Cm Math Evidence

Let $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$ be a probability space (Probability measures and probability spaces), and fix integers $m,d\ge 1$.

Measurement model. A measurement model with input dimension $m$ and output dimension $d$ consists of

1. an open set $U\subseteq {\mathbb{R}}^m$, the input domain, and a map $f:U\to {\mathbb{R}}^d$, the model function, which is Borel measurable as a map of measurable spaces $f:(U,\mathcal{B}(U))⟶({\mathbb{R}}^d,\mathcal{B}({\mathbb{R}}^d))$ (A measurable function between measurable spaces, The Borel sigma-algebra of a topological space);
2. a random element $X:\mathrm{\Omega }\to {\mathbb{R}}^m$, the input vector, whose values lie in $U$, that is, $X(\omega )\in U$ for every $\omega \in \mathrm{\Omega }$ (Random elements and real random variables, with target $({\mathbb{R}}^m,\mathcal{B}({\mathbb{R}}^m))$); write $X_U:\mathrm{\Omega }\to U$, $X_U(\omega ):=X(\omega )$, for its corestriction;
3. the output quantity $Y:=f\circ X_U:\mathrm{\Omega }\to {\mathbb{R}}^d$.

The corestriction $X_U$ is measurable for the relative Borel sigma-algebra $\mathcal{B}(U)$: it is generated by the relative open sets $U\cap O$ with $O\subseteq {\mathbb{R}}^m$ open, so every $A\in \mathcal{B}(U)$ has the form $A=U\cap B$ for some $B\in \mathcal{B}({\mathbb{R}}^m)$, and $X_U^{-1}(A)=X^{-1}(B)$ because $X(\omega )\in U$. For every Borel $C\subseteq {\mathbb{R}}^d$, measurability of $f$ then gives $Y^{-1}(C)=X_U^{-1}(f^{-1}(C))\in \mathfrak{F},$ so $Y$ is a random element.

A point $\mu \in U$ is a nominal input of the model. The model function is continuous, of class $C^k$, or totally differentiable at $\mu$ when $f$ has the corresponding property as a map on the open set $U$ ($C^k$ Euclidean maps and diffeomorphisms, The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder); such a regularity hypothesis must be stated whenever a derivative of $f$ is used. The linearization of $f$ at $\mu$ is the affine map $T_{\mu }(x):=f(\mu )+Df(\mu )(x-\mu )$ when $Df(\mu )$ exists.

Additive-error observation model. Let $\mathrm{\Theta }$ be a set, the parameter set; let $Z$ be a set, the apparatus and preparation settings; let $V\subseteq {\mathbb{R}}^d$ be an open set with $0\in V$, the error domain; let $h:\mathrm{\Theta }\times Z\to {\mathbb{R}}^d$ be a map, the prediction map; and let $\epsilon :\mathrm{\Omega }\to {\mathbb{R}}^d$ be a random element with $\epsilon (\omega )\in V$ for every $\omega$, the error. For each $(\theta ,z)\in \mathrm{\Theta }\times Z$ the observation is the random element $Y_{\theta ,z}:=h(\theta ,z)+\epsilon :\mathrm{\Omega }⟶{\mathbb{R}}^d\mathrm{.}$ Well-definedness: for fixed $\theta$ and $z$ the map ${\tau }_{\theta ,z}(x):=h(\theta ,z)+x$ is a translate of ${\mathbb{R}}^d$, hence continuous, so ${\tau }_{\theta ,z}^{-1}(B)$ is Borel for every Borel $B\subseteq {\mathbb{R}}^d$ (A continuous map has Borel preimages of Borel sets); therefore $Y_{\theta ,z}^{-1}(B)={\epsilon }^{-1}({\tau }_{\theta ,z}^{-1}(B))\in \mathfrak{F}$ by the measurability of $\epsilon$, so $Y_{\theta ,z}$ is a random element in the sense of Random elements and real random variables.

Conventions, hypotheses and scope

The following are additional hypotheses and are not part of either definition: that the error has mean zero, $E[\epsilon ]=0$; that $\epsilon$ is square-integrable, $E\mathrm{\mid }\epsilon {\mathrm{\mid }}^2<+\mathrm{\infty }$; that the error is bounded; that a stated probability law for $\epsilon$ is specified; or that the input vector is of the additive form $X=\mu +\epsilon$ with a deterministic nominal input $\mu$. Any of these must be stated, with its exact quantifiers, at each use.

