The multivariate trimmed-mean rate follows from a correct-scale slab event

Since $K_{i_0}\mathrm{\ne }\mathrm{\varnothing }$, $i_0\in S$. If $S=\mathbb{Z}$, [F1] puts the algorithm output in ${\bigcap }_{i\in \mathbb{Z}}\mathrm{\Gamma }(2^i)\subseteq K_{i_0}$. If $S$ is nonempty and proper, its least element $j_{\ast }$ satisfies $j_{\ast }\le i_0$, so nesting in [F1] gives $K_{j_{\ast }}\subseteq K_{i_0}$ and the output again belongs to $K_{i_0}$. Thus the event condition implies $\mathrm{\parallel }\widehat{\mu }-\mu {\mathrm{\parallel }}_2\le b\epsilon Q_0$.

The definition of $Q_0$ gives $b\epsilon Q_0\le 256b\sqrt{\mathrm{tr}\mathrm{\Sigma }\mathrm{/}N}+16b\sqrt{{\lambda }_{\max }(\mathrm{\Sigma })\epsilon }$. Since $\epsilon \le 10\eta +2560\log (2\mathrm{/}\delta )\mathrm{/}N$, square-root subadditivity and [F3] bound the second term by $16b\sqrt{10{\lambda }_{\max }(\mathrm{\Sigma })\eta }+16b\sqrt{2560{\lambda }_{\max }(\mathrm{\Sigma })\log (1\mathrm{/}\delta )\mathrm{/}N}+16b\sqrt{2560{\lambda }_{\max }(\mathrm{\Sigma })\log 2\mathrm{/}N}$. By [F2] the final term is at most $16b\sqrt{2560\log 2}\sqrt{\mathrm{tr}\mathrm{\Sigma }\mathrm{/}N}$. These three coefficients are bounded by $c_b$, giving the stated implication. At $b=4$ this is exactly the previous constant.