Sparse q<2 detection regime split

The finite coordinate density of $X_i$ with mean ${\mu }_i$ divided by its null density is $\exp ({\mu }_i{X}_i-{\mu }_i^2\mathrm{/}2)$. Thus a finite uniform mixture of the two signs at a coordinate of absolute mean $z$ has likelihood $e^{-z^2\mathrm{/}2}\cosh (z{X}_i)$, and the prior with probabilities $1-g,g\mathrm{/}2,g\mathrm{/}2$ at $0,z,-z$ has likelihood $1+g{\xi }_z(X_i)$, where ${\xi }_z(t)=e^{-z^2\mathrm{/}2}\cosh (zt)-1$. Multiplication over finitely many coordinates is justified by [F7]. Off these coordinates the law is unchanged; equality first on finite cylinders and then [F9] proves the likelihood identity on the whole sequence sample space. Every prior below has finite coordinate support; no new countable product probability law is constructed.

Completing the square and applying [F5] to the given unit density gives $\mathbb{E}{e}^{tG}=e^{t^2\mathrm{/}2}$. Consequently, by expanding the hyperbolic functions, $\mathbb{E}{\xi }_z(G+\mu )=\cosh (z\mu )-1,\mathrm{Var}{\xi }_z(G+\mu )=\cosh (z^2)-1+(e^{z^2}-1){\sinh }^2(z\mu )\mathrm{.}$ For fixed $B$, $\mathbb{E}{\xi }_z(G{)}^4\le C_B(\cosh (z^2)-1{)}^2$ when $0<z\le B$: the bound $\mathrm{\mid }{\xi }_z(G)\mathrm{\mid }\le C_B{z}^2(1+G^2)e^{B\mathrm{\mid }G\mathrm{\mid }}$ follows from the Taylor bound for $\cosh$ and $e^{-z^2\mathrm{/}2}$, its fourth-power integral is finite by completing the square, and $\cosh (z^2)-1\ge z^4\mathrm{/}2$. We will also use $\mathbb{P}(\mathrm{\mid }G\mathrm{\mid }>t)\le 2e^{-t^2\mathrm{/}2}$, obtained from [F6] and the same exponential moment, and the elementary Mills bounds $\bar{\mathrm{\Phi }}(t)\le \varphi (t)\mathrm{/}t$ and $\bar{\mathrm{\Phi }}(t)\ge \varphi (t+1\mathrm{/}t)\mathrm{/}t$ for $t\ge 1$. The first follows by integrating $x\varphi (x)\mathrm{/}t$ over $[t,\mathrm{\infty })$, and the second by integrating $\varphi$ over $[t,t+1\mathrm{/}t]$.

First suppose $\lambda >0$ and set $L=qs-2\beta >0$, $\kappa =L\mathrm{/}(2-q)$. Use [F2] to define the unique positive $n,h_0$ by $c_1{h}_0{n}^{2\beta +1}=r^2{\epsilon }^{-2},c_2{h}_0{n}^{qs+1}={\epsilon }^{-q},U^2=c_0{h}_0^{2}n,$ and put $z_i=z(i\mathrm{/}n)$, $h_i=h_{0}d(i\mathrm{/}n)$. Then $n^L=(c_1\mathrm{/}{c}_2){\epsilon }^{2-q}\mathrm{/}{r}^2$. The scalar certificate in [F2] gives ${\sum }_{i\le m}{h}_i(\cosh (z_i{v}_i)-1)\ge \frac{h_0}{2}(n^{-2\beta }{\sum }_{i\le m}{i}^{2\beta }{v}_i^2-n^{-qs}{\sum }_{i\le m}{i}^{qs}\mathrm{\mid }{v}_i{\mathrm{\mid }}^q)\mathrm{.}$ Also $\sum h_i<\mathrm{\infty }$ and $V:=\sum h_i^2(\cosh (z_i^2)-1)\sim U^2$ as $n\to \mathrm{\infty }$. These are score coefficients, which need not be at most one. On compact subsets of $0<r\mathrm{/}{r}_{\epsilon }^{\ast }<\mathrm{\infty }$, $U$ stays bounded and bounded away from zero, $n$ grows as a positive power of ${\epsilon }^{-1}$, and $h_0\asymp n^{-1\mathrm{/}2}$.

In the Gaussian branch, with the profile notation of step 1.3, the extremal value in the Statement is exactly $J_{\epsilon }(\tilde{r})$ in [F2], because $\cosh (z^2)-1=2{\sinh }^2(z^2\mathrm{/}2)$. That lemma proves its power equivalent whenever $\tilde{r}=o({\epsilon }^{a_c})$, equivalently when its associated $n\to \mathrm{\infty },h_0\to 0$. Since $\tilde{r}\mathrm{/}r\to 1$, this is the asserted restriction and the equivalent may be written with $r$. In particular on each compact transition range, $u_{\epsilon }\sim U=c_{\alpha ,\beta ,q}{\epsilon }^{-A\mathrm{/}D}{r}^{B\mathrm{/}D}\mathrm{.}$ No equivalent outside the stated sufficient zone is used.

