Lévy limit for the IID Monte Carlo mean: Under AC and $0<{\sigma }^2<\mathrm{\infty }$, $\sqrt{n}({\bar{Y}}_n-\mu )\Rightarrow N(0,{\sigma }^2)$.

IID Monte Carlo sample mean, variance, and standard error: ${\widehat{se}}_n=s_n\mathrm{/}\sqrt{n}$ for $n\ge 2$ under the finite-second-moment contract.

Given: Countable Choice, the displayed function $F$, and a real IID sequence $(Y_i{)}_{i\ge 1}$ with the probability law constructed from $F$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): Countable Choice says every countable family of nonempty sets has a choice function.

Continuity and derivatives of positive-base real powers: For rational $\alpha$, $x\mapsto x^{\alpha }$ is continuous and differentiable on $(0,\mathrm{\infty })$ with derivative $\alpha x^{\alpha -1}$.

Monotonicity of $r\mapsto a^r$ and of $a\mapsto a^r$: For positive rational $r$, $a\mapsto a^r$ is strictly increasing on positive reals.

Probability laws correspond to distribution functions: Under Countable Choice, every nondecreasing right-continuous $F$ with limits $0$ and $1$ at the two infinities is the CDF of a unique Borel probability measure on $\mathbb{R}$.

Random elements and real random variables: A real random variable is a measurable map to $\mathbb{R}$; the identity map on a Borel probability space is measurable.

Expectation of a nonnegative or integrable random variable: A nonnegative random variable may have extended expectation $\mathbb{E}[Y]=\int Yd\mathbb{P}$.

Layer-cake formulas for random variables: For nonnegative $Y$, $\mathbb{E}[Y]={\int }_0^{\mathrm{\infty }}\mathbb{P}(Y>t)dt$, with infinity allowed.

For 0 < p < infinity, the layer-cake formula computes the integral of |f|^p from the distribution function: For measurable $Y$ and $p>0$, $\mathbb{E}[\mathrm{\mid }Y{\mathrm{\mid }}^p]=p{\int }_0^{\mathrm{\infty }}{t}^{p-1}\mathbb{P}(\mathrm{\mid }Y\mathrm{\mid }>t)dt$, with infinity allowed.

Truncated integrals of rational powers: For rational $p\mathrm{\ne }1$ and $0<A<B$, ${\int }_A^B{t}^{-p}dt=(B^{1-p}-A^{1-p})\mathrm{/}(1-p)$.

A nonnegative improper Riemann integral on a half-line agrees with the Lebesgue integral: Under Countable Choice, if a nonnegative function is Riemann integrable on compact intervals and its improper Riemann integral is finite, that value equals its Lebesgue integral on the half-line.

A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral: Under Countable Choice, a bounded Riemann-integrable function on a compact interval has the same Lebesgue and Riemann integrals.

Continuous functions on Euclidean spaces are Borel measurable: Under Countable Choice, continuous maps between Euclidean spaces are Borel measurable.

Arithmetic and lattice operations preserve measurability whenever they are defined: Products of measurable real functions are measurable.

Monotone convergence for the integral: Integrals of an increasing sequence of nonnegative measurable functions increase to the integral of their pointwise limit.

Strong and weak consistency of the IID Monte Carlo mean: For a real IID sequence with finite first absolute moment, its Monte Carlo sample mean converges in probability and almost surely to the common mean.

IID Monte Carlo sample mean, variance, and standard error: The empirical $s_n\mathrm{/}\sqrt{n}$ is a statistic for real samples; its interpretation as an estimate of a finite population standard error is under the finite-second-moment condition.

Change of variables for expectation: Expectations of measurable functions of a random variable are integrals against its law, including extended expectations for nonnegative functions.

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: A continuous real function on a compact interval is bounded and Riemann integrable there.