Gaussian Vectors, Processes, Random Distributions, and Wick Structure

A real Gaussian object is one whose every finite family of linear probes has a jointly Gaussian law. For a vector probe $t$, that law has characteristic function $\exp(i t^{\mathsf T}m-\tfrac12 t^{\mathsf T}\Sigma t)$, so its mean and covariance determine the law. For centered variables, every odd moment vanishes and every even moment is a sum over pairings of covariances: the Wick–Isserlis theorem.

The same algebra survives in infinite dimensions, but existence changes its meaning. A mean function together with a positive-semidefinite covariance kernel determines consistent finite-dimensional Gaussian laws, not regular sample paths. A Gaussian random distribution also requires a test-function topology and a theorem that constructs a probability measure on the continuous dual. Continuum formulas therefore begin with smeared variables $\Phi(f)$; point values, coincident products, reflection positivity, and Lorentzian reconstruction require further arguments.

Required background. Characteristic Functions, Moments, Cumulants, and Generating Functionals supplies uniqueness of characteristic functions, justified source differentiation, and the distinction between moments and cumulants; Bilinear and Hermitian Forms, Adjoints, and Isometries supplies positive-semidefinite forms and Gram matrices used for covariance.

Helpful background. Test-Function Spaces, Distributions, Support, and Convergence explains why a distribution is defined by its action on probes rather than by point values.

Gaussian laws and the local setting

All random variables below are real and commuting. Degenerate one-dimensional Gaussians are allowed: $N(\mu,0)$ is the point mass at $\mu$. The continuum test space is the real Schwartz space $\mathcal S_{\mathbb R}(\mathbb R^d)$ with continuous dual $\mathcal S'_{\mathbb R}(\mathbb R^d)$. The QFT-facing calculation is Euclidean, finite dimensional, and normalized; it does not assume that a Lorentzian functional integral is a probability measure.

Gaussian vectors include singular laws

A random vector $X:\Omega\to\mathbb R^N$ is Gaussian when $a^{\mathsf T}X$ is a possibly degenerate one-dimensional Gaussian for every $a\in\mathbb R^N$. Define

$$\begin{aligned} m&=\mathbb E[X],\\ \Sigma &=\mathbb E\!\left[ (X-m)(X-m)^{\mathsf T} \right]. \end{aligned}$$

The covariance matrix is symmetric and positive semidefinite because

$$a^{\mathsf T}\Sigma a =\operatorname{Var}(a^{\mathsf T}X) \geq0.$$

The joint characteristic function is

$$\varphi_X(t) =\mathbb E[e^{i t^{\mathsf T}X}] =\exp\!\left( i t^{\mathsf T}m -\frac12t^{\mathsf T}\Sigma t \right).$$

This proves that $m$ and $\Sigma$ determine a Gaussian law. It also proves the converse existence statement. Given any $m\in\mathbb R^N$ and symmetric $\Sigma\succeq0$, choose a matrix $B$ with $BB^{\mathsf T}=\Sigma$, take a standard Gaussian vector $Z$ of the required rank, and set

$$X=m+BZ.$$

Then $X$ has the displayed characteristic function. No inverse covariance is needed.

When $\Sigma\succ0$, the law has the Lebesgue density

$$p(x) = \frac{ \exp\!\left[ -\tfrac12(x-m)^{\mathsf T} \Sigma^{-1}(x-m) \right] }{ (2\pi)^{N/2}(\det\Sigma)^{1/2} }.$$

This density formula is not the definition. If $\Sigma$ is singular, the law is supported on the affine subspace $m+\operatorname{im}B$ and has no density with respect to $N$-dimensional Lebesgue measure. For example, $X=(Z,Z)^{\mathsf T}$ is Gaussian with

$$\Sigma= \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix},$$

but its support is the diagonal of $\mathbb R^2$.

Gaussianity is the decisive hypothesis. A standard normal variable and a Rademacher variable taking values $\pm1$ with equal probability both have mean zero and variance one, but their fourth moments are respectively $3$ and $1$. Mean and covariance do not determine an arbitrary law.