In the Gaussian branch, with the profile notation of step 1.3, for the Gaussian lower bound, fix a compact transition range. The feasible construction in [F2] can be made with expected energy $(1+\tau )r^2{\epsilon }^{-2}$ and expected budget $(1-\tau ){\epsilon }^{-q}$, on a finite window $a\le i\mathrm{/}{n}^{\mathrm{\prime }}\le b$, with probabilities $g_i=h_0^{\mathrm{\prime }}d(i\mathrm{/}{n}^{\mathrm{\prime }})\le 1$ and amplitudes $z(i\mathrm{/}{n}^{\mathrm{\prime }})$. Let the margins tend slowly to zero and windows tend slowly to $(0,\mathrm{\infty })$. This can be specified without a rate assumption: use windows $[1\mathrm{/}k,k]$, margin $1\mathrm{/}k$, and pass to each next integer $k$ only after its finite Riemann errors, feasibility bounds and the concentration errors below are smaller than the prescribed margins; the preceding compact-window estimates hold uniformly on the chosen compact transition range. The profile moments $M_1,M_2$ give relative variances $O((n{h}_0{)}^{-1})=O(n^{-1\mathrm{/}2})$ for both random constraint sums before the slowly varying window constants. Choose the numerical diagonal slowly enough that those relative variances divided by ${\tau }^2$ tend to zero. Chebyshev then puts prior probability $1-o(1)$ on the actual alternative. Each draw has finite coordinate support and hence is in ${\mathrm{\ell }}^2$. The prior cost tends to $U^2$, its largest coordinate cost tends to zero, and its cost on $z_i>B$ tends uniformly to zero as $B\to \mathrm{\infty }$, by the bounded small-end and exponential large-end envelopes in [F2]. Let $\mathrm{\Pi }$ denote this finite-support prior on parameter sequences, let $\mathcal{A}$ be the event that a draw satisfies both alternative constraints, and put $p=\mathrm{\Pi }(\mathcal{A})=1-o(1)$. If $p=1$, conditioning changes no mixture law. Otherwise, eventually $0<p<1$ and the unconditioned observation law decomposes as $P_{\mathrm{\Pi }}=p{P}_{\mathrm{\Pi }\mid \mathcal{A}}+(1-p)P_{\mathrm{\Pi }\mid {\mathcal{A}}^c}$. For every measurable event $E$, this gives $\mathrm{\mid }{P}_{\mathrm{\Pi }}(E)-P_{\mathrm{\Pi }\mid \mathcal{A}}(E)\mathrm{\mid }=(1-p)\mathrm{\mid }{P}_{\mathrm{\Pi }\mid {\mathcal{A}}^c}(E)-P_{\mathrm{\Pi }\mid \mathcal{A}}(E)\mathrm{\mid }\le 1-p$, so [F11] yields $\mathrm{\parallel }{P}_{\mathrm{\Pi }}-P_{\mathrm{\Pi }\mid \mathcal{A}}{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le 1-p$. For every randomized test $\psi$ with values in $[0,1]$, the functions $N^{-1}{\sum }_{k=1}^N{\mathbf{1}}_{\{\psi \ge k\mathrm{/}N\}}$ approximate $\psi$ uniformly within $1\mathrm{/}N$; applying the event bound to each summand and letting $N\to \mathrm{\infty }$ gives $\mathrm{\mid }{\mathbb{E}}_{P_{\mathrm{\Pi }}}\psi -{\mathbb{E}}_{P_{\mathrm{\Pi }\mid \mathcal{A}}}\psi \mathrm{\mid }\le 1-p=o(1)$ uniformly over tests.

In the Gaussian branch, with the profile notation of step 1.3, for the matching upper bounds define the product-Borel score $F(X)={\sum }_{i\ge 1}{h}_i{e}^{-z_i^2\mathrm{/}2}\cosh (z_i{X}_i)-{\sum }_{i\ge 1}{h}_i\mathrm{.}$ The first series is nonnegative and may be infinite; then the test rejects. Under the null its expectation is $\sum h_i<\mathrm{\infty }$, so it is finite almost surely by [F6], and its centered partial sums converge in mean square with variance $V$. The bounded-amplitude fourth-moment calculation in step 1.2 and [F3], followed by the null variance tail bound in [F2], give $F\Rightarrow N(0,U_0^2)$. For $n$ large put $Q_i^2=2\log (i+1)+4\log \log (i+3)+4\log \log n,\mathcal{Y}=\{\mathrm{\exists }i:\mathrm{\mid }{X}_i\mathrm{\mid }>Q_i\}\mathrm{.}$ Step 1.2 and a countable union bound give ${\mathbb{P}}_0(\mathcal{Y})\le 2(\log n{)}^{-2}{\sum }_{i\ge 1}[(i+1)(\log (i+3){)}^2{]}^{-1}=o(1)$. For every fixed $K$, ${\max }_{i\le n(\log n{)}^K}{Q}_i=\sqrt{2\log n}+o(\sqrt{\log n})$. Reject on $\{F>H_{\mathrm{\ell }}\sqrt{V}\}\cup \mathcal{Y}$ for the significance problem, and on $\{F>V\mathrm{/}2\}\cup \mathcal{Y}$ for total error. These are unions with a centered-cosh score, rather than an intersection or a positive-exponential uncentered score.

