Severely ill-posed analytic q=2 detection rate

Let $m$ be the largest active index and put $t=A{e}^{2\alpha m}\in [e^{-2\alpha },1)$. For $j=m-\mathrm{\ell }$, the profile factor is ${\rho }_{\mathrm{\ell }}=1-t{e}^{-2\alpha \mathrm{\ell }}$. For every $\mathrm{\ell }\ge 1$, $1-e^{-2\alpha }\le {\rho }_{\mathrm{\ell }}\le 1$, while $0\le {\rho }_0\le 1$. The sums for $J_1$ and $J_0$ are geometric in $\mathrm{\ell }$; their $\mathrm{\ell }=1$ terms give positive lower bounds and the full geometric sums give upper bounds, so $J_1\asymp J_0\asymp e^{4\beta m}$. In $J_2$, the summands are $t{e}^{-(2\alpha +4\beta )\mathrm{\ell }}{\rho }_{\mathrm{\ell }}{e}^{4\beta m}$; the $\mathrm{\ell }=1$ term is bounded below by a positive constant times $e^{4\beta m}$, and the geometric series gives the matching upper bound. Thus $J_2\asymp e^{4\beta m}$. By [F1], $r^2\asymp A\asymp e^{-2\alpha m},u_{\epsilon }^2\asymp {\epsilon }^{-4}{r}^4{e}^{-4\beta m}\asymp {\epsilon }^{-4}{r}^{4+4\beta \mathrm{/}\alpha }\mathrm{.}$

The last expression is comparable to $(r\mathrm{/}{\epsilon }^{\alpha \mathrm{/}(\alpha +\beta )}{)}^{4+4\beta \mathrm{/}\alpha }$. Thus radii below the displayed scale by a factor tending to zero have $u_{\epsilon }\to 0$, and radii above it by a factor tending to infinity have $u_{\epsilon }\to \mathrm{\infty }$ when $r_{\epsilon }\to 0$. If $r_{\epsilon }$ does not tend to zero, reduce to ${\bar{r}}_{\epsilon }=\sqrt{r_{\epsilon }{r}_{\epsilon }^{\ast }}$. Nonempty alternatives satisfy $r_{\epsilon }^2\le a_1^{-2}$ by the two q=2 constraints, so ${\bar{r}}_{\epsilon }\to 0$ and ${\bar{r}}_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to \mathrm{\infty }$. The inclusion ${\mathrm{\Theta }}_2(r_{\epsilon })\subseteq {\mathrm{\Theta }}_2({\bar{r}}_{\epsilon })$ transfers consistent tests to the original class. The risk implications in [F1] establish the separation rate.