One useful Gaussian-only consequence concerns independence. If the jointly Gaussian vector $(X,Y)$ has zero cross-covariance, its characteristic function factors into the characteristic functions of $X$ and $Y$, so the two subvectors are independent. Without joint Gaussianity this fails: if $U\sim N(0,1)$, then $U$ and $U^2-1$ have zero covariance but are plainly dependent.

The Wick–Isserlis pairing theorem

Let $Y_1,\ldots,Y_q$ be a centered jointly Gaussian family and write

$$C_{ab}=\mathbb E[Y_aY_b].$$

Let $\mathcal P_2(2n)$ denote the set of partitions of $\{1,\ldots,2n\}$ into unordered pairs.

Theorem (Wick–Isserlis). If $q$ is odd, then

$$\mathbb E[Y_1\cdots Y_q]=0.$$

If $q=2n$ is even, with $n\geq1$, then

$$\mathbb E[Y_1\cdots Y_{2n}] = \sum_{P\in\mathcal P_2(2n)} \prod_{\{a,b\}\in P} C_{ab}.$$

The formula allows a singular covariance. There are

$$|\mathcal P_2(2n)| =\frac{(2n)!}{2^n n!} =(2n-1)!!$$

pairings. McCullagh’s McCullagh 2018, § 3.9.1, pp. 85–86, PDF states the result in set-partition language and traces it to Isserlis.

Full finite-dimensional proof

The moment-generating function of the centered vector is finite for every $t\in\mathbb R^q$ and equals

$$M(t) =\exp\!\left( \frac12\sum_{a,b=1}^q C_{ab}t_at_b \right).$$

Moments are therefore obtained by differentiating at the origin. Expanding the exponential gives

$$M(t) = \sum_{k=0}^{\infty} \frac{1}{2^k k!} \left( \sum_{a,b=1}^q C_{ab}t_at_b \right)^k.$$

Only even total degrees occur, so every centered odd moment is zero. To obtain a derivative of total order $2n$ at $t=0$, only the term $k=n$ can contribute. Choosing its $n$ quadratic factors pairs the $2n$ differentiated labels. Every unordered pairing occurs $n!$ times from ordering the factors and $2^n$ times from orienting the two entries of each factor. These $2^n n!$ copies cancel the prefactor, leaving each covariance product exactly once. This proves the theorem.

At fourth order the theorem reads

$$\begin{aligned} \mathbb E[Y_1Y_2Y_3Y_4] ={}&C_{12}C_{34} +C_{13}C_{24}\\ &+C_{14}C_{23}. \end{aligned}$$

Equivalently, the centered Gaussian cumulant generator is quadratic, so every cumulant above second order vanishes and only two-element blocks survive in the moment–cumulant partition formula. This is an independent check of the coefficient proof, not a converse: a finite list of Gaussian-looking moments does not establish Gaussianity.

For noncentered variables, write $X_j=m_j+Y_j$ and expand. Only singleton factors $m_j$ and paired covariance factors survive. Thus mean and covariance also determine every noncentered Gaussian moment.

Gaussian processes: existence is not path regularity

For an index set $T$, a Gaussian process $(X_t)_{t\in T}$ is a family for which every finite vector $(X_{t_1},\ldots,X_{t_n})$ is jointly Gaussian. Its mean and covariance kernel are

$$\begin{aligned} m(t)&=\mathbb E[X_t],\\ K(s,t) &=\mathbb E[(X_s-m(s))(X_t-m(t))]. \end{aligned}$$

Rasmussen and Williams give this finite-family definition and its marginal-consistency requirement in Rasmussen and Williams 2006, § 2.2, p. 13, PDF. The kernel must be symmetric and positive semidefinite: for every finite choice of indices and coefficients,

$$\sum_{j,k=1}^n a_ja_kK(t_j,t_k)\geq0.$$

Cited existence theorem. Conversely, any function $m:T\to\mathbb R$ and any symmetric positive-semidefinite kernel $K$ determine a Gaussian process law on $\mathbb R^T$ equipped with its product, or cylinder, $\sigma$-algebra.