When a model function is merely continuous, continuity and A continuous map has Borel preimages of Borel sets show that its preimages of Borel sets are Borel, so clause 1 holds automatically for such $f$; the explicit measurability in clause 1 keeps the definition meaningful for discontinuous model functions as well, where required.

The model is an input–output relation between random quantities and their images. It contains no unit assignment, no physical identification of $\mathrm{\Theta }$, $Z$ or $U$ with measured quantities, no sampling protocol and no distributional assumption beyond the stated probability space. A physical consumer must supply the units of the coordinates, the preparation and apparatus protocol, and the empirical status of every distributional or moment hypothesis that it invokes.

Let $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$ be a probability space and let $p,d\ge 1$ be integers. A calibration family is a family $(h_c{)}_{c\in C},h_c:\mathrm{\Theta }⟶{\mathbb{R}}^d,$ of prediction maps on a nonempty parameter set $\mathrm{\Theta }\subseteq {\mathbb{R}}^p$, in fixed numerical coordinates, indexed by a nonempty set $C$ of apparatus settings (Measurement models and additive-error observation models).

A calibration datum for this family consists of

1. a finite family $({\theta }^{(r)}{)}_{1\le r\le n}$, $n\ge 1$, of reference inputs, either fixed values ${\theta }^{(r)}\in \mathrm{\Theta }$ or maps ${\theta }^{(r)}:\mathrm{\Omega }\to \mathrm{\Theta }$, according to the branch specified in clause 4, representing values supplied by a reference standard under stated conditions;
2. a set $Y\subseteq ({\mathbb{R}}^d{)}^n$ of admissible reference observations and the observed vector $y=(y_1,\dots ,y_n)\in Y$;
3. a calibration rule, that is, a map $\mathrm{\Gamma }:D⟶C,(y,{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}})⟼c,D\subseteq Y\times {\mathrm{\Theta }}^n;$ the nominal reference tuple ${\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}=({\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(r)}{)}_{r=1}^n$ belongs to ${\mathrm{\Theta }}^n$ and $(y,{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}})\in D$;
4. either, in the fixed-value branch, a stated bound $\mathrm{\mid }{\theta }^{(r)}-{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(r)}\mathrm{\mid }\le {\delta }_r$ with ${\delta }_r\ge 0$ for each $r$, or, in the random-input branch, a Borel random element $\eta :\mathrm{\Omega }\to ({\mathbb{R}}^p{)}^n$ of deviations with a stated joint law (Random elements and real random variables), where ${\eta }_r:\mathrm{\Omega }\to {\mathbb{R}}^p$ is its $r$th component and ${\theta }^{(r)}(\omega ):={\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(r)}+{\eta }_r(\omega )\in \mathrm{\Theta }$ for every $r$ and every $\omega \in \mathrm{\Omega }$. The bound uses the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ on the fixed coordinates; deviations need not themselves belong to $\mathrm{\Theta }$.

Given the datum, the calibrated setting is $c:=\mathrm{\Gamma }(y,{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}})$ and the calibrated prediction map is $h_c$.

Well-definedness and conventions

The rule $\mathrm{\Gamma }$ and the sets $C$, $\mathrm{\Theta }$, $Y$ are data: nothing in the definition requires $\mathrm{\Gamma }$ to be injective, continuous, measurable, or to return an apparatus setting for which the calibrated prediction is accurate. The reference inputs are values of a physical standard; their uncertainty is recorded in clause 4 and must be carried into any quantitative conclusion about $h_c$ by the argument that uses it. The definition states the structure of a calibration only; properties such as consistency, unbiasedness or attainment of a stated accuracy require separate hypotheses and proofs, and no such property is asserted here.

Reference inputs are explicit arguments of the rule: a fixed-reference rule ${\mathrm{\Gamma }}_{{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}}(y)$ is only a slice of $\mathrm{\Gamma }$. This mathematical definition attaches no units to the coordinates. A physical consumer must specify one unit and, when a norm compares different coordinates, one positive reference scale per coordinate; subtraction and addition then operate on numerical coordinates in those fixed units.

