Characteristic Functions, Moments, Cumulants, and Generating Functionals

A characteristic function exists for every probability law and determines that law. A moment-generating function is more restrictive: it is useful only where the relevant exponential moment is finite. Whenever differentiation is justified, derivatives of the generator give moments, while derivatives of its normalized logarithm give cumulants. In several variables the same construction gives joint cumulants; with a finite Euclidean source, it also packages connected correlations and nonlinear response.

Those statements have different hypotheses, and keeping them separate is the main task of this page. A finite list of moments need not determine a law, a characteristic function can have no useful derivatives, and a formal Lorentzian path integral is not a probability measure. The logarithm is also only local unless zeros and branches have been controlled.

Required background. Probability Spaces, Random Variables, and Conditional Expectation supplies expectation, independence, integrability criteria, and dominated convergence.

Helpful background. Fourier Series, Fourier Transforms, and Plancherel Theory supplies the transform convention and inversion language. Readers needing a shorter review can use the statistical ensembles and probability repair or the Fourier, distributions, and Green functions repair.

Characteristic, moment, cumulant, and generating functions

Let $X$ be an $\mathbb R^d$-valued random vector with law $\mu$. We use the characteristic-function convention

$$\varphi_X(t) =\mathbb E[e^{i t^{\mathsf T}X}] =\int_{\mathbb R^d}e^{i t^{\mathsf T}x}\,\mu(\mathrm dx), \qquad t\in\mathbb R^d.$$

The sign is explicit because the inverse transform below uses $e^{-i t^{\mathsf T}x}$. Four related objects should not be conflated:

Object	Definition	Domain	What it encodes
Characteristic function	$\varphi_X(t)=\mathbb E[e^{i t^{\mathsf T}X}]$	Every real $t$	The complete probability law
Moment-generating function	$M_X(s)=\mathbb E[e^{s^{\mathsf T}X}]$	Only the set where the expectation is finite	Moments and exponential tilts, under additional hypotheses
Cumulant-generating function	$K_X(s)=\log M_X(s)$	Real points where $M_X$ is finite; a neighborhood of $0$ is an extra hypothesis	Joint cumulants and additive structure
Generating functional	$Z[J]=\mathbb E[e^{\langle J,\Phi\rangle}]$ or $\mathbb E[e^{i\langle J,\Phi\rangle}]$	Depends on the field, test space, and source	Smeared moments or cumulants through directional derivatives

The first row is automatic because $|e^{i t^{\mathsf T}X}|=1$. The other rows require domain, differentiability, normalization, and sometimes infinite-dimensional existence arguments.

Characteristic functions determine probability laws

Several basic facts follow directly from the definition:

$$\varphi_X(0)=1, \qquad |\varphi_X(t)|\leq 1, \qquad \varphi_X(-t)=\overline{\varphi_X(t)}.$$

They require no moment assumption. Characteristic functions are uniformly continuous, since

$$\begin{aligned} |\varphi_X(t+h)-\varphi_X(t)| &\leq \mathbb E\!\left[ |e^{i h^{\mathsf T}X}-1| \right]\\ &\longrightarrow 0 \qquad (h\to0) \end{aligned}$$

by dominated convergence, with a bound independent of $t$.

They also obey a positivity condition. For any points $t_1,\ldots,t_N\in\mathbb R^d$ and coefficients $c_1,\ldots,c_N\in\mathbb C$,

$$\begin{aligned} \sum_{j,k=1}^N \overline{c_j}c_k\, \varphi_X(t_k-t_j) &= \mathbb E\!\left[ \left| \sum_{k=1}^N c_k e^{i t_k^{\mathsf T}X} \right|^2 \right]\\ &\geq 0. \end{aligned}$$

Thus every characteristic function is normalized, continuous, and positive definite. These properties are valuable diagnostics, although uniqueness of the underlying probability law requires the inversion result below.