The proof architecture is short, although the extension theorem itself is not reproved here. For each finite set $F=\{t_1,\ldots,t_n\}$, form the possibly singular Gaussian law with mean $m_F$ and covariance matrix $K_F$. Permuting indices merely permutes this law. If $F\subset G$, setting the omitted dual variables to zero in the characteristic function for $G$ gives the characteristic function for $F$; the finite laws are therefore consistent. Kolmogorov’s extension theorem then supplies the product-space measure. Tumulka states the arbitrary-index version for Borel coordinate spaces in Tumulka 2008, § 1.1, p. 2. Durrett proves the countable-coordinate case in Durrett 2019, Appendix A.3, pp. 464–465, PDF and shows in the Brownian construction, pp. 355–357, why a second argument is needed to obtain continuous paths.

The distinction is visible in a one-line counterexample. On $T=[0,1]$, set

$$K(s,t)=\mathbf 1_{\{s=t\}}.$$

This kernel is positive semidefinite and produces independent standard normals at distinct indices. Yet for $s\neq t$,

$$X_t-X_s\sim N(0,2).$$

Along any sequence of distinct $t_n\to t$, the increments fail to converge to zero even in probability. Hence this process has no continuous modification. Continuity, differentiability, stationarity, and the Markov property are not consequences of covariance positivity. They require separate hypotheses; Stochastic Processes and Correlation Functions develops the general process language.

Gaussian random distributions

A random tempered distribution is modeled here as a measurable map

$$\Phi:\Omega\longrightarrow \mathcal S'_{\mathbb R}(\mathbb R^d),$$

where the $\sigma$-algebra on the dual is generated by the evaluation maps $u\mapsto u(f)$ for $f\in\mathcal S_{\mathbb R}$. Its smeared variables are

$$\Phi(f)=\langle\Phi,f\rangle.$$

The random distribution is Gaussian when every finite vector $(\Phi(f_1),\ldots,\Phi(f_n))$ is Gaussian. For a centered field, its covariance is the symmetric bilinear form

$$C(f,g)=\mathbb E[\Phi(f)\Phi(g)].$$

Positivity is necessary, but algebraic positivity alone does not construct a distribution-valued random object. The missing input is continuity in the test-function topology.

Cited theorem (Bochner–Minlos). Let $E$ be a real nuclear locally convex space, such as $\mathcal S_{\mathbb R}(\mathbb R^d)$. If $\chi:E\to\mathbb C$ is normalized, continuous, and positive definite, then $\chi$ is the characteristic functional of a unique probability measure on $E'$. Positive definiteness means

$$\sum_{j,k=1}^n \overline{z_j}z_k\, \chi(f_k-f_j) \geq0$$

for all finite choices of $f_j\in E$ and $z_j\in\mathbb C$. Fageot, Amini, and Unser state the theorem and the generalized-process setup in Fageot, Amini, and Unser 2014, §§ 2.1–2.2, pp. 4–5.

Suppose now that $m:E\to\mathbb R$ is continuous linear and that $C:E\times E\to\mathbb R$ is continuous, symmetric, bilinear, and positive semidefinite. Then

$$\chi(f) =\exp\!\left( i m(f)-\frac12C(f,f) \right)$$

meets the theorem’s hypotheses. To see positive definiteness, restrict to the span of finitely many $f_j$ and construct the finite Gaussian vector with mean $m(f_j)$ and covariance $C(f_j,f_k)$. Its characteristic function gives the displayed quadratic form as an expectation of an absolute square. Bochner–Minlos then produces a Gaussian random distribution with mean $m$ and covariance $C$.

Kolmogorov extension by itself would construct a compatible family indexed by test functions. It would not ensure that $f\mapsto\Phi(f)$ is almost surely a continuous linear functional. That is the extra conclusion supplied by the nuclear-space theorem.