Let $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$ be a probability space (Probability measures and probability spaces) and let $n\ge 1$ be a natural number. A repeated-observation sampling model for a real quantity consists of random variables $Y_0,\dots ,Y_{n-1}:\mathrm{\Omega }⟶\mathbb{R}$ (Random elements and real random variables) together with the following model hypotheses, each of which is adopted as part of the model and is not a consequence of the data:

1. (IID hypothesis) the family $(Y_k{)}_{k<n}$ is independent and identically distributed in the sense of Identical distribution and IID families; that is, the whole family is mutually independent and the laws of the $Y_k$ all agree;
2. (second-moment hypothesis) $E\left[Y_0^2\right]<+\mathrm{\infty }$.

Under hypotheses 1 and 2 the numbers $\mu :=E[Y_0]\in \mathbb{R},{\sigma }^2:=\mathrm{Var}(Y_0)\in [0,+\mathrm{\infty })$ are finite by Moments, variance, and covariance on a probability space, and hypothesis 1 makes them the common mean and variance of the family; in particular $E[Y_k]=\mu$ and $\mathrm{Var}(Y_k)={\sigma }^2$ for every $k<n$. The sample mean of the model is the random variable $\bar{Y}:=\frac{1}{n}{\sum }_{k=0}^{n-1}{Y}_k\mathrm{.}$

Conventions and scope

The data of the model are the probability space, the variables $Y_k$, and the adopted hypotheses; hypotheses 1 and 2 can fail for the same family of variables on the same probability space, so they must be stated, not inferred from the notation. The definition assumes no particular law for the $Y_k$: it neither assumes a normal law, nor a finite fourth moment, nor a continuous distribution. Hypotheses of that kind, and any conclusion needing them, must be stated separately at their point of use.

A sampling model is a mathematical description of repeated observation. It does not assert that any physical sequence of measurements satisfies the IID and second-moment hypotheses, and it attaches no value to $\mu$ or ${\sigma }^2$ beyond the one supplied by the model.

For a finite repeated-observation model of Repeated-observation sampling models with IID and finite-variance hypotheses with $n\ge 1$, write its variables as $Y_1,\dots ,Y_n$, where $Y_i$ here denotes the supplier's $Y_{i-1}$ for $1\le i\le n$. They are IID square-integrable real random variables on one probability space, with common mean $\mu$ and variance ${\sigma }^2$. Put $\bar{Y}=n^{-1}{\sum }_{k=1}^n{Y}_k$. Then $E[\bar{Y}]=\mu ,E\mathrm{\mid }\bar{Y}-\mu {\mathrm{\mid }}^2=\frac{{\sigma }^2}{n},\mathbb{P}(\mathrm{\mid }\bar{Y}-\mu \mathrm{\mid }\ge a)\le \frac{{\sigma }^2}{n{a}^2}(a>0)\mathrm{.}$ No extension to an infinite IID sequence is required; no convergence assertion is made for a single finite family.

Let $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$ be a probability space (Probability measures and probability spaces), let $m\ge 1$, and let $X:\mathrm{\Omega }\to {\mathbb{R}}^m$ be a random element with target $({\mathbb{R}}^m,\mathcal{B}({\mathbb{R}}^m))$ (Random elements and real random variables). Use the statistical one-based coordinate labels: for every $v\in {\mathbb{R}}^m$ write $v_i:=v(i-1)$ for $1\le i\le m$, so $X_i(\omega )=X(\omega )(i-1)$. Each $X_i$ is a real random variable because the coordinate projection $x\mapsto x(i-1)$ is continuous and therefore has Borel preimages of Borel sets (A continuous map has Borel preimages of Borel sets). For every integer $r\ge 1$ and matrix $A\in M_{r\times m}(\mathbb{R})$, the notation $A_{ij}$ below means its zero-based matrix entry $A(i-1,j-1)$ for $1\le i\le r$ and $1\le j\le m$; likewise ${\mathrm{\Sigma }}_{ij}$ means the entry at $(i-1,j-1)$ in the covariance matrix below in the matrix space of The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations. These are labels for the root library's zero-based function entries, not a different matrix convention. If $U\subseteq {\mathbb{R}}^m$ is open, $h:U\to {\mathbb{R}}^r$ has all first partials at $a\in U$, and $A=Jh(a)$, then $A_{ij}$ in this notation is ${\mathrm{\partial }}_{j-1}{h}_{i-1}(a)$, with the zero-based Jacobian convention of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. Assume $X$ is square-integrable: $E\left[\mathrm{\mid }X{\mathrm{\mid }}^2\right]={\sum }_{i=1}^{m}E\left[X_i^2\right]<+\mathrm{\infty },$ where $\mathrm{\mid }\cdot \mathrm{\mid }$ is the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Its mean vector is $\mu =E[X]:=(E[X_1],\dots ,E[X_m])\in {\mathbb{R}}^m,$ the coordinates being the expectations of Moments, variance, and covariance on a probability space. Its covariance matrix is the $m\times m$ real matrix of The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations $\mathrm{\Sigma }=\mathrm{Cov}(X):=({\mathrm{\Sigma }}_{ij}{)}_{i,j=1}^m,{\mathrm{\Sigma }}_{ij}:=E[(X_i-{\mu }_i)(X_j-{\mu }_j)]\in \mathbb{R}\mathrm{.}$ For the random vector $X$ the entries ${\mathrm{\Sigma }}_{ij}$ are also written $\mathrm{Cov}(X_i,X_j)$, and ${\mathrm{\Sigma }}_{ii}=\mathrm{Var}(X_i)$.