Now suppose $\lambda \le 0$, put $e=2\beta -qs\ge 0$, $n=r^{-1\mathrm{/}\alpha }$, $m=r^{s\mathrm{/}\alpha }$, $A_{\epsilon }=m\mathrm{/}\epsilon$ and $D_{\epsilon }=A_{\epsilon }-\sqrt{2\log n}$. Every alternative has an index $i\le \lfloor n\rfloor$ with $\mathrm{\mid }{\eta }_i\mathrm{\mid }\ge m$. Indeed with $w_i=i^{qs}\mathrm{\mid }{\eta }_i{\mathrm{\mid }}^q$, its energy summand is $w_i{i}^e\mathrm{\mid }{\eta }_i{\mathrm{\mid }}^{2-q}$. If $i\le n$ and $\mathrm{\mid }{\eta }_i\mathrm{\mid }<m$, this multiplier is strictly below $n^e{m}^{2-q}=r^2$. If $i>n$, the termwise budget gives $\mathrm{\mid }{\eta }_i\mathrm{\mid }\le i^{-s}$, so the multiplier is at most $i^{-2\alpha }<r^2$. If all the indicated coordinates were smaller than $m$, summing with $\sum w_i\le 1$ would give energy strictly below $r^2$, a contradiction. Let $N=\lfloor n\rfloor$ and $t_N=\sqrt{2\log N}$. The finite maximum test $R=\mathbf{1}\{{\max }_{i\le N}\mathrm{\mid }{X}_i\mathrm{\mid }>t_N\}$ has size $p_0\le 2N\bar{\mathrm{\Phi }}(t_N)=O(1\mathrm{/}{t}_N)=o(1)$ by step 1.2. Its maximal type-II error is at most $\mathrm{\Phi }(t_N-A_{\epsilon })=\mathrm{\Phi }(-D_{\epsilon })+o(1)$, uniformly in $A_{\epsilon }$, since $t_N-\sqrt{2\log n}\to 0$ and ${\mathrm{\Phi }}^{\mathrm{\prime }}$ is bounded. For significance $\mathrm{\ell }$, the randomized test $\rho +(1-\rho )R$ with $\rho =(\mathrm{\ell }-p_0)\mathrm{/}(1-p_0)$ is eventually valid and has size exactly $\mathrm{\ell }$; its type-II error is at most $(1-\mathrm{\ell })\mathrm{\Phi }(-D_{\epsilon })+o(1)$. The unrandomized maximum gives the total-error upper bound.

In the Gaussian branch, with the profile notation of step 1.3, under the null put $A_i=g_i{\xi }_{z_i}(G_i)$ for the finite prior in step 2.2. They are centered and independent, and $\sum \mathbb{E}{A}_i^2\to U_0^2$ along any subsequence with $U\to U_0\in (0,\mathrm{\infty })$. On $z_i\le B$, step 1.2 and the vanishing maximum coordinate variance give $\sum \mathbb{E}{A}_i^4\to 0$. The omitted linear sum has variance tending uniformly to zero as $B\to \mathrm{\infty }$; the omitted quadratic sum has expectation tending to zero. Apply [F3] to the finite bounded-amplitude arrays and then let $B\to \mathrm{\infty }$, using Chebyshev for the linear tail. The same fourth-moment bound gives concentration of their quadratic sums. Thus $\sum A_i\Rightarrow N(0,U_0^2),\sum A_i^2\to U_0^2,{\max }_i\mathrm{\mid }{A}_i\mathrm{\mid }\to 0$ in probability for the last two limits. Each factor $1+A_i=1-g_i+g_i{e}^{-z_i^2\mathrm{/}2}\cosh (z_i{G}_i)$ is positive. On $\max \mathrm{\mid }{A}_i\mathrm{\mid }\le \delta <1\mathrm{/}2$, [F10] gives $\mathrm{\mid }\sum \log (1+A_i)-\sum A_i+\frac{1}{2}\sum A_i^2\mathrm{\mid }\le C\delta \sum A_i^2$. Let $\delta \downarrow 0$. The likelihood from step 1.1 therefore has $\log L\Rightarrow N(-U_0^2\mathrm{/}2,U_0^2)$. The addition of terms converging in probability and the successive tail limits preserve the displayed distribution limit by squeezing distribution functions between thresholds differing by an arbitrary positive tolerance.