White noise and the Hilbert-space warning

Gaussian white noise on $\mathbb R^d$ has

$$C(f,g) =\int_{\mathbb R^d} f(x)g(x)\,\mathrm d^d x, \qquad \chi(f) =\exp\!\left(-\frac12\|f\|_2^2\right).$$

The $L^2$ pairing is continuous on Schwartz space, so Bochner–Minlos gives an $\mathcal S'$-valued random object. It is not an $L^2$-valued random variable. Indeed, enumerate the Hermite-function orthonormal basis $(e_n)$ of $L^2$; each $e_n$ is Schwartz. The coefficients $\Phi(e_n)$ are independent $N(0,1)$ variables, and the strong law gives

$$\frac1N\sum_{n=1}^N |\Phi(e_n)|^2 \longrightarrow1 \quad\text{almost surely}.$$

Parseval’s sum therefore diverges. More generally, Hairer’s Hairer 2026, Proposition 3.18, p. 17, PDF shows that the covariance of a Gaussian measure on a separable Hilbert space must be trace class, and that every positive symmetric trace-class covariance does define such a measure. The identity is not trace class in infinite dimension.

Point evaluation fails for the same reason. If $\rho_\varepsilon(x)=\varepsilon^{-d}\rho(x/\varepsilon)$ is a normalized mollifier approaching a delta distribution, then

$$\operatorname{Var}[\Phi(\rho_\varepsilon)] =\|\rho_\varepsilon\|_2^2 =\varepsilon^{-d}\|\rho\|_2^2.$$

The variance diverges as $\varepsilon\downarrow0$, so this approximation does not yield a point random variable $\Phi(x)$.

Wick structure remains a smeared statement

Apply the finite theorem to any probes $f_1,\ldots,f_{2n}$. For a centered Gaussian random distribution,

$$\mathbb E\!\left[ \prod_{j=1}^{2n}\Phi(f_j) \right] = \sum_{P\in\mathcal P_2(2n)} \prod_{\{a,b\}\in P} C(f_a,f_b).$$

For fixed probes, covariance-relative Wick subtraction is also defined. At second order,

$$:\!\Phi(f)\Phi(g)\!:_C =\Phi(f)\Phi(g)-C(f,g),$$

which has mean zero. This formula does not define $:\!\Phi(x)^2\!:$ merely by setting $f=g=\delta_x$. Local Wick powers require a controlled limiting or extension procedure. Nor is this classical probability identity the operator time-ordering theorem or the fermionic sign rule.

Controlled QFT application: a regulated free Euclidean field

Take $N<\infty$ real field coordinates $\phi_x$, representing a fixed finite lattice or mode cutoff. Let $L\succeq0$ be the regulated kinetic matrix and choose $m>0$. Then

$$A=L+m^2I\succ0.$$

The normalized positive Euclidean measure is

$$\mathrm d\mu_A(\phi) = \frac{(\det A)^{1/2}}{(2\pi)^{N/2}} \exp\!\left( -\frac12\phi^{\mathsf T}A\phi \right) \mathrm d^N\phi.$$

With $C=A^{-1}$ and a real source $J$, completing the square gives

$$\begin{aligned} Z_E[J] &=\mathbb E_{\mu_A} [e^{J^{\mathsf T}\phi}]\\ &=\exp\!\left( \frac12J^{\mathsf T}CJ \right),\\ W_E[J] &=\log Z_E[J] =\frac12J^{\mathsf T}CJ. \end{aligned}$$

The calculation is finite, $Z_E[0]=1$, and every derivative is justified. The covariance and four-point function are

$$\langle\phi_x\phi_y\rangle_E=C_{xy},$$

and

$$\begin{aligned} \langle \phi_x\phi_y\phi_z\phi_w \rangle_E ={}&C_{xy}C_{zw} +C_{xz}C_{yw}\\ &+C_{xw}C_{yz}. \end{aligned}$$

Source differentiation and the Wick–Isserlis theorem give the same three terms. The quadratic $W_E$ independently confirms that connected cumulants above second order vanish. Schwartz derives the Lorentzian source formula and the same free four-point pairing in Schwartz 2014, § 14.3, pp. 261–263. Here the factors of $i$ have not been copied: the local convention is the positive Euclidean weight $e^{-S_E+J^{\mathsf T}\phi}$.