Well-definedness and elementary properties

Integrability of the entries. Square-integrability gives $E[X_i^2]<+\mathrm{\infty }$ for each $i$. Since the underlying measure is a probability measure, the elementary inequality $\mathrm{\mid }{X}_i\mathrm{\mid }\le \frac{1}{2}(1+X_i^2)$ gives $E\mathrm{\mid }{X}_i\mathrm{\mid }\le \frac{1}{2}(1+E[X_i^2])<+\mathrm{\infty }$ by monotonicity and homogeneity of the nonnegative integral (Monotonicity and nonnegative homogeneity of the nonnegative integral); hence $X_i$ is integrable and ${\mu }_i=E[X_i]$ is finite (Moments, variance, and covariance on a probability space). Also $(X_i-{\mu }_i{)}^2\le 2X_i^2+2{\mu }_i^2,$ so $X_i-{\mu }_i$ is square-integrable; the probability-space inequality $\mathrm{\mid }{X}_i-{\mu }_i\mathrm{\mid }\le \frac{1}{2}(1+(X_i-{\mu }_i{)}^2)$ then makes it integrable as well. Finally $\mathrm{\mid }(X_i-{\mu }_i)(X_j-{\mu }_j)\mathrm{\mid }\le \frac{1}{2}((X_i-{\mu }_i{)}^2+(X_j-{\mu }_j{)}^2),$ and the right-hand side is integrable by additivity, monotonicity and homogeneity of the nonnegative integral (Additivity of the nonnegative Lebesgue integral, Monotonicity and nonnegative homogeneity of the nonnegative integral); hence every ${\mathrm{\Sigma }}_{ij}$ is a finite real number.

Symmetry. For all $i,j$ the real numbers $(X_i-{\mu }_i)(X_j-{\mu }_j)$ and $(X_j-{\mu }_j)(X_i-{\mu }_i)$ are equal pointwise, so ${\mathrm{\Sigma }}_{ij}={\mathrm{\Sigma }}_{ji}$; thus ${\mathrm{\Sigma }}^{\mathsf{T}}=\mathrm{\Sigma }$.

For $M\in M_{m\times m}(\mathbb{R})$ and $z\in {\mathbb{R}}^m$, write $z^{\mathsf{T}}Mz:={\sum }_{i=1}^m{\sum }_{j=1}^m{z}_i{M}_{ij}{z}_j\mathrm{.}$ For a real symmetric $m\times m$ matrix $M$, positive semidefinite means $z^{\mathsf{T}}Mz\ge 0$ for every $z\in {\mathbb{R}}^m$, and positive definite means $z^{\mathsf{T}}Mz>0$ for every nonzero $z\in {\mathbb{R}}^m$. For a real random variable $Z$, almost surely constant means that $Z=c$ outside a measurable $\mathbb{P}$-null set for some $c\in \mathbb{R}$ (Measure-null sets and almost-everywhere statements relative to a measure).