Inversion and uniqueness

For a real-valued $X$, the inversion theorem states that at continuity points $a<b$ of the distribution function,

$$\mathbb P(a<X\leq b) = \lim_{T\to\infty} \frac{1}{2\pi} \int_{-T}^{T} \frac{e^{-ita}-e^{-itb}}{it}\, \varphi_X(t)\,\mathrm dt.$$

The quotient at $t=0$ is interpreted by continuity. Without the continuity assumption, the right-hand side is

$$\mathbb P(a<X<b) +\frac12\mathbb P(X=a) +\frac12\mathbb P(X=b).$$

Thus endpoint atoms contribute half their mass. The formula proves the uniqueness result

$$\varphi_X=\varphi_Y \quad\Longrightarrow\quad \mathcal L(X)=\mathcal L(Y).$$

This is the inversion theorem in Durrett’s Durrett 2019, § 3.3.1, pp. 128–30, PDF.

When $\varphi_X\in L^1(\mathbb R^d)$, the law has the bounded continuous density

$$f_X(x) = \frac{1}{(2\pi)^d} \int_{\mathbb R^d} e^{-i t^{\mathsf T}x}\varphi_X(t)\,\mathrm d^dt.$$

This is a sufficient condition, not a characterization of all distributions with densities.

Characteristic functions also connect this page to the preceding treatment of weak convergence. Lévy’s continuity theorem says that $X_n\Rightarrow X$ implies pointwise convergence $\varphi_{X_n}(t)\to\varphi_X(t)$. Conversely, a pointwise limit of characteristic functions is a weak limit when the limiting function is continuous at the origin. The last condition is essential: for $X_n\sim\mathcal N(0,n)$,

$$\varphi_{X_n}(t)=e^{-nt^2/2} \longrightarrow \begin{cases} 1,&t=0,\\ 0,&t\neq0, \end{cases}$$

and the discontinuity records probability mass escaping to infinity. The two directions and the continuity-at-zero condition are stated in Durrett’s Durrett 2019, Theorem 3.3.17, p. 132, PDF. The full convergence framework belongs to Probabilistic Convergence, Laws of Large Numbers, and Central Limit Theorems.

Moments come from derivatives only under hypotheses

Let $\alpha=(\alpha_1,\ldots,\alpha_d)$ be a multi-index and $|\alpha|=\alpha_1+\cdots+\alpha_d$. If

$$\mathbb E\|X\|^{|\alpha|}<\infty,$$

then dominated differentiation gives

$$\partial^\alpha\varphi_X(t) = \mathbb E\!\left[ (iX)^\alpha e^{i t^{\mathsf T}X} \right],$$

and hence

$$\partial^\alpha\varphi_X(0) =i^{|\alpha|}\mathbb E[X^\alpha].$$

The integrable dominating function is $\|X\|^{|\alpha|}$. This is a clean sufficient condition; it is not a license to infer every absolute moment from the mere existence of a derivative at zero. Durrett’s Durrett 2019, Theorem 3.3.18, p. 134, PDF gives the scalar dominated-differentiation statement.

The Cauchy law makes the boundary visible. Its characteristic function is

$$\varphi_X(t)=e^{-|t|},$$

which exists for all $t$ but is not differentiable at the origin. The mean does not exist, and $\mathbb E[e^{sX}]=\infty$ for every $s\neq0$.

The domain of the moment-generating function

Define

$$D_X = \left\{ s\in\mathbb R^d: M_X(s)=\mathbb E[e^{s^{\mathsf T}X}]<\infty \right\}.$$

The set $D_X$ contains $0$ and is convex by Hölder’s inequality, but $0$ can lie on its boundary. If $M_X$ is finite on an open neighborhood of $0$, then all absolute moments exist, $M_X$ is analytic on a possibly smaller neighborhood, and

$$\partial^\alpha M_X(0)=\mathbb E[X^\alpha].$$

In that case the local MGF determines the law. None of these conclusions follows from a formal Taylor series alone. McCullagh’s McCullagh 2017, § 2.2.1, pp. 29–30, PDF explicitly separates a convergent MGF from a finite-order or divergent formal moment expansion.

