Hellinger entropy for compact-location normal mixtures in a scale band

Put $K:=\lceil C_0(A\mathrm{/}\tau \vee 1)\log (e\mathrm{/}ϵ)\rceil$. By [F1], every $p_{F,\sigma }\in {\mathcal{P}}_{A,\tau }$ is within $L^1$ distance ${ϵ}^2\mathrm{/}{C}_1$ of a mixture with at most $K$ atoms, for a fixed sufficiently large $C_1$. Pad shorter mixtures by zero-weight atoms.

Differentiating the density in [F3] gives $\mathrm{\parallel }{\mathrm{\partial }}_z{\varphi }_{\sigma }(\cdot -z){\mathrm{\parallel }}_1={\sigma }^{-1}E\mathrm{\mid }Z\mathrm{\mid }\le {\sigma }^{-1}$, since $2\mathrm{\mid }Z\mathrm{\mid }\le 1+Z^2$ and $E{Z}^2=1$ by [F5], and $\mathrm{\parallel }{\mathrm{\partial }}_{\sigma }{\varphi }_{\sigma }(\cdot -z){\mathrm{\parallel }}_1={\sigma }^{-1}E\mathrm{\mid }{Z}^2-1\mathrm{\mid }\le 2{\sigma }^{-1}$ for $Z\sim N(0,1)$. Integrating these derivatives along the location and scale segments by [F7] yields $\mathrm{\parallel }{\varphi }_{\sigma }(\cdot -z)-{\varphi }_{\sigma }(\cdot -z^{\mathrm{\prime }}){\mathrm{\parallel }}_1\le \mathrm{\mid }z-z^{\mathrm{\prime }}\mathrm{\mid }\mathrm{/}\sigma$ and $\mathrm{\parallel }{\varphi }_{\sigma }-{\varphi }_{{\sigma }^{\mathrm{\prime }}}{\mathrm{\parallel }}_1\le 2\mathrm{\mid }\sigma -{\sigma }^{\mathrm{\prime }}\mathrm{\mid }\mathrm{/}(\sigma \wedge {\sigma }^{\mathrm{\prime }})$. For the first class use a scale grid in $[\tau ,2\tau ]$; for an actual band, instead grid $[\tau ,\min \{2\tau ,s_n\}]$, omitting $2\tau$ when that endpoint is excluded. In either case choose grid mesh at most $c_2\tau {ϵ}^2$, with every grid point admitted by the corresponding scale constraint. Use a location grid in $[-A,A]$ with mesh at most $c_2\tau {ϵ}^2$ and an $L^1$ grid on the $K$-simplex of mesh at most $c_2{ϵ}^2$. Replacing the scale, locations and weights changes the mixture density in $L^1$ by at most $C{c}_2{ϵ}^2$: integrate the componentwise location and scale bounds against the finite mixing weights, and use Tonelli [F6] for the componentwise integrals. Choose $c_2$ so this plus the approximation from step 1.1 is at most ${ϵ}^2\mathrm{/}4$. Thus the resulting finite set, with centers in the class being covered, is an $L^1$ cover of radius ${ϵ}^2\mathrm{/}4$, hence an $h$-cover of radius $ϵ\mathrm{/}2$ by [F2].

The scale grid has at most $C{ϵ}^{-2}$ points and the location grid at most $C(1+A\mathrm{/}(\tau {ϵ}^2))$ points. Rounding each of the first $K-1$ weights down to a mesh proportional to ${ϵ}^2\mathrm{/}K$ leaves the last weight nonnegative after setting it to make the sum one, gives at most $(CK\mathrm{/}{ϵ}^2{)}^K$ weight vectors, and changes their $L^1$ distance by at most $2c_2{ϵ}^2$. Therefore the logarithm of the cover size is at most $C\log (e\mathrm{/}ϵ)+K[\log \text{\hspace{0.17em}⁣}\left(\frac{C(A\mathrm{/}\tau \vee 1)}{{ϵ}^2}\right)+\log \text{\hspace{0.17em}⁣}\left(\frac{CK}{{ϵ}^2}\right)]\le C(A\mathrm{/}\tau \vee 1){[\log \text{\hspace{0.17em}⁣}\left(\frac{C(A\mathrm{/}\tau \vee 1)}{ϵ}\right)]}^2\mathrm{.}$ The constructed finite set is an $h$-net of radius $ϵ\mathrm{/}2$, hence also a net of radius $ϵ$; its cardinality therefore bounds the covering number and proves the first bound.

If $r:=F([-A,A{]}^c)=0$, then $F$ already belongs to the compact-location class. If $0<r\le \rho <1$, let $F_A$ and $F_{A^c}$ be the normalized restrictions to $[-A,A]$ and its complement. Then $p_{F,\sigma }=(1-r)p_{F_A,\sigma }+r{p}_{F_{A^c},\sigma }$, so $\mathrm{\parallel }{p}_{F,\sigma }-p_{F_A,\sigma }{\mathrm{\parallel }}_1\le 2r$ by integrating the mixture densities (Tonelli [F6]). By [F2], $h(p_{F,\sigma },p_{F_A,\sigma })\le \sqrt{2r}$; the triangle inequality in [F4] then transfers the cover to the tail-constrained class. The scale grid is restricted as in step 2.1, so every center has $F([-A,A{]}^c)=0$ and satisfies the band and cutoff constraints. For sufficiently small universal $c$, $\rho \le c{ϵ}^2$ makes the truncation error at most $ϵ\mathrm{/}2$, so the radius-$ϵ\mathrm{/}2$ cover from steps 2.1 and 3.1 becomes a radius-$ϵ$ cover after the constant rescaling used in the entropy bound.