Positive semidefiniteness. Let $z=(z_1,\dots ,z_m)\in {\mathbb{R}}^m$. The random variable $W:={\sum }_{i=1}^m{z}_i{X}_i$ is a finite linear combination of square-integrable random variables, hence square-integrable, and bilinearity of covariance (Variance and covariance identities for random variables) gives $z^{\mathsf{T}}\mathrm{\Sigma }z={\sum }_{i,j=1}^m{z}_i{z}_j\mathrm{Cov}(X_i,X_j)=\mathrm{Cov}(W,W)=\mathrm{Var}(W)=E[(W-E[W]{)}^2]\ge 0,$ the inequality holding because $(W-E[W]{)}^2\ge 0$ pointwise and expectation is monotone (Linearity, monotonicity, and the modulus bound for expectation). If $z^{\mathsf{T}}\mathrm{\Sigma }z=\mathrm{Var}(W)=0$, Chebyshev's inequality gives $\mathbb{P}(\mathrm{\mid }W-E[W]\mathrm{\mid }\ge 1\mathrm{/}r)=0$ for every integer $r\ge 1$; the union of these events is $\{W\mathrm{\ne }E[W]\}$, so $W=E[W]$ almost surely (Chebyshev's inequality for random variables). Conversely, if $W=c$ almost surely, invariance of the integral under almost-everywhere equality gives $E[W]=c$ and $E[(W-E[W]{)}^2]=0$ (Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree). Hence $z^{\mathsf{T}}\mathrm{\Sigma }z=0$ exactly when $W$ is almost surely constant, and $\mathrm{\Sigma }$ is positive definite exactly when for every $z\mathrm{\ne }0$ the random variable ${\sum }_{i=1}^m{z}_i{X}_i$ fails to be almost surely constant.

No independence hypothesis is used anywhere in this definition; $\mathrm{\Sigma }$ records the pairwise covariances of the coordinates.

Let $m,d\ge 1$, let $U\subseteq {\mathbb{R}}^m$ be open, let $\mu \in U$, and let $h:U\to {\mathbb{R}}^d$ be Borel measurable and totally differentiable at $\mu$ with total derivative $A:=Dh(\mu ):{\mathbb{R}}^m\to {\mathbb{R}}^d$ (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, A measurable function between measurable spaces); for instance this holds when $h$ is $C^1$ on $U$, since continuous maps have Borel preimages of Borel sets (A continuous map has Borel preimages of Borel sets). Let $X:\mathrm{\Omega }\to {\mathbb{R}}^m$ be a random element with values in $U$, square-integrable with mean vector $E[X]=\mu$ and covariance matrix $\mathrm{\Sigma }$ (Mean vector and covariance matrix of a square-integrable random vector), and define the linearization $T:U⟶{\mathbb{R}}^d,T(x):=h(\mu )+A(x-\mu )\mathrm{.}$ Then $T\circ X$ is a random element of ${\mathbb{R}}^d$, its mean vector is $E[T(X)]=h(\mu ),$ and its covariance matrix is $\mathrm{Cov}(T(X))=A\mathrm{\Sigma }{A}^{\mathsf{T}},$ where, in the one-based statistical coordinate notation of Mean vector and covariance matrix of a square-integrable random vector, $A_{ij}:={\mathrm{\partial }}_{j-1}{h}_{i-1}(\mu ),1\le i\le d,1\le j\le m,$ and $A\mathrm{\Sigma }{A}^{\mathsf{T}}$ is the matrix product of Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication in the matrix space of The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations.

The formula is exact for the affine map $T$; it is a statement about $T\circ X$, not about $h\circ X$, unless a remainder hypothesis controls the difference (Mean, mean-square and covariance bounds for second-order linearization error).