The lognormal law supplies two distinct warnings. Let $X=e^Y$ with $Y\sim\mathcal N(0,1)$. Then

$$\mathbb E[X^n]=e^{n^2/2}, \qquad n=0,1,2,\ldots,$$

so every nonnegative integer moment is finite, but $M_X(s)=\infty$ for every $s>0$. Moreover, if $f_0$ is the lognormal density, then

$$f_\varepsilon(x) =f_0(x) \left[1+\varepsilon\sin(2\pi\log x)\right], \qquad |\varepsilon|\leq1,$$

defines a family of distinct densities with the same nonnegative integer moments. Indeed,

$$\begin{aligned} \mathbb E\!\left[ e^{nY}\sin(2\pi Y) \right] &= \operatorname{Im} \exp\!\left[ \frac12(n+2\pi i)^2 \right]\\ &=0 \qquad (n=0,1,2,\ldots). \end{aligned}$$

Thus “all moments exist,” “the MGF exists near zero,” and “the moments determine the law” are three different statements. Durrett gives the lognormal moment-indeterminacy construction in Durrett 2019, § 3.3.5, p. 140, PDF.

Cumulants are logarithmic coordinates

Suppose first that $M_X$ is finite on a neighborhood of $0$. Because $M_X(s)>0$ for real $s$, define

$$K_X(s)=\log M_X(s), \qquad K_X(0)=0.$$

The joint cumulant tensor is

$$\kappa_{i_1\cdots i_n} = \left. \frac{\partial^n K_X(s)} {\partial s_{i_1}\cdots\partial s_{i_n}} \right|_{s=0}.$$

One can instead work with the characteristic function. Continuity and $\varphi_X(0)=1$ give a zero-free ball around the origin. On that ball choose the local logarithm satisfying $\log\varphi_X(0)=0$. If $\mathbb E\|X\|^n<\infty$, then the dominated-differentiation result applies through order $n$, and for $1\leq r\leq n$,

$$\kappa_{i_1\cdots i_r} = i^{-r} \left. \frac{\partial^r\log\varphi_X(t)} {\partial t_{i_1}\cdots\partial t_{i_r}} \right|_{t=0}.$$

If such log derivatives happen to exist without the corresponding absolute moments, they are derivative-defined coefficients; they should not be silently identified with the moment cumulants below.

The word local matters. For a symmetric Rademacher variable, $\varphi_X(t)=\cos t$, so the characteristic function has zeros and no single global logarithm.

For a scalar variable with raw moments $m_r=\mathbb E[X^r]$, coefficient comparison in $M=e^K$ gives

$$\begin{aligned} \kappa_1&=m_1,\\ \kappa_2&=m_2-m_1^2,\\ \kappa_3&=m_3-3m_2m_1+2m_1^3,\\ \kappa_4&= m_4-4m_3m_1-3m_2^2 +12m_2m_1^2-6m_1^4. \end{aligned}$$

Equivalently, if $\mu=\mathbb E[X]$ and $\sigma^2=\operatorname{Var}(X)$,

$$\kappa_2=\sigma^2, \qquad \kappa_3=\mathbb E[(X-\mu)^3], \qquad \kappa_4=\mathbb E[(X-\mu)^4]-3\sigma^4.$$

The third and fourth cumulants are not themselves the standardized skewness and excess kurtosis; those divide by appropriate powers of $\sigma$.

The partition formula and connected blocks

Let $\Pi_n$ be the set of partitions of $\{1,\ldots,n\}$. If every product appearing below is integrable, the joint cumulant has the finite-order definition

$$\kappa(X_1,\ldots,X_n) = \sum_{\pi\in\Pi_n} (-1)^{|\pi|-1}(|\pi|-1)! \prod_{B\in\pi} \mathbb E\!\left[ \prod_{j\in B}X_j \right].$$

Möbius inversion on the partition lattice gives the inverse identity

$$\mathbb E[X_1\cdots X_n] = \sum_{\pi\in\Pi_n} \prod_{B\in\pi} \kappa(X_j:j\in B).$$