In the Gaussian branch, with the profile notation of step 1.3, fix $d\in (0,1)$. Alternatives with ${\mathbb{P}}_{\mu }({\mathcal{Y}}^c)<d$ already have type-II error below $d$. For every other alternative, writing $p_i={\mathbb{P}}_{\mu }(\mathrm{\mid }{X}_i\mathrm{\mid }>Q_i)$, finite independence gives ${\prod }_{i\le m}(1-p_i)\ge d$ for each $m$. Hence ${\sum }_i{p}_i\le -\log d$ and $\mathrm{\mid }{\mu }_i\mathrm{\mid }\le Q_i+C_d$, the latter because the probability of the inside interval is at most $\mathrm{\Phi }(Q_i-\mathrm{\mid }{\mu }_i\mathrm{\mid })$. Choose $K>(2-q)\mathrm{/}(2L)$ and $N=\lceil n(\log n{)}^K\rceil$. The energy of indices $i>N$ is at most ${\epsilon }^{-q}{\sup }_{i>N}{i}^{-L}(Q_i+C_d{)}^{2-q}\le C_d{\epsilon }^{-q}{N}^{-L}(\log N{)}^{(2-q)\mathrm{/}2}=o(r^2{\epsilon }^{-2})\mathrm{.}$ The envelope is eventually decreasing and ${\epsilon }^{-q}{n}^{-L}$ is a fixed multiple of $r^2{\epsilon }^{-2}$. Among $i\le N$ delete those with $\mathrm{\mid }{\mu }_i\mathrm{\mid }>c\sqrt{\log n}$, where $c=0.45\in (\sqrt{2}-1,1\mathrm{/}2)$. The lower Mills bound in step 1.2 implies $p_i\ge n^{-a+o(1)}$ there, with $a=(\sqrt{2}-c{)}^2\mathrm{/}2<1\mathrm{/}2$, uniformly over these indices. Thus at most $n^{a+o(1)}$ are deleted. Their energy is at most $n^{2\beta +a+o(1)}=o(n^{2\beta +1\mathrm{/}2})=o(r^2{\epsilon }^{-2})$, including the polynomial logarithmic factors from $N$ and the means. Define ${\mu }_i^{\ast }={\mu }_i$ for $i\le N$ not deleted and ${\mu }_i^{\ast }=0$ for every deleted index and every $i>N$. This vector has finite support, satisfies $\mathrm{\mid }{\mu }_i^{\ast }\mathrm{\mid }\le c\sqrt{\log n}$, and has no increased $q$-budget. Its energy loss is at most the sum of the already bounded tail energy and deleted-coordinate energy, so its retained energy is $(1-o(1))r^2{\epsilon }^{-2}$. Its score mean $M={\mathbb{E}}_{{\mu }^{\ast }}F$ is therefore at least $U^2-o(1)$ by the certificate in step 1.3 and $c_0=(c_1-c_2)\mathrm{/}2$. Coordinatewise $\mathrm{\mid }{\mu }_i^{\ast }\mathrm{\mid }\le \mathrm{\mid }{\mu }_i\mathrm{\mid }$, and $\mathrm{\mid }G+\mu \mathrm{\mid }$ is stochastically increasing in $\mathrm{\mid }\mu \mathrm{\mid }$, since $\mathrm{\Phi }(t-\mathrm{\mid }\mu \mathrm{\mid })-\mathrm{\Phi }(-t-\mathrm{\mid }\mu \mathrm{\mid })$ decreases with $\mathrm{\mid }\mu \mathrm{\mid }$. Independence and the positive coefficients imply the score comparison for every finite partial sum. The full score event is the decreasing intersection of these partial-sum events after adding the fixed $\sum h_i$; [F12] passes the comparison to the full nonnegative series, giving ${\mathbb{P}}_{\mu }(F\le t)\le {\mathbb{P}}_{{\mu }^{\ast }}(F\le t)$.

