Statistical, resampling, simulation, and numerical error ledger

Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space as in Probability measures and probability spaces, let $\theta \in \mathbb{R}$ be a fixed target, and let $m\in \mathbb{N}\cup \{0\}$. Take $r_1,\dots ,r_m$ and $\tilde{r}$ to be real random variables on this space, meaning measurable maps $(\mathrm{\Omega },\mathcal{F})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ as in Random elements and real random variables. Set $r_0(\omega ):=\theta$ for every $\omega \in \mathrm{\Omega }$. Deterministic values are included by using a one-point probability space. The reference chain $r_0,r_1,\dots ,r_m$ must put every quantity on the same scale and state which ideal analysis stage each transition represents. Define the signed stage errors pointwise by $e_j:=r_j-r_{j-1}(1\le j\le m)$ and the final numerical error $e_{\mathrm{f}\mathrm{p}}:=\tilde{r}-r_m\mathrm{.}$

Each error is measurable because finite sums and differences of measurable real-valued functions are measurable (Arithmetic and lattice operations preserve measurability whenever they are defined).

When the stages are present, $e_1$ may record data-sampling error, a later transition may record a bootstrap or permutation approximation, a discretized target may record numerical-model bias, and a finite simulation may record computation-randomization error. The labels describe the declared adjacent references; they do not by themselves assert that errors are independent, centered, unbiased, or even random. If a stage is absent, omit it or set its two adjacent references equal. If no floating-point computation is involved, set $\tilde{r}=r_m$.

For a probability statement about these errors, the required joint probability law and any conditioning convention must also be specified. For example, fixing the observed data and varying independent randomizations concerns computation error conditional on those data; it does not include variation across new data samples. A deterministic quadrature rule has no replicate-based error estimate unless an actual randomization is added. Any numerical roundoff bound must state its arithmetic and normal-range hypotheses.

Let $\theta =r_0,r_1,\dots ,r_m,\tilde{r}$ be the finite scalar reference chain and returned value of Successive statistical and computational error targets, with $m\ge 0$. Then, pointwise, $\tilde{r}-\theta ={\sum }_{j=1}^m(r_j-r_{j-1})+(\tilde{r}-r_m),$ and consequently $\mathrm{\mid }\tilde{r}-\theta \mathrm{\mid }\le {\sum }_{j=1}^m\mathrm{\mid }{r}_j-r_{j-1}\mathrm{\mid }+\mathrm{\mid }\tilde{r}-r_m\mathrm{\mid }\mathrm{.}$ The sum is empty when $m=0$. No independence, unbiasedness, or distributional assumption is needed.

Let $E_1,\dots ,E_m$ be square-integrable real random variables on one probability space, where $m\ge 1$. Write $b_j=\mathbb{E}{E}_j$. Then $\mathbb{E}\left[{\left({\sum }_{j=1}^m{E}_j\right)}^2\right]={\left({\sum }_{j=1}^m{b}_j\right)}^2+{\sum }_{j=1}^m\mathrm{Var}(E_j)+2{\sum }_{1\le j<k\le m}\mathrm{Cov}(E_j,E_k)\mathrm{.}$ The correction to the sum of component variances is exactly $({\sum }_j{b}_j{)}^2+2{\sum }_{j<k}\mathrm{Cov}(E_j,E_k)$. Thus the mean square equals the sum of component variances exactly when this combined correction is zero. In particular, if the total bias is zero, pairwise independence suffices.

Let $Z_1,\dots ,Z_R$ be IID real random variables with mean $\mu$ and finite variance ${\tau }^2$, where $R\ge 2$, and let ${\bar{Z}}_R=R^{-1}{\sum }_{r=1}^R{Z}_r$. Then ${\widehat{V}}_R:=\frac{1}{R(R-1)}{\sum }_{r=1}^R(Z_r-{\bar{Z}}_R{)}^2$ is unbiased for $\mathrm{Var}({\bar{Z}}_R)={\tau }^2\mathrm{/}R$. The probability law may be the computation-randomization law conditional on fixed observed data; in that use, this estimate concerns computation variability only.

Assume AC. Let $(Z_r{)}_{r\ge 1}$ be IID real computation-replicate outputs with mean $\mu$ and variance $0<{\tau }^2<\mathrm{\infty }$. Then, as $R\to \mathrm{\infty }$, $\frac{\sqrt{R}({\bar{Z}}_R-\mu )}{\tau }\Rightarrow N(0,1),{\bar{Z}}_R:=\frac{1}{R}{\sum }_{r=1}^R{Z}_r\mathrm{.}$ This is a limit for the replicate-randomization law under the stated fixed-target setup; it is not a data-sampling confidence statement and does not by itself justify a finite-$R$ Student interval.