This formula is the precise algebraic meaning of a full correlation being a sum of products of connected blocks. It can define a cumulant through a fixed finite order even when no neighborhood-valued MGF exists: expand the finite polynomial

$$\mathbb E\!\left[ \prod_{j=1}^n(1+u_jX_j) \right]$$

and read the coefficient of $u_1\cdots u_n$ in its formal logarithm. This is finite-order algebra, not a claim that an infinite moment series converges. McCullagh derives both partition identities and this finite-order interpretation in McCullagh 2017, § 2.3.4, p. 38, PDF.

Direct partition enumeration has combinatorial cost equal to the Bell number $|\Pi_n|$. For low orders this makes every subtraction transparent; for high orders, recursive differentiation or symbolic partition algorithms are less error-prone.

Independence, joint structure, and the Gaussian boundary

For random vectors $X$ and $Y$, independence is equivalent to factorization of the joint characteristic function:

$$X\perp Y \quad\Longleftrightarrow\quad \varphi_{(X,Y)}(s,t) =\varphi_X(s)\varphi_Y(t)$$

for all $s,t$. The reverse implication uses uniqueness of characteristic functions. If two nonempty subfamilies of the arguments of a joint cumulant are independent, that mixed cumulant vanishes. One vanishing covariance, or any finite list of vanishing mixed cumulants, does not generally prove independence.

For independent scalar variables $X$ and $Y$,

$$M_{X+Y}(s)=M_X(s)M_Y(s), \qquad K_{X+Y}(s)=K_X(s)+K_Y(s)$$

on their common local domain. Consequently,

$$\kappa_n(X+Y)=\kappa_n(X)+\kappa_n(Y).$$

Affine transformations obey

$$\kappa_1(aX+b)=a\kappa_1(X)+b, \qquad \kappa_n(aX+b)=a^n\kappa_n(X), \quad n\geq2.$$

Thus, for iid variables with the required cumulants,

$$\kappa_r(\overline X_n) =n^{1-r}\kappa_r(X_1).$$

Bazant 2006, pp. 3–6 develops the same cumulant additivity and iid scaling from characteristic functions.

This scaling explains why higher standardized cumulants are suppressed in classical central-limit regimes. More precisely, if $0<\kappa_2(X_1)<\infty$ and $r>2$, then

$$\frac{\kappa_r(\overline X_n)} {[\kappa_2(\overline X_n)]^{r/2}} = n^{1-r/2} \frac{\kappa_r(X_1)} {[\kappa_2(X_1)]^{r/2}}.$$

This scaling does not replace the hypotheses of a central limit theorem.

A Gaussian vector with mean $m$ and covariance $\Sigma\succeq0$ has

$$\varphi_X(t) = \exp\!\left( i t^{\mathsf T}m -\frac12t^{\mathsf T}\Sigma t \right).$$

Its first two cumulants are $m$ and $\Sigma$, and every higher cumulant vanishes. Conversely, an actual cumulant-generating function that is quadratic on a neighborhood of the origin determines a Gaussian law. A finite list of zero higher cumulants does not: the distribution

$$\mathbb P(X=0)=\frac23, \qquad \mathbb P(X=\sqrt3) =\mathbb P(X=-\sqrt3)=\frac16$$

has mean zero, variance one, $\kappa_3=0$, and $\kappa_4=0$, but is discrete. The full Gaussian characterization, Gaussian processes, random distributions, and Wick structure belong to Gaussian Vectors, Processes, Random Distributions, and Wick Structure.

A reliable calculation procedure

The method applies to probability laws, finite regulated Euclidean integrals, and characteristic or exponential functionals whose domains have been established.