In the degenerate branch, with the notation of step 2.4, for the matching degenerate lower bound first take $D_{\epsilon }\to d\in \mathbb{R}$. Put $M=\lfloor n\mathrm{/}(\log n{)}^2\rfloor$, $I=\{\lfloor n\rfloor -M+1,\dots ,\lfloor n\rfloor \}$, and mix uniformly over indices $i\in I$ and signs with ${\eta }_i=\pm r{i}^{-\beta }$, all other coordinates zero. Every point has energy $r^2$ and budget $(r{i}^{\alpha }{)}^q\le 1$. Write $T=\sqrt{2\log M}$ and ${\mu }_i=r\mathrm{/}(\epsilon i^{\beta })$; uniformly on $I$, ${\mu }_i=T+d+o(1)$ since $M\mathrm{/}n=O((\log n{)}^{-2})$ and $T-\sqrt{2\log n}\to 0$. The null likelihood from step 1.1 is $L=M^{-1}{\sum }_{i\in I}{e}^{-{\mu }_i^2\mathrm{/}2}\cosh ({\mu }_i{G}_i)$. Truncate each summand to $\mathrm{\mid }{G}_i\mathrm{\mid }\le T$, obtaining $\tilde{L}$. Completing the square gives its summand mean $\mathrm{\Phi }(T-{\mu }_i)-\mathrm{\Phi }(-T-{\mu }_i)\to c=\mathrm{\Phi }(-d)$ and its second moment at most $\frac{1}{2}{e}^{{\mu }_i^2}\mathrm{\Phi }(T-2{\mu }_i)+\frac{1}{2}{e}^{-{\mu }_i^2}$. The upper Mills bound and ${\mu }_i^2-\frac{1}{2}(2{\mu }_i-T{)}^2-\log M=-({\mu }_i-T{)}^2$ bound this second moment divided by $M$ by $O(1\mathrm{/}T)$. Independence gives $\mathrm{Var}\tilde{L}=O(1\mathrm{/}T)\to 0$, and ${\mathbb{P}}_0(L\mathrm{\ne }\tilde{L})\le 2M\bar{\mathrm{\Phi }}(T)=O(1\mathrm{/}T)\to 0$. Thus ${\mathbb{E}}_0\mathrm{\mid }\tilde{L}-c\mathrm{\mid }\to 0$ and $L\to c$ in null probability. Since $L\ge \tilde{L}$, every size-$\mathrm{\ell }$ test has mixture type-II error at least $c(1-\mathrm{\ell })-\mathbb{E}\mathrm{\mid }\tilde{L}-c\mathrm{\mid }$; every total error is at least $\mathbb{E}\min (1,L)\to c$. Bounded convergence in probability here follows directly by splitting at any fixed positive deviation; no likelihood uniform integrability is asserted. This proves both lower bounds for finite $d$.

In the Gaussian branch, with the profile notation of step 1.3, for any likelihood $L$, randomized test $\chi$ and $c>0$, scalar algebra gives $L(1-\chi )\ge \min (L,c)-c\chi$. The limit law in step 3.1 has no positive atoms, so ${\mathbb{E}}_0\min (L,c)={\int }_0^c{\mathbb{P}}_0(L>t)dt$ converges by [F8]; the identity itself follows first for step functions and then by increasing finite subdivisions. Choose $c=\exp (-U_0^2\mathrm{/}2+U_0{H}_{\mathrm{\ell }})$. Its limiting upper-tail probability is $\mathrm{\ell }$, and completing the square gives the limiting lower bound $\mathrm{\Phi }(H_{\mathrm{\ell }}-U_0)$ for every test of size at most $\mathrm{\ell }$. Likewise every total error is at least ${\mathbb{E}}_0\min (1,L)\to 2\mathrm{\Phi }(-U_0\mathrm{/}2)$. Conditioning in step 2.2 costs only $o(1)$ uniformly over tests. This proves both minimax lower bounds on compact transitions without an additional Neyman--Pearson or unrestricted-choice supplier.

In the Gaussian branch, with the profile notation of step 1.3, for the residual vector set $m_i=h_i(\cosh (z_i{\mu }_i^{\ast })-1)$, so $M=\sum m_i\ge 0$. The exact variance formula in step 1.2 shows that the excess above $V$ is $\sum h_i^2(e^{z_i^2}-1){\sinh }^2(z_i{\mu }_i^{\ast })\le \sum K_i{m}_i,K_i=h_i(e^{z_i^2}-1)(\cosh (z_i{\mu }_i^{\ast })+1)\mathrm{.}$ For $z_i\le \sqrt{\log n}$, the global bound $\psi (z)(e^{z^2}-1)\le C$ in [F2], its relation $x\le C(1+z^{1\mathrm{/}\kappa })$ at large $z$, and $h_0\asymp n^{-1\mathrm{/}2}$ give $K_i\le n^{-1\mathrm{/}2+c+o(1)}=o(1)$. For $z_i>\sqrt{\log n}$ the other global bound in [F2] gives an absolute excess bound $C{h}_0^2{\sum }_{z_i>\sqrt{\log n}}(i\mathrm{/}n{)}^{4\beta }\exp (-z_i^2+2c{z}_i\sqrt{\log n})\le C{h}_0^2{\sum }_{z_i>\sqrt{\log n}}(i\mathrm{/}n{)}^{4\beta }{e}^{-(1-2c)z_i^2}=o(1)\mathrm{.}$ The final estimate follows from the exponential profile tail and $h_0^{2}n=O(1)$; its discrete tail is bounded by the integral of an eventually decreasing polynomial-times-exponential envelope plus a mesh term, as in [F2]. Thus uniformly over residual alternatives, ${\mathrm{Var}}_{{\mu }^{\ast }}F\le V+o(1)M+o(1)$. Unbounded $z_i$ have been controlled by an absolute variance bound, not a uniform bounded-amplitude assumption.