Let $m\ge 1$, let $U\subseteq {\mathbb{R}}^m$ be open and convex (A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points), let $h:U\to \mathbb{R}$ be of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms); for $x\in U$, $H_h(x)$ is its Hessian (The Hessian matrix and critical points of a scalar field). For $M=(M_{ij})\in M_{m\times m}(\mathbb{R})$ (The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations), define $\widehat{M}:{\mathbb{R}}^m\to {\mathbb{R}}^m$ by $(\widehat{M}v{)}_i:={\sum }_{j=0}^{m-1}{M}_{ij}{v}_j,0\le i<m;$ for all $u,v\in {\mathbb{R}}^m$ and $\alpha ,\beta \in \mathbb{R}$, termwise distributivity gives $\widehat{M}(\alpha u+\beta v)=\alpha \widehat{M}u+\beta \widehat{M}v$, so $\widehat{M}$ is linear in the sense of A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates; write $Mv:=\widehat{M}v$. For $v\in {\mathbb{R}}^m$, the quadratic-form notation $v^{\mathsf{T}}Mv$ means $⟨v,Mv⟩$ in this Euclidean inner product. Define $\parallel M\parallel :=\parallel \widehat{M}{\parallel }_{\mathrm{o}\mathrm{p}}$, the operator norm of this linear map (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||) using the Euclidean inner product on ${\mathbb{R}}^m$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$); the maximum in this definition exists by The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector. Let $B\in [0,+\mathrm{\infty })$ satisfy $\parallel H_h(x)\parallel \le B\text{for every }x\in U\mathrm{.}$ Then for all $\mu ,y\in U$, $\mid h(y)-h(\mu )-\mathrm{\nabla }h(\mu )\cdot (y-\mu )\mid \le \frac{B}{2}\mid y-\mu {\mid }^2,$ where $\mathrm{\nabla }h(\mu )\cdot (y-\mu )$ is the Euclidean pairing (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

Let $m\ge 1$, let $U\subseteq {\mathbb{R}}^m$ be open and convex, let $h:U\to \mathbb{R}$ be of class $C^2$, and let $B\in [0,+\mathrm{\infty })$ satisfy $\parallel H_h(x)\parallel \le B$ for every $x\in U$ (A uniform Hessian bound controls the second-order Taylor remainder). Let $X:\mathrm{\Omega }\to {\mathbb{R}}^m$ be a random element with $X(\omega )\in U$ for every $\omega$, with $E[X]=\mu \in U$ and $E\mid X-\mu {\mid }^2<+\mathrm{\infty }$, and put $R:=h(X)-h(\mu )-\mathrm{\nabla }h(\mu )\cdot (X-\mu )\mathrm{.}$ Then $R$ is integrable and $\mid E[R]\mid \le E\mid R\mid \le \frac{B}{2}E\mid X-\mu {\mid }^2\mathrm{.}$ If moreover $E\mid X-\mu {\mid }^4<+\mathrm{\infty }$, then $R^2$ is integrable and $E\left[R^2\right]\le \frac{B^2}{4}E\mid X-\mu {\mid }^4\mathrm{.}$

Vector covariance error. Let $f=(f_1,\dots ,f_d):U\to {\mathbb{R}}^d$, $d\ge 1$, have $C^2$ components satisfying $\mathrm{\parallel }{H}_{f_i}(x)\mathrm{\parallel }\le B_i$ on $U$, $B_i\ge 0$. Keep the same input hypotheses and assume $M_4:=E\mathrm{\mid }X-\mu {\mathrm{\mid }}^4<\mathrm{\infty }$. Put $A=Df(\mu )$, $L=f(\mu )+A(X-\mu )$, $S=A\mathrm{Cov}(X)A^{\mathsf{T}}$ and $q_i=B_i^2{M}_4\mathrm{/}4$. Then $f(X)$ is square-integrable and for every $1\le i,j\le d$, $\mathrm{\mid }\mathrm{Cov}(f(X){)}_{ij}-S_{ij}\mathrm{\mid }\le \sqrt{S_{ii}{q}_j}+\sqrt{S_{jj}{q}_i}+\sqrt{q_i{q}_j}\mathrm{.}$ In particular, for the scalar function $h$ above, writing $s^2=\mathrm{Var}(h(\mu )+\mathrm{\nabla }h(\mu )\cdot (X-\mu ))$ and $q=B^2{M}_4\mathrm{/}4$ gives $\mathrm{\mid }\mathrm{Var}(h(X))-s^2\mathrm{\mid }\le 2\sqrt{s^{2}q}+q=B\sqrt{s^2{M}_4}+\frac{B^2}{4}{M}_4\mathrm{.}$ No independence of input coordinates or remainders is assumed. These covariance bounds require the fourth moment; the second-moment mean bound alone does not control a covariance difference.