1. Choose the generator. Use a characteristic function when law determination or weak convergence is the goal. Use an MGF or Euclidean source generator only after checking exponential integrability.
2. Normalize and state the domain. Verify $\varphi(0)=1$ or divide an unnormalized Euclidean source integral by its value at zero. Record the neighborhood in which it is finite and nonzero.
3. Justify every derivative. Supply an integrable dominating function, analyticity from exponential integrability, or a finite-order algebraic definition. Stop at the highest justified order.
4. Take the logarithm locally. Fix the branch by $\log1=0$. Derivatives of the generator give full moments; derivatives of the logarithm give cumulants.
5. Validate independently. Check normalization, conjugation symmetry or positive definiteness, covariance positivity, independence factorization, and at least one direct moment calculation.

The output is a law, a collection of moments or connected cumulants, or a source-response relation. Differentiation is usually inexpensive at low order; converting all order-$n$ moments and cumulants by partitions grows with the Bell number. The calculation must stop when the proposed source is outside the finite domain, the needed moment is absent, a logarithm crosses a zero, normalization fails, or a formal functional has not been given a mathematical definition.

Source tilting turns cumulants into responses

Let $\Phi\in\mathbb R^N$ have an MGF finite on an open set. Write

$$Z_E[J]=\mathbb E[e^{J^{\mathsf T}\Phi}], \qquad W_E[J]=\log Z_E[J].$$

For $J$ in the interior of the finite domain, define the tilted probability measure

$$\frac{\mathrm d\mathbb P_J}{\mathrm d\mathbb P} = \exp\!\left[ J^{\mathsf T}\Phi-W_E[J] \right].$$

Differentiating gives

$$\begin{aligned} \frac{\partial Z_E}{\partial J_a} &=Z_E[J]\,\mathbb E_J[\Phi_a],\\ \frac{\partial W_E}{\partial J_a} &=\mathbb E_J[\Phi_a],\\ \frac{\partial^2 W_E} {\partial J_a\partial J_b} &=\operatorname{Cov}_J(\Phi_a,\Phi_b). \end{aligned}$$

Higher derivatives are the joint cumulants in the tilted measure. The first derivative is the sourced mean response; the Hessian is the linear response and is positive semidefinite. Therefore $W_E$ is convex wherever this differentiation is valid. Fithian, n.d., “Differential identities” derives the gradient-as-mean and Hessian-as-covariance formulas on the interior of the finite natural-parameter domain. Positivity of the Hessian is a useful independent check on signs and source conventions.

For a random distribution $\Phi$ acting on test functions $f$, the characteristic functional

$$\mathcal C[f]=\mathbb E[e^{i\Phi(f)}]$$

exists for every admissible $f$. The exponential functional $Z[f]=\mathbb E[e^{\Phi(f)}]$ still needs exponential integrability. When directional derivatives are justified,

$$\begin{aligned} D^nZ[0](f_1,\ldots,f_n) &=\mathbb E[\Phi(f_1)\cdots\Phi(f_n)],\\ D^n\log Z[0](f_1,\ldots,f_n) &=\kappa(\Phi(f_1),\ldots,\Phi(f_n)). \end{aligned}$$

The smeared variables $\Phi(f)$ are essential. A continuum field $\Phi(x)$ may be a distribution rather than a pointwise random variable, so unsmeared functional derivatives need a separate kernel or distributional argument.

Euclidean and Lorentzian conventions

For a positive finite-dimensional Euclidean weight, this page uses

$$\mathcal Z_E[J] = \int e^{-S_E(\phi)+\langle J,\phi\rangle}\,\mathrm d\phi, \qquad Z_E[J]=\frac{\mathcal Z_E[J]}{\mathcal Z_E[0]}, \qquad W_E[J]=\log Z_E[J].$$

Then $Z_E[0]=1$. At $J=0$, derivatives of $Z_E$ give unsourced full Euclidean correlations, and derivatives of $W_E$ give their cumulants. At nonzero $J$, the same derivatives give correlations in the tilted measure.