In the degenerate branch, with the notation of step 2.4, if $D_{\epsilon }\to +\mathrm{\infty }$, step 2.4 settles both errors by nonnegativity. If $D_{\epsilon }\to -\mathrm{\infty }$, fix $K>0$. The continuous strictly increasing function $r\mapsto {\epsilon }^{-1}{r}^{s\mathrm{/}\alpha }-\sqrt{(2\mathrm{/}\alpha )\log (1\mathrm{/}r)}$ ranges from $-\mathrm{\infty }$ to a positive value on $(0,1)$, so it has a unique radius $r_{\epsilon ,K}$ at value $-K$, and this radius tends to zero. Eventually $r<r_{\epsilon ,K}$; inclusion of alternative sets and step 3.3 give lower limits at least $\mathrm{\Phi }(K)$ for total error and $(1-\mathrm{\ell })\mathrm{\Phi }(K)$ for type-II error. Let $K\to \mathrm{\infty }$. Subsequences in the extended real line now establish the two degenerate formulas for arbitrary $D_{\epsilon }$. Finally put $p=s\mathrm{/}\alpha$ and $\mathrm{\Lambda }=(2\mathrm{/}s{)}^{\alpha \mathrm{/}(2s)}$, so ${\mathrm{\Lambda }}^p=\sqrt{2\mathrm{/}s}$. For each fixed $c>0$, direct substitution into the displayed $r_{\epsilon }^{\ast }$ gives $D_{\epsilon }(c{r}_{\epsilon }^{\ast })=(c^p-1)\sqrt{2\log ({\epsilon }^{-1})\mathrm{/}s}+o(1)\mathrm{.}$ The logarithmic correction is $O(\log \log ({\epsilon }^{-1})\mathrm{/}\sqrt{\log ({\epsilon }^{-1})})$, which tends to zero. Thus the stated sharp constant, zero/infinite-ratio separation limits and the assignment of $\lambda =0$ all follow, with no strict inequality imposed on $e$.

In the Gaussian branch, with the profile notation of step 1.3, if $M$ stays bounded, fix $B>0$. For $z_i\le B$, $h_i{z}_i^2\le C_B{h}_0(i\mathrm{/}n{)}^{2\beta }$ by [F2]. Writing the hyperbolic addition formula and using $\mathrm{\mid }\sinh (z\mu )\mathrm{\mid }\le z\mathrm{\mid }\mu \mathrm{\mid }{e}^{B\mathrm{\mid }\mu \mathrm{\mid }}$ gives $\mathbb{E}{\mid h_i({\xi }_{z_i}(G_i+{\mu }_i^{\ast })-\mathbb{E}{\xi }_{z_i}(G_i+{\mu }_i^{\ast }))\mid }^4\le C_B(h_i{z}_i^2{)}^4(1+\mathrm{\mid }{\mu }_i^{\ast }{\mathrm{\mid }}^4)e^{4B\mathrm{\mid }{\mu }_i^{\ast }\mathrm{\mid }}\mathrm{.}$ Here the fourth moments of $\cosh (zG)-\mathbb{E}\cosh (zG)$ and $\sinh (zG)$ are bounded respectively by $C_B{z}^8$ and $C_B{z}^4$, by the same Gaussian exponential integration as in step 1.2. Since $z_i\le B$ implies $i\le C_{B}n$, the summed fourth moments are $O_B(n{h}_0^4)n^{o(1)}=n^{-1+o(1)}\to 0$. The variance of the omitted $z_i>B$ centered sum is at most its null variance tail plus $o(1)$, by step 4.2 and bounded $M$. Apply [F3] to the retained finite arrays and then let $B\to \mathrm{\infty }$ to obtain $F-M\Rightarrow N(0,U_0^2)$. The assertion is uniform in the sense that every sequence of residual alternatives with bounded $M$ obeys this limit. If $M\to \mathrm{\infty }$, step 4.2 and Chebyshev instead show ${\mathbb{P}}_{{\mu }^{\ast }}(F\le t)\to 0$ for every bounded threshold. These conclusions and $M\ge U^2-o(1)$ prove the upper type-II limits $\mathrm{\Phi }(H_{\mathrm{\ell }}-U_0)$ and $\mathrm{\Phi }(-U_0\mathrm{/}2)$. To justify passage to the supremum, any contrary sequence has either a bounded-mean subsequence (then a convergent-mean subsequence) or a subsequence with $M\to \mathrm{\infty }$; both cases give the asserted bound. Let $d\downarrow 0$ for the alternatives excluded in step 3.2. Under the null step 2.3 gives the corresponding errors $\mathrm{\ell }+o(1)$ and $\mathrm{\Phi }(-U_0\mathrm{/}2)+o(1)$. If the first test has size $p_{\epsilon }>\mathrm{\ell }$, multiply its indicator by $\mathrm{\ell }\mathrm{/}{p}_{\epsilon }$; otherwise leave it unchanged. Its size is then at most $\mathrm{\ell }$ and every power changes by $o(1)$. This proves the exact-level Gaussian upper bound and the total-error upper bound.

