Compactly supported Gaussian mixtures admit logarithmic-size finite approximations

Partition $[-A,A]$ into $K\le 2A\mathrm{/}\sigma +1$ consecutive Borel intervals $I_i$ of length at most $\sigma$, with left endpoints $a_i$. Ignore intervals of zero $F$-mass and let $G_i$ be the law of $W_i=(Z-a_i)\mathrm{/}\sigma$ conditional on $Z\in I_i$. Let ${\mathrm{\ell }}_i$ be the length of $I_i$ and put $b_i={\mathrm{\ell }}_i\mathrm{/}\sigma \le 1$, so $G_i$ is supported on $[0,b_i]$. Approximate the moment vector $(\int wd{G}_i,\dots ,\int w^{m}d{G}_i)$ by finite sums obtained from partitions of $[0,b_i]$; it is therefore in the closure of the convex hull of the compact moment curve $w\mapsto (w,\dots ,w^m)$ on that interval. Every finite convex combination with more than $m+1$ points can be reduced: affine dependence supplies coefficients $c_j$, not all zero, with $\sum c_j=0$ and $\sum c_j{x}_j=0$; moving the weights along this relation until one becomes zero preserves nonnegativity and the represented point, so iteration gives at most $m+1$ points. Thus the convex hull is the continuous image of a compact set consisting of $m+1$ curve points and simplex weights; it is compact and closed. The moment vector belongs to it, giving a probability $G_i^{\mathrm{\prime }}$ on at most $m+1$ points of $[0,b_i]$ matching the first $m$ moments of $G_i$.

The Hermite recurrence [F3] and differentiation of $\varphi$ give ${\varphi }^{(r)}=(-1{)}^r{H}_r\varphi$ for every integer $r\ge 0$. The case $r=0$ is immediate. If it holds at orders $r$ and $r-1$, differentiating $H_r$'s recurrence gives $H_{r+1}^{\mathrm{\prime }}=H_r+x{H}_r^{\mathrm{\prime }}-r{H}_{r-1}^{\mathrm{\prime }}=H_r+r(x{H}_{r-1}-(r-1)H_{r-2})=(r+1)H_r$; the base cases $H_0^{\mathrm{\prime }}=0$ and $H_1^{\mathrm{\prime }}=1$ start this induction. Thus differentiating $(-1{)}^r{H}_r\varphi$ gives $(-1{)}^{r+1}{H}_{r+1}\varphi$. Since $H_r$ is monic, $H_r^{(r)}=r!$. Apply integration by parts [F7] repeatedly on $[-T,T]$ and let $T\to \mathrm{\infty }$; the boundary terms vanish because each derivative is a polynomial times the Gaussian. This gives $\int H_r^2\varphi =(-1{)}^r\int H_r{\varphi }^{(r)}=\int H_r^{(r)}\varphi =r!$. Cauchy--Schwarz [F4] therefore gives $\mathrm{\parallel }{\varphi }^{(r)}{\mathrm{\parallel }}_1\le \sqrt{r!}$. Also ${\varphi }^{(r)}$ tends to zero at $-\mathrm{\infty }$, so [F8] on $[-T,x]$, followed by $T\to \mathrm{\infty }$, gives $\mathrm{\mid }{\varphi }^{(r)}(x)\mathrm{\mid }\le {\int }_{-\mathrm{\infty }}^x\mathrm{\mid }{\varphi }^{(r+1)}(t)\mathrm{\mid }dt\le \sqrt{(r+1)!}$ and $\mathrm{\parallel }{\varphi }^{(r)}{\mathrm{\parallel }}_{\mathrm{\infty }}\le \sqrt{(r+1)!}$.

Repeated application of [F8] gives the Taylor integral remainder for $w\mapsto \varphi (x-w)$ through degree $m-1$: $R_m(x,w)=\frac{(-w{)}^m}{(m-1)!}{\int }_0^1(1-t{)}^{m-1}{\varphi }^{(m)}(x-tw)dt$. By translation invariance and Tonelli [F6], $\mathrm{\parallel }{R}_m(\cdot ,w){\mathrm{\parallel }}_1\le w^m\mathrm{\parallel }{\varphi }^{(m)}{\mathrm{\parallel }}_1\mathrm{/}m!\le 1\mathrm{/}\sqrt{m!}$; the same formula gives $\mathrm{\parallel }{R}_m(\cdot ,w){\mathrm{\parallel }}_{\mathrm{\infty }}\le w^m\mathrm{\parallel }{\varphi }^{(m)}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{/}m!\le \sqrt{m+1}\mathrm{/}\sqrt{m!}$ by step 1.2. Moment matching in step 1.1 cancels the polynomial terms after integration against $G_i-G_i^{\mathrm{\prime }}$. Integrating the remainder bounds against these two probability laws gives $\mathrm{\parallel }{p}_{G_i,1}-p_{G_i^{\mathrm{\prime }},1}{\mathrm{\parallel }}_1,\mathrm{\parallel }{p}_{G_i,1}-p_{G_i^{\mathrm{\prime }},1}{\mathrm{\parallel }}_{\mathrm{\infty }}\le 2\sqrt{m+1}\mathrm{/}\sqrt{m!}$. The bound $\log (m!)={\sum }_{k=1}^m\log k\ge {\int }_1^m\log xdx=m\log m-m+1$ shows this error is at most $\eta$ when $m=\lceil c\log (e\mathrm{/}\eta )\rceil$ for a sufficiently large universal $c$, with $\eta \in (0,1\mathrm{/}2)$ absorbed by increasing $c$.

Let ${\nu }_i$ and ${\nu }_i^{\mathrm{\prime }}$ be the pushforwards of $G_i$ and $G_i^{\mathrm{\prime }}$, respectively, under $w\mapsto a_i+\sigma w$, and set $F^{\mathrm{\prime }}={\sum }_{i}F(I_i){\nu }_i^{\mathrm{\prime }}$. Since $G_i^{\mathrm{\prime }}$ is supported on $[0,b_i]$, every ${\nu }_i^{\mathrm{\prime }}$ is supported in the closure of $I_i$, so $F^{\mathrm{\prime }}$ is a discrete probability measure on $[-A,A]$ with at most $K(m+1)\le C(A\mathrm{/}\sigma \vee 1)\log (e\mathrm{/}\eta )$ support points. Translation and scaling give $p_{{\nu }_i,\sigma }(x)={\sigma }^{-1}{p}_{G_i,1}((x-a_i)\mathrm{/}\sigma )$ and the same identity for ${\nu }_i^{\mathrm{\prime }}$ and $G_i^{\mathrm{\prime }}$. Summing these component errors with weights $F(I_i)$ shows the $L^1$ error is at most $\eta$ and the supremum error at most $\eta \mathrm{/}\sigma$.