Several checks expose the boundary of the example:

• $AC=I$, and $C\succ0$ as a covariance must be.
• Odd moments vanish both by the theorem and by the symmetry $\phi\mapsto-\phi$.
• On a periodic lattice at $m=0$, the constant vector is a zero mode of $L$. Then $\det A=0$, $A^{-1}$ does not exist, and the displayed density is not normalizable. One must keep $m>0$, remove the zero mode, or impose boundary conditions that eliminate it.
• Removing the regulator requires a separate topology and limiting argument; the finite determinant cannot simply be renamed a continuum determinant.

There is nevertheless a precise continuum statement at the level established above. For real Schwartz probes and $m>0$, define, using the site’s Fourier signs in Euclidean space,

$$C_m(f,g) =\int\frac{\mathrm d^d p}{(2\pi)^d}\, \frac{ \widetilde f(-p)\widetilde g(p) }{p^2+m^2}.$$

This is continuous and positive semidefinite; for example,

$$0\leq C_m(f,f) \leq m^{-2}\|f\|_2^2.$$

Bochner–Minlos therefore constructs the centered massive free Euclidean field as a Gaussian random tempered distribution. Sheffield describes this random-distribution interpretation and the covariance $(-\Delta+m^2)^{-1}$ in Sheffield 2007, § 3.3, p. 19. At $m=0$, the infrared integral already fails for generic Schwartz probes in $d\leq2$; the dimension and zero-mode prescription must be stated before claiming a measure.

This construction proves neither reflection positivity nor Lorentzian reconstruction. For the regulated physical derivation, inverse-kernel prescriptions, determinants, and source normalization, continue to Gaussian Fields and Sources. For support, Cameron–Martin directions, reflection positivity, and clustering, continue to Gaussian Euclidean Fields as Measures.

Common pitfalls and stop rules

Using the density as the definition. A singular covariance still defines a Gaussian law, but not a full-dimensional density. Use the characteristic function or a linear image of a standard Gaussian.

Dropping the Gaussian hypothesis. Mean and covariance determine Gaussian laws, not arbitrary laws. Zero covariance also implies independence only for jointly Gaussian subvectors.

Reading paths from a kernel. Positive semidefiniteness constructs finite-dimensional laws. Continuity, measurability in the index, stationarity, and Markov structure are separate properties.

Confusing a family of probes with a random distribution. Consistent laws for $\Phi(f)$ do not by themselves make $f\mapsto\Phi(f)$ almost surely continuous and linear. State the test space and invoke an appropriate existence theorem.

Replacing smearing by point substitution. Wick pairings of $\Phi(f)$ do not license $\Phi(x)$, $C(x,x)$, or local powers. Diagonal restriction and renormalized composite fields need additional control.

Equating positivity conditions. Covariance positivity constructs an ordinary Gaussian measure. Osterwalder–Schrader reflection positivity is a stronger, different condition and does not follow from it.

Importing operator or fermionic Wick rules. The theorem on this page is a commuting probability-moment identity. Operator ordering, contractions, and fermionic signs are treated in Wick’s Theorem and Free Gaussian Factorization.

Check your understanding

1. Diagnose a singular Gaussian

Let $Z\sim N(0,1)$ and $X=(Z,Z)^{\mathsf T}$. Find the characteristic function, covariance, support, and $\mathbb E[X_1^2X_2^2]$.

Solution

For $t=(t_1,t_2)^{\mathsf T}$,

$$\varphi_X(t) =\mathbb E[e^{i(t_1+t_2)Z}] =\exp\!\left[-\frac12(t_1+t_2)^2\right].$$

Thus

$$\Sigma= \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}$$

has rank one, and the law is supported on $\{(x_1,x_2):x_1=x_2\}$. Wick–Isserlis gives

$$\mathbb E[X_1^2X_2^2] =C_{11}C_{22}+2C_{12}^2 =3,$$

in agreement with $\mathbb E[Z^4]=3$.