In the Gaussian branch, write $\rho =r\mathrm{/}{r}_{\epsilon }^{\ast }$, $p=B\mathrm{/}D>0$ and $c=c_{\alpha ,\beta ,q}$. For a comparison radius $R=t{r}_{\epsilon }^{\ast }$, let $T_{\epsilon ,t}^{\mathrm{\ell }}$ be the score-and-maximum significance test in steps 2.3 and 5.1, with its exact-level scaling, and let $T_{\epsilon ,t}^{\mathrm{t}\mathrm{o}\mathrm{t}}$ be their unscaled total-error test. The proved estimates are uniform over each fixed $t\in [1\mathrm{/}k,k]$: otherwise a countersequence has a convergent subsequence of $t$ in that compact interval, and all the compact-transition estimates already proved apply with a positive finite limiting $U=c{t}^p$, contradicting the countersequence. Consequently, for every integer $k\ge 2$ there is a sufficiently small noise threshold such that for all smaller $\epsilon$ and every $t\in [1\mathrm{/}k,k]$, the profile dimension is $n>e$, $\mathrm{\mid }{u}_{\epsilon }(t{r}_{\epsilon }^{\ast })-c{t}^p\mathrm{\mid }\le 1\mathrm{/}k$, the maximal type-II error of $T_{\epsilon ,t}^{\mathrm{\ell }}$ at radius $t{r}_{\epsilon }^{\ast }$ is at most $\mathrm{\Phi }(H_{\mathrm{\ell }}-c{t}^p)+1\mathrm{/}k$, and the total error of $T_{\epsilon ,t}^{\mathrm{t}\mathrm{o}\mathrm{t}}$ there is at most $2\mathrm{\Phi }(-c{t}^p\mathrm{/}2)+1\mathrm{/}k$. Both tests are therefore defined on this whole compact range, and the significance test has size at most $\mathrm{\ell }$ for every such noise level. Fix $j_1=1$ and recursively let $j_k$ be the least integer $j>j_{k-1}$, $j\ge k$, for which all these uniform properties hold for every $0<\epsilon \le 2^{-j}$. These integers exist by the uniform convergence just proved. For $0<\epsilon \le 2^{-j_2}$ define $k_{\epsilon }=\max \{k\ge 2:\epsilon \le 2^{-j_k}\}$. This is a nonempty finite set because $j_k\ge k$; $k_{\epsilon }\to \mathrm{\infty }$ as $\epsilon \downarrow 0$. The least-integer rule and this finite maximum specify one deterministic diagonal, depending only on the fixed model, significance level and the given ${\delta }_{\epsilon }$ family, not on the unknown truth or the observations.

For this single diagonal, define the Gaussian-regime test family for every known radius $r>0$ as follows. When $\rho <1\mathrm{/}{k}_{\epsilon }$, use the constant randomized significance test ${\chi }_{\epsilon ,r}^{\mathrm{\ell }}\equiv \mathrm{\ell }$ and the never-reject total-error test ${\chi }_{\epsilon ,r}^{\mathrm{t}\mathrm{o}\mathrm{t}}\equiv 0$. When $1\mathrm{/}{k}_{\epsilon }\le \rho \le k_{\epsilon }$, use $T_{\epsilon ,\rho }^{\mathrm{\ell }}$ and $T_{\epsilon ,\rho }^{\mathrm{t}\mathrm{o}\mathrm{t}}$. When $\rho >k_{\epsilon }$, use $T_{\epsilon ,k_{\epsilon }}^{\mathrm{\ell }}$ and $T_{\epsilon ,k_{\epsilon }}^{\mathrm{t}\mathrm{o}\mathrm{t}}$, constructed at the smaller comparison radius $k_{\epsilon }{r}_{\epsilon }^{\ast }$, never at the possibly invalid profile dimension associated with $r$. For $\epsilon >2^{-j_2}$ set the family to the same two trivial tests; this finite-noise convention does not affect an asymptotic statement. Every test is product-Borel, each significance test has size at most $\mathrm{\ell }$, and every formula $\log \log n$ actually evaluated has $n>e$. Write $k=k_{\epsilon }$. In the middle case the upper risk bounds in step 6.1 hold with error $1\mathrm{/}k$, and $\mathrm{\mid }{u}_{\epsilon }(r)-c{\rho }^p\mathrm{\mid }\le 1\mathrm{/}k$, so the attained errors are bounded by the stated Gaussian profiles plus a quantity tending to zero. In the small-radius case, monotonicity of the relaxed extremal problem gives $0\le u_{\epsilon }(r)\le c{k}^{-p}+1\mathrm{/}k\to 0$; the two trivial tests have respective errors $1-\mathrm{\ell }$ and $1$ whenever the alternative is nonempty, and no larger errors if it is empty, matching the profile upper bounds up to a vanishing quantity. In the large-radius case, the actual alternative is a subset of the comparison-radius alternative, so both achieved errors are no larger than their comparison-radius errors, bounded by $\mathrm{\Phi }(H_{\mathrm{\ell }}-c{k}^p)+1\mathrm{/}k$ and $2\mathrm{\Phi }(-c{k}^p\mathrm{/}2)+1\mathrm{/}k$, respectively, and these tend to zero. Also $u_{\epsilon }(r)\ge c{k}^p-1\mathrm{/}k\to \mathrm{\infty }$, as required for the displayed profiles. These bounds apply to all three cases with the same deterministic $k_{\epsilon }$; hence they also apply to ratios oscillating between them, without selecting a test by a subsequence.