The site’s Lorentzian convention is instead

$$Z_L[J] = \int\mathcal D\phi\, \exp\!\left( iS[\phi]+i\int J\phi \right), \qquad W_L[J]=-i\log Z_L[J].$$

Writing $\widehat Z_L[J]=Z_L[J]/Z_L[0]$, the corresponding formal translation is

$$\begin{aligned} \langle T\phi(x_1)\cdots\phi(x_n)\rangle &= (-i)^n \left. \frac{\delta^n\widehat Z_L[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0},\\ \langle T\phi(x_1)\cdots\phi(x_n)\rangle_{\mathrm c} &= (-i)^{n-1} \left. \frac{\delta^n W_L[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0}. \end{aligned}$$

These factors belong to the displayed source convention; they must not be copied into the Euclidean probability formulas. Nor does the Lorentzian symbol $\mathcal D\phi$ by itself construct a probability measure. Vacuum normalization, ordering, and the developed physical interpretation are handled in Connected Correlators and Cumulants. For the displayed convention and the full-versus-connected derivative formulas, see Schwartz 2014, § 14.3, pp. 261–63 and § 34.1.2, pp. 737–39.

Worked application: a normalized two-mode Euclidean source

Consider the finite regulated field $\Phi=(\phi,\chi)^{\mathsf T}$ with

$$S_E(\Phi) =\frac12\Phi^{\mathsf T}A\Phi, \qquad A= \begin{pmatrix} a&b\\ b&c \end{pmatrix}, \qquad a>0,\quad c>0,\quad \Delta=ac-b^2>0.$$

The inequalities make $A$ positive definite, so the weight $e^{-S_E}$ is integrable and positive. For a real source $J=(J_\phi,J_\chi)^{\mathsf T}$, define

$$\mathcal Z_E[J] = \int_{\mathbb R^2} \exp\!\left[ -\frac12\Phi^{\mathsf T}A\Phi +J^{\mathsf T}\Phi \right]\,\mathrm d^2\Phi.$$

Set $C=A^{-1}$. Completing the square gives

$$-\frac12\Phi^{\mathsf T}A\Phi+J^{\mathsf T}\Phi = -\frac12(\Phi-CJ)^{\mathsf T}A(\Phi-CJ) +\frac12J^{\mathsf T}CJ,$$

with

$$C =\frac1\Delta \begin{pmatrix} c&-b\\ -b&a \end{pmatrix}.$$

Translation invariance of the ordinary Lebesgue integral yields

$$\mathcal Z_E[J] =\frac{2\pi}{\sqrt\Delta} \exp\!\left( \frac12J^{\mathsf T}CJ \right).$$

After normalization,

$$Z_E[J] =\frac{\mathcal Z_E[J]}{\mathcal Z_E[0]} =\exp\!\left( \frac12J^{\mathsf T}CJ \right), \qquad W_E[J]=\frac12J^{\mathsf T}CJ.$$

This is the finite positive Euclidean counterpart of the free quadratic source completion in Schwartz 2014, § 14.3, pp. 261–263, with normalization and all signs stated locally here.

The source responses are therefore

$$\mathbb E_J[\Phi]=CJ, \qquad \operatorname{Cov}_J(\Phi)=C.$$

At zero source,

$$\begin{aligned} \mathbb E[\phi]&=\mathbb E[\chi]=0,\\ \kappa(\phi,\phi)&=\frac{c}{\Delta},\\ \kappa(\phi,\chi)&=-\frac{b}{\Delta},\\ \kappa(\chi,\chi)&=\frac{a}{\Delta}, \end{aligned}$$

and every cumulant of order three or higher vanishes because $W_E$ is quadratic.

The fourth derivative of $Z_E$, however, is not zero. For component indices $i,j,k,\ell$,

$$\mathbb E[\Phi_i\Phi_j\Phi_k\Phi_\ell] = C_{ij}C_{k\ell} +C_{ik}C_{j\ell} +C_{i\ell}C_{jk}.$$

The three terms are exactly the three two-block partitions of four centered variables; the connected four-point cumulant is the fourth derivative of $W_E$ and vanishes. This calculation shows why the logarithm selects the one-block contribution without yet invoking a diagrammatic linked-cluster theorem.