2. Build a process with no continuous modification

For $K(s,t)=\mathbf1_{\{s=t\}}$, verify positive semidefiniteness and explain why the Gaussian process cannot have a continuous modification.

Solution

For distinct indices,

$$\sum_{j,k}a_ja_kK(t_j,t_k) =\sum_j a_j^2\geq0.$$

If an index is repeated, first combine its coefficients; the result is still a sum of squares. Thus $K$ is a valid covariance kernel. For distinct $s,t$, the variables are independent standard normals and $X_t-X_s\sim N(0,2)$. Along distinct $t_n\to t$, these increments do not converge to zero in probability, whereas almost-sure continuity would imply such convergence. No continuous modification exists.

3. Test the regulated zero mode

Let $L$ be the graph Laplacian of a connected periodic lattice. Why does the massless measure with $A=L$ fail, and why does $A=L+m^2I$ work for $m>0$?

Solution

The constant vector $\mathbf1$ satisfies $L\mathbf1=0$, so $\det L=0$ and the quadratic weight is constant along $\phi\mapsto\phi+c\mathbf1$. Integration in that direction diverges. For $m>0$,

$$v^{\mathsf T}(L+m^2I)v =v^{\mathsf T}Lv+m^2\|v\|^2 >0$$

for every nonzero $v$. The matrix is positive definite, the normalized Gaussian measure exists, and its covariance is $(L+m^2I)^{-1}$.

Synthesis and continuations

Gaussian correlations are fixed by mean and covariance because every finite Gaussian characteristic function is the exponential of that linear–quadratic data; differentiating it gives the Wick–Isserlis pairing sum. In infinite dimensions, covariance positivity still fixes all finite laws, but it does not choose regular paths or construct a continuous linear random functional. Kolmogorov extension, path-regularity criteria, Hilbert-space trace-class conditions, and Bochner–Minlos answer different existence questions and must not be interchanged.

Continue to Stochastic Processes and Correlation Functions for general finite-dimensional laws, stationarity, and correlation structure. Continue to Gaussian Fields and Sources for the regulated free-field integral and its physical source conventions. For operator ordering and bosonic or fermionic contraction signs, continue to Wick’s Theorem and Free Gaussian Factorization.

References

• Rick Durrett, Probability: Theory and Examples, fifth edition, PDF, Cambridge University Press, 2019. Appendix A.3, pp. 464–465, proves the countable-coordinate case; § 7.1, pp. 355–357, exposes the separate work required to obtain continuous paths.

• Julien Fageot, Arash Amini, and Michael Unser, “On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling”, arXiv:1401.6850v2, 2014. §§ 2.1–2.2, pp. 4–5, support generalized stochastic processes, characteristic functionals, and the Bochner–Minlos theorem.

• Martin Hairer, Advanced Stochastic Analysis, PDF, notes dated May 19, 2026. Proposition 3.18, p. 17, supports the trace-class characterization of Hilbert-space Gaussian covariances.

• Peter McCullagh, Tensor Methods in Statistics, Dover edition, 2018. § 3.9.1, pp. 85–86, is the structural source for the Wick–Isserlis pairing formula and its pairing count.

• Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, MIT Press, 2006. § 2.2, pp. 13–14, is the teaching source for Gaussian processes, their mean and covariance functions, and finite-law consistency.

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. Section 14.3, pp. 261–263, derives source differentiation, the free quadratic generator, and four-point pairings; its Lorentzian factors are translated to the positive Euclidean convention used here.

• Scott Sheffield, “Gaussian Free Fields for Mathematicians”, Probability Theory and Related Fields 139 (2007), 521–541. §§ 2.5 and 3.3, pp. 12 and 19, support the random-distribution interpretation, failure of pointwise Fourier sums, and the massive covariance $(-\Delta+m^2)^{-1}$.

• Roderich Tumulka, “A Kolmogorov Extension Theorem for POVMs”, Letters in Mathematical Physics 84 (2008), 41–46. § 1.1, p. 2, states the classical arbitrary-index theorem for Borel coordinate spaces.