The compact-transition lower bounds in step 4.1 extend globally by monotonicity of the minimax errors and of $J_{\epsilon }(r(1-{\delta }_{\epsilon }))$. Comparing $r\mathrm{/}{r}_{\epsilon }^{\ast }\to 0$ with each fixed small positive transition constant gives $u_{\epsilon }\to 0$, $\mathrm{lim\hspace{0.17em}inf}{\gamma }_{\epsilon }\ge 1$ and $\mathrm{lim\hspace{0.17em}inf}{\beta }_{\epsilon }(\mathrm{\ell })\ge 1-\mathrm{\ell }$; the corresponding upper bounds are the trivial tests. Comparing $r\mathrm{/}{r}_{\epsilon }^{\ast }\to \mathrm{\infty }$ with each fixed large transition constant gives $u_{\epsilon }\to \mathrm{\infty }$ and vanishing minimax errors. For any other sequence of positive ratios, every subsequence has a further subsequence tending to zero, a finite positive constant or infinity, so the same lower-profile conclusion applies. Together with the globally defined upper family in step 7.1 this proves both sharp Gaussian formulas and asymptotic minimax attainment for every $r_{\epsilon }\to 0$, including oscillating ratios. In particular the displayed $r_{\epsilon }^{\ast }$ is a separation rate with both impossibility and consistency halves. In the degenerate regime, the finite maximum and exact-level randomized tests in step 2.4 already apply along every $r_{\epsilon }\to 0$, because $N\to \mathrm{\infty }$ and $p_0\to 0$ irrespective of the noise/radius ratio; to define them also before $N\ge 2$ and $p_0<\mathrm{\ell }$, use the constant significance test and never-reject total-error test until those inequalities hold. Their global sharp attainment follows from step 4.3 and its matching upper bounds. No unrestricted power equivalent or undefined transition threshold is used in either family.

The separation-rate definition in [F1] allows every positive radius sequence, not only vanishing ones. In either regime the displayed positive rate $r_{\epsilon }^{\ast }$ tends to zero. If $r_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to 0$, then $r_{\epsilon }\to 0$, so the impossibility half already proved applies. For the consistency half, let $r_{\epsilon }>0$ satisfy $r_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to \mathrm{\infty }$ and set ${\bar{r}}_{\epsilon }=\min \{r_{\epsilon },\sqrt{r_{\epsilon }^{\ast }}\}$. Then $0<{\bar{r}}_{\epsilon }\le r_{\epsilon }$, ${\bar{r}}_{\epsilon }\to 0$, and ${\bar{r}}_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }=\min \{r_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast },1\mathrm{/}\sqrt{r_{\epsilon }^{\ast }}\}\to \mathrm{\infty }$. The Gaussian tests of step 7.1 and the degenerate tests of steps 2.4 and 8.1, evaluated at ${\bar{r}}_{\epsilon }$, therefore have vanishing total error and vanishing maximal type-II error at each fixed significance level on ${\mathrm{\Theta }}_q({\bar{r}}_{\epsilon })$. The defining energy inequality gives ${\mathrm{\Theta }}_q(r_{\epsilon })\subseteq {\mathrm{\Theta }}_q({\bar{r}}_{\epsilon })$, so using these same tests at the requested radius leaves their null size unchanged and cannot increase their maximal type-II error. Thus both minimax errors at $r_{\epsilon }$ are bounded above by their counterparts at ${\bar{r}}_{\epsilon }$ and tend to zero. If ${\mathrm{\Theta }}_q(r_{\epsilon })$ is empty, both minimax errors are zero by the explicit convention in [F1]; the never-reject test also realizes zero total error in that case. This proves the rate implications for all positive radius sequences, including nonvanishing or oscillating radii, in both regimes. All cutoffs and profile parameters used in this reduction are evaluated at the vanishing comparison radius.

The Gaussian and degenerate bounds agree with their respective lower bounds, and the globally specified tests in steps 7.1 and 8.1 attain them asymptotically, including exact significance adjustment. Step 2.1 proves the qualified extremal power equivalent, step 8.1 proves the full Gaussian separation claim, and step 4.3 proves the degenerate sharp normalization. Step 9.1 supplies the separation-rate implications for all positive radius sequences. Each test is product-Borel, using a specified nonnegative countable series or finitely many coordinates and deterministic randomization. All Gaussian laws were given by [F4], all auxiliary probability products were finite, and the finite-array approximation used [F3] under Countable Choice. This proves every stated clause without a stronger choice assumption or a citation in place of a local proof.