A matrix-depth maximizer exists on the bounded covariance classes

Identify the symmetric matrices with their coordinates in ${\mathbb{R}}^{p^2}$. The dense class $K={\mathcal{F}}_p(\beta M)$ is bounded because each entry satisfies $\mathrm{\mid }{\mathrm{\Gamma }}_{ij}\mathrm{\mid }=\mathrm{\mid }{e}_i^T\mathrm{\Gamma }{e}_j\mathrm{\mid }\le \mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \beta M$ by [F3]. It is closed: if ${\mathrm{\Gamma }}_k\to \mathrm{\Gamma }$ coordinatewise with ${\mathrm{\Gamma }}_k\in K$, then $\mathrm{\Gamma }$ is symmetric, $v^T\mathrm{\Gamma }v={\lim }_k{v}^T{\mathrm{\Gamma }}_{k}v\ge 0$ for every $v$, and $\mathrm{\parallel }\mathrm{\Gamma }v{\mathrm{\parallel }}_2={\lim }_k\mathrm{\parallel }{\mathrm{\Gamma }}_{k}v{\mathrm{\parallel }}_2\le \beta M\mathrm{\parallel }v{\mathrm{\parallel }}_2$ for every $v$. By the operator-norm definition in [F2], this gives $\mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \beta M$. Thus $K$ is closed and bounded, so [F3] makes it compact.

Fix $u$ and write $q(\mathrm{\Gamma })=u^T\mathrm{\Gamma }u$ and $r_i=(u^T{x}_i{)}^2$. If ${\mathrm{\Gamma }}_k\to \mathrm{\Gamma }$, then $q({\mathrm{\Gamma }}_k)\to q(\mathrm{\Gamma })$. For each $i$, if $r_i<q(\mathrm{\Gamma })$ or $r_i>q(\mathrm{\Gamma })$, the two weak-tail indicators eventually keep their limiting membership; if $r_i=q(\mathrm{\Gamma })$, the observation is counted in both weak tails at $\mathrm{\Gamma }$. Therefore the limsup of each finite tail count is at most its value at $\mathrm{\Gamma }$, and the limsup of their minimum is at most the minimum of their values there. The minimum of the two empirical tails is upper semicontinuous in $\mathrm{\Gamma }$. The infimum over $u$ remains upper semicontinuous because each strict sublevel set of the infimum is the union of the corresponding open strict sublevel sets from [F4]. Hence $\mathrm{\Gamma }\mapsto D_U(\mathrm{\Gamma },P_n)$ is upper semicontinuous on either search class.

For each fixed $S\subseteq [p]$ with $\mathrm{\mid }S\mathrm{\mid }\le s$, requiring the off-diagonal entries outside $S\times S$ to vanish is a finite collection of closed coordinate equalities. Its intersection with the dense compact class is compact by [F5]. There are finitely many such $S$, and their union is ${\mathcal{F}}_{p,s}(\beta M)$ by [F2]. A finite union of compact sets is compact, since any open cover has a finite subcover on each of its finitely many members.

Both applicable search classes are nonempty because they contain the zero matrix. By steps 1.1–1.2 they are compact Euclidean sets. By step 1.2, their empirical depth functions are upper semicontinuous, so [F5] gives a maximizer on each applicable class. Its argmax is the nonempty weak superlevel set at that maximum, closed by [F4] and compact by [F5]. This proves the claim.