There are several independent checks:

• $Z_E[0]=1$, so the source object is normalized.
• $AC=I$, and $C\succ0$, as a covariance matrix must be.
• The response equation $A\mathbb E_J[\Phi]=J$ follows both from completing the square and from differentiating $W_E$.
• Directly differentiating $e^{J^{\mathsf T}CJ/2}$ four times reproduces the three pairings above.

The stop condition is equally explicit. As $\Delta\downarrow0$, one eigenvalue of $A$ approaches zero, the covariance diverges, and $\mathcal Z_E[0]$ ceases to define a normalizable probability measure at the boundary. Interacting continuum fields require a regulator, existence arguments, and renormalization beyond this example.

Common pitfalls and stop rules

Calling every transform an MGF. A characteristic function always exists; an MGF may be finite only at zero or on a one-sided domain. State which exponential and which domain are being used.

Differentiating before checking integrability. A formal derivative of an expectation is not a moment theorem. Supply domination or local exponential integrability, and stop at the highest justified order.

Replacing an actual function by its formal moment series. The lognormal example has all integer moments but no two-sided MGF neighborhood, and its moments do not determine its law. Formal coefficient identities remain finite-order algebra only.

Taking a global logarithm without checking zeros. The branch normalized by $\log1=0$ always exists near the origin, but a characteristic function can vanish elsewhere. Never continue the cumulant logarithm through a zero without a separate branch analysis.

Equating zero covariance with independence. Independence kills every mixed cumulant that straddles independent groups, but a single vanishing mixed cumulant proves very little. Use factorization of the full joint characteristic function for an exact criterion.

Declaring Gaussianity from a few cumulants. Vanishing third and fourth cumulants is not enough. Use the full Gaussian characteristic function or a quadratic generator on a domain that determines the law.

Ignoring normalization or signature. Divide a Euclidean source integral by its zero-source value before reading it as an MGF. Translate the factors of $i$ when moving to the Lorentzian convention, and do not call an oscillatory path integral a probability measure.

Using point fields when only smeared fields exist. In an infinite-dimensional theory, establish the test-function space and the continuity of the functional before interpreting functional derivatives as correlation kernels.

Check your understanding

1. Find a local logarithm that cannot be global

Let $X$ take the values $\pm1$ with equal probability. Compute its characteristic function, MGF, and first four cumulants. Why can the characteristic logarithm not be defined globally on the real line?

Solution

Direct averaging gives

$$\varphi_X(t)=\cos t, \qquad M_X(s)=\cosh s.$$

Near zero,

$$K_X(s)=\log\cosh s =\frac{s^2}{2}-\frac{s^4}{12}+O(s^6).$$

Since $K_X(s)=\sum_{n\geq1}\kappa_n s^n/n!$,

$$\kappa_1=0, \qquad \kappa_2=1, \qquad \kappa_3=0, \qquad \kappa_4=-2.$$

The characteristic function vanishes at $t=\pi/2+k\pi$, so a logarithm normalized at the origin cannot be continued as one finite branch through all real $t$.

2. Track cumulants of an iid average

Suppose $X_1,\ldots,X_n$ are iid and have cumulants through order $r$. Derive the cumulants of $\overline X_n=n^{-1}\sum_{j=1}^nX_j$.

Solution

Independence first gives

$$\kappa_r\!\left( \sum_{j=1}^nX_j \right) =n\kappa_r(X_1).$$

Scaling by $n^{-1}$ then gives

$$\kappa_r(\overline X_n) =n^{-r}n\kappa_r(X_1) =n^{1-r}\kappa_r(X_1).$$

For $r=1$ the mean is unchanged, for $r=2$ the variance falls as $1/n$, and higher unstandardized cumulants fall faster. The calculation uses independence and the existence of the stated cumulants.

3. Separate a centered four-point function

For centered variables $X_1,X_2,X_3,X_4$, use the partition identity to express $\mathbb E[X_1X_2X_3X_4]$ in terms of second and fourth cumulants.

