Smeared Wick powers have a two-dimensional L² limit

For fixed $ϵ,\delta >0$, the smoothed fields at $x,y$ are jointly centered Gaussian by [F1], and their cross-covariance is $C_{ϵ,\delta }(x-y)$ by [F2]. Apply [F3] to obtain $\mathbb{E}[\overline{:\text{\hspace{0.17em}⁣}{\mathrm{\Phi }}_{ϵ}(x{)}^n\text{\hspace{0.17em}⁣}:}:\text{\hspace{0.17em}⁣}{\mathrm{\Phi }}_{\delta }(y{)}^n\text{\hspace{0.17em}⁣}:]=n!C_{ϵ,\delta }(x-y{)}^n$. For fixed cutoffs, each variance is finite and independent of $x$ by translation invariance; Cauchy–Schwarz [F7] therefore bounds the absolute expected product by a finite constant on the bounded support of $f(x)g(y)$. Fubini [F4] applies and yields $\mathbb{E}[\overline{W_{ϵ,n}(f)}{W}_{\delta ,n}(g)]=n!\int \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\int \overline{f(x)}g(y)C_{ϵ,\delta }(x-y{)}^n{d}^{2}x{d}^{2}y$. For measurability of the samplewise smearing, on a fixed bounded box containing the support replace the continuous function $x\mapsto :\text{\hspace{0.17em}⁣}{\mathrm{\Phi }}_{ϵ}(x{)}^n\text{\hspace{0.17em}⁣}:$ on each fine grid cell by its value at one fixed point of that cell. Its integral against $f$ is a finite sum of measurable evaluations with deterministic coefficients ${\int }_{\text{cell}}f$; uniform continuity on the box makes the sums converge to the samplewise integral. The same approximations make the field jointly measurable in sample and position, so Fubini applies to measurable functions.

Let $K_f,K_g$ be bounded supports of $f,g$. By [F2], $C_{ϵ,\delta }(z{)}^n\to G_m(z{)}^n$ for every $z\mathrm{\ne }0$ and, on the bounded difference set $K_f-K_g$, its absolute value is bounded by $K(1+\mathrm{\mid }\log \mathrm{\mid }z\mathrm{\mid }\mathrm{\mid }{)}^n$ for all sufficiently small cutoffs. This majorant is locally integrable: on the annulus $2^{-j-1}<\mathrm{\mid }z\mathrm{\mid }\le 2^{-j}$ its integral is at most $\pi 2^{-2j}(1+(j+1)\log 2{)}^n$, and the resulting series converges by the ratio test. Consequently $(x,y)\mapsto \mathrm{\mid }f(x)g(y)\mathrm{\mid }(1+\mathrm{\mid }\log \mathrm{\mid }x-y\mathrm{\mid }\mathrm{\mid }{)}^n$ is integrable on $K_f\times K_g$: for fixed $x$, the translated bounded set $x-K_g$ lies in one fixed disk on which the logarithmic power is integrable, uniformly in $x$, and $K_f$ has finite area. For every sequence $({ϵ}_k,{\delta }_k)\to (0,0)$, apply [F5] on this product to get $\mathbb{E}[\overline{W_{{ϵ}_k,n}(f)}{W}_{{\delta }_k,n}(g)]\to n!\int \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\int \overline{f(x)}g(y)G_m(x-y{)}^n{d}^{2}x{d}^{2}y$. Sequential characterization of limits in ${\mathbb{R}}^2$ gives the same joint limit as $ϵ,\delta \downarrow 0$.

Take $g=f$. The right side of step 2.1 is a fixed finite real number, being the limit of the nonnegative diagonal covariances. Expanding the squared norm gives $\mathrm{\parallel }{W}_{ϵ,n}(f)-W_{\delta ,n}(f){\mathrm{\parallel }}_2^2=\mathbb{E}\mathrm{\mid }{W}_{ϵ,n}(f){\mathrm{\mid }}^2+\mathbb{E}\mathrm{\mid }{W}_{\delta ,n}(f){\mathrm{\mid }}^2-2\mathrm{Re}\mathbb{E}[\overline{W_{ϵ,n}(f)}{W}_{\delta ,n}(f)]$, which tends to zero by step 2.1. Thus the cutoff family is Cauchy in $L^2({\mu }_C)$, and [F6] supplies its limit $W_n(f)$. Applying the mixed covariance convergence in step 2.1 and [F7] to pass both factors to their $L^2$ limits proves the displayed covariance formula for $W_n(f),W_n(g)$. For real $f$, each cutoff smearing is real; continuity of the imaginary-part map in $L^2$ gives $\mathrm{\parallel }\mathrm{Im}{W}_n(f){\mathrm{\parallel }}_2\le \mathrm{\parallel }{W}_n(f)-W_{ϵ,n}(f){\mathrm{\parallel }}_2\to 0$, so the limit class is real.