Covariance, diagonal, and normalized-kernel criteria for Gaussian entire functions

Definitions and mathematical statement

A centered proper complex Gaussian entire function $G$ is a random entire function whose finite evaluation vectors are centered complex Gaussian and satisfy ${\mathbb{E}}[G(z)G(w)]=0$. Its covariance kernel is $K(z,w)={\mathbb{E}}[G(z)\overline{G(w)}]$. A kernel is Hermitian positive semidefinite if $K(w,z)=\overline{K(z,w)}$ and \[ \sum_{i,j=1}^m c_i\overline{c_j}K(z_i,z_j)\geq0 \] for every finite choice of points and complex coefficients. It is sesquientire if it is entire in $z$ and anti-entire in $w$.

Question 7 (Steven Evans). Is there a necessary-and-sufficient condition for a given correlation function to be the correlation function of an entire Gaussian function? The original prints "enire"; the evident word correction is "entire." "Correlation" is not defined: covariance values, normalized covariance, nonproper Gaussian data, and correlations of the zero point process are different targets.

Reported covariance criterion. Under the centered proper covariance interpretation, the aid states \[ K\text{ is the covariance of a Gaussian entire function} \quad\Longleftrightarrow\quad K\text{ is Hermitian positive semidefinite and sesquientire}. \] For a sesquientire expansion $K(z,w)=\sum_{m,n\geq0}a_{mn}z^m\overline w^{,n}$, positivity is equivalently positivity of every finite leading Taylor matrix $(a_{mn})_{0\leq m,n\leq M}$. The expansion and global analyticity remain required; positivity of formal matrices alone does not supply a globally entire kernel. The reported realization is an almost surely locally uniformly convergent series \[ G(z)=\sum_j\xi_j e_j(z), \] where $(e_j)$ is an orthonormal basis for the reproducing-kernel Hilbert space of $K$ and the $\xi_j$ are independent standard proper complex Gaussians.

Reported normalized-covariance criterion. Suppose $R:{\mathbb{C}}\times{\mathbb{C}}\to{\mathbb{C}}$ is continuous, Hermitian positive semidefinite, and $R(z,z)=1$. Fix $z_0\in{\mathbb{C}}$ with $r(z)=R(z,z_0)\ne0$ for all $z$, and define \[ C(z,w)=\frac{R(z,w)}{r(z)\overline{r(w)}}. \] The aid asserts that $R$ is of the form \[ R(z,w)=\frac{K(z,w)}{\sqrt{K(z,z)K(w,w)}} \] for a centered proper Gaussian entire function with $K(z,z)>0$ everywhere if and only if $C$ is sesquientire and a continuous argument $\theta$ of $r$, meaning $r(z)=|r(z)|e^{i\theta(z)}$, is harmonic on ${\mathbb{C}}$. This harmonic-phase requirement is not replaced merely by positive semidefiniteness.

The supplied explicit obstruction is the rank-one kernel \[ R_*(z,w)=\exp\big(i(|z|^2-|w|^2)\big). \] It is continuous, Hermitian positive semidefinite, and unit-diagonal, and its transformed $C$ is identically $1$. Nevertheless its base-section phase $|z|^2-|z_0|^2$ is not harmonic, so the report says it cannot be such a normalized covariance.

The summary additionally reports exact diagonal and real-translation-stationary criteria without printing their statements; these cannot be reconstructed from the supplied data. For nonproper centered complex Gaussians, a covariance alone does not specify the pseudo-covariance ${\mathbb{E}}[G(z)G(w)]$. If the question concerns zero-process correlation functions, the preceding covariance criteria do not answer that distinct realizability question. The registry therefore records partial progress rather than a resolution of every reading.

Short English statement

The original correlation-function question is ambiguous. For ordinary proper Gaussian covariance, the reported answer is positivity plus holomorphic/antiholomorphic dependence. For normalized covariance with a nowhere-zero base section, the aid adds an exact harmonic-phase test and a rank-one example showing why positivity alone is insufficient. Zero-process and nonproper interpretations remain separate.

Sources

• Local UnsolvedMath registry, permanent ID $20000632$, catalogue entry; canonical aim-analysis-notes.json, notes[163].
• AIM workshop "Random analytic functions," April 2006, Question 7 attributed to Steven Evans: https://aimath.org/WWN/randomzeros/arcc1.pdf. The supplied verification says the two-page PDF was inspected, while registry statement status remains reconstructed_unverified; no independent PDF inspection is claimed here.
• Research attempt 1: partial_result, run status partially_solved, internal rigor proved, low novelty confidence. Explicitly a machine-generated research aid, not peer reviewed; independently verify before citation. No original literature entry or full external theorem references are supplied.