Solution

Every partition containing a singleton contributes a factor $\kappa(X_j)=\mathbb E[X_j]=0$. The surviving partitions are the one-block partition and the three pair partitions, so

$$\begin{aligned} \mathbb E[X_1X_2X_3X_4] ={}&\kappa(X_1,X_2,X_3,X_4)\\ &+\kappa(X_1,X_2)\kappa(X_3,X_4)\\ &+\kappa(X_1,X_3)\kappa(X_2,X_4)\\ &+\kappa(X_1,X_4)\kappa(X_2,X_3). \end{aligned}$$

In the Gaussian example the fourth cumulant is zero, leaving the three pairings. For a non-Gaussian law the one-block term need not vanish.

4. Translate the Lorentzian source factors

Under the convention $Z_L[J]=\int\mathcal D\phi\,e^{iS+i\int J\phi}$, compute the first two derivatives of $W_L[J]=-i\log Z_L[J]$ and identify the connected two-point function.

Solution

At a general source,

$$\frac{\delta W_L}{\delta J(x)} = \langle\phi(x)\rangle_J.$$

Differentiating again gives

$$\frac{\delta^2W_L} {\delta J(x)\delta J(y)} = i\left[ \langle T\phi(x)\phi(y)\rangle_J -\langle\phi(x)\rangle_J \langle\phi(y)\rangle_J \right].$$

Therefore, at zero source,

$$\langle T\phi(x)\phi(y)\rangle_{\mathrm c} = -i \left. \frac{\delta^2W_L} {\delta J(x)\delta J(y)} \right|_{J=0}.$$

The minus-$i$ is the $n=2$ case of $(-i)^{n-1}$; it would be absent in the positive Euclidean convention used in the worked example.

Synthesis and continuations

Use a characteristic function when an always-defined, law-determining transform is required. Use an MGF only on its finite domain, and take moments from derivatives only after justifying differentiation. Normalize before taking a logarithm: the derivatives of the original generator give full moments, while derivatives of the logarithm give the one-block cumulants. Independence turns products into sums and makes cumulants additive. A source then upgrades the same algebra into response theory, with the Hessian of the Euclidean log generator equal to a positive-semidefinite covariance.

Continue to Gaussian Vectors, Processes, Random Distributions, and Wick Structure for the full Gaussian and infinite-dimensional theory. Continue to Connected Correlators and Cumulants for vacuum normalization, physical source derivatives, and the linked-cluster interpretation in QFT.

References

• Martin Z. Bazant, “Lecture 2: Moments, Cumulants, and Scaling”, MIT OpenCourseWare 18.366, Fall 2006, pp. 3–6. This is the teaching source for the progression from characteristic functions to tensor moments, low-order cumulants, and additivity under independent sums.

• Rick Durrett, Probability: Theory and Examples, fifth edition, PDF, Cambridge University Press, 2019. This is the structural probability source. § 3.3.1, pp. 125–31, supports the definition, basic properties, inversion, and uniqueness of characteristic functions; § 3.3.2, pp. 132–33, supports Lévy’s continuity theorem; § 3.3.3, pp. 134–36, supports moment differentiation; and § 3.3.5, pp. 140–43, supports the moment-problem and lognormal counterexamples.

• William Fithian, “Exponential Families”, undated Statistics 210A course notes, University of California, Berkeley, accessed August 11, 2026. The sections “Exponential family structure” and “Differential identities” support the finite natural-parameter domain, exponential tilting, and the gradient-as-mean and Hessian-as-covariance identities.

• Peter McCullagh, Tensor Methods in Statistics, Dover edition, 2017. This is the structural cumulant source. Chapter 2, §§ 2.2–2.7, pp. 29–44, supports the distinction between convergent and finite-order cumulant expansions, the moment–cumulant set-partition formulas, Gaussian cumulants, mixed cumulants across independent blocks, and additivity for independent sums.

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. Section 14.3, pp. 261–63, supports normalized source derivatives and the free quadratic generator; § 34.1.2, pp. 737–39, supports $W[J]=-i\log Z[J]$ and the separation of full and connected Green functions.