Sharp Gaussian asymptotics under a vanishing largest weight

Under an alternative $\eta$, write ${\mu }_j={\eta }_j\mathrm{/}\epsilon$ and let $G_j={\xi }_j$ denote the independent standard-normal noise coordinates, so $Y_j\mathrm{/}\epsilon ={\mu }_j+G_j$. Then $T_{\epsilon }-m_{\eta }={\sum }_j{w}_{\epsilon ,j}(G_j^2-1)+2{\sum }_j{w}_{\epsilon ,j}{\mu }_j{G}_j\mathrm{.}$ The first sum converges to $N(0,1)$ by [F2]. The second has mean zero and variance $4{\sum }_j{w}_{\epsilon ,j}^2{\mu }_j^2\le 4w_{\epsilon ,0}{m}_{\eta }$. Thus for any sequence of alternatives with bounded $m_{\eta }$, the second sum tends to zero in probability by [F7], and [F5] gives $T_{\epsilon }-m_{\eta }\Rightarrow N(0,1)$.

Put ${\delta }_j={\tilde{\eta }}_{\epsilon ,j}^2\mathrm{/}{\epsilon }^2=2u_{\epsilon }{w}_{\epsilon ,j}$. Since $u_{\epsilon }$ is bounded and $w_{\epsilon ,0}\to 0$, ${\max }_j{\delta }_j\to 0$ and ${\sum }_j{\delta }_j^2=2u_{\epsilon }^2=O(1)$. For $\varphi (x)=(2\pi {)}^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}2}$, direct differentiation gives ${\varphi }^{\mathrm{\prime }}(x)=-x\varphi (x)$. For each $R>0$ and $k=3,4$, [F9] applied on $[-R,R]$ to $u(x)=x^{2k-1}$ and $v(x)=\varphi (x)$ gives ${\int }_{-R}^R{x}^{2k}\varphi (x)dx=[-x^{2k-1}\varphi (x){]}_{-R}^R+(2k-1){\int }_{-R}^R{x}^{2k-2}\varphi (x)dx\mathrm{.}$ The boundary term tends to zero by [F10], and [F11] passes the nonnegative truncated integrals to the full real line. Starting from $\mathbb{E}{G}^4=3$ in [F4], the recurrence gives $\mathbb{E}{G}^6=5\cdot 3=15$ and $\mathbb{E}{G}^8=7\cdot 15=105$. Taylor's theorem applied to $f(x)=\log \cosh x$ gives $\mathrm{\mid }f(x)-x^2\mathrm{/}2+x^4\mathrm{/}12\mathrm{\mid }\le C\mathrm{\mid }x{\mathrm{\mid }}^6$ for all real $x$: on $[-1,1]$ use the bounded sixth derivative, and outside that interval use $\mathrm{\mid }\log \cosh x\mathrm{\mid }\le \mathrm{\mid }x\mathrm{\mid }$. Therefore the summed Taylor remainder in $\log L_{\epsilon }$ has expected absolute value at most $C{\sum }_j{\delta }_j^3\mathbb{E}\mathrm{\mid }G{\mathrm{\mid }}^6\le C({\max }_j{\delta }_j){\sum }_j{\delta }_j^2\to 0$.

Put $F_{\epsilon }(t)={\mathbb{P}}_0(T_{\epsilon }\le t)$ and $p=1-\alpha$. Since $T_{\epsilon }$ is real-valued, the events $\{T_{\epsilon }\le n\}$ increase to the whole space and $\{T_{\epsilon }\le -n\}$ decrease to the empty set; [F8] shows that $F_{\epsilon }(n)\to 1$ and $F_{\epsilon }(-n)\to 0$. Thus $S=\{t:F_{\epsilon }(t)\ge p\}$ is nonempty and bounded below, so $c_{\epsilon ,\alpha }=\inf S$ is finite. For every $x>c_{\epsilon ,\alpha }$, the definition of the infimum gives some $s\in S$ with $s<x$, and monotonicity gives $F_{\epsilon }(x)\ge F_{\epsilon }(s)\ge p$. Hence $F_{\epsilon }(c_{\epsilon ,\alpha }+1\mathrm{/}n)\ge p$ for every positive integer $n$. The events $\{T_{\epsilon }\le c_{\epsilon ,\alpha }+1\mathrm{/}n\}$ decrease to $\{T_{\epsilon }\le c_{\epsilon ,\alpha }\}$, so [F8] gives $F_{\epsilon }(c_{\epsilon ,\alpha })\ge p$ and therefore ${\mathbb{P}}_0(T_{\epsilon }>c_{\epsilon ,\alpha })\le \alpha$. By [F2] and [F6], $c_{\epsilon ,\alpha }\to H_{\alpha }$, since the limiting normal distribution is continuous and strictly increasing. To bound maximal type-II error, consider any sequence ${\eta }_{\epsilon }\in {\mathrm{\Theta }}_2(r_{\epsilon })$. If its means $m_{{\eta }_{\epsilon }}$ are unbounded, pass to a subsequence on which they tend to infinity; the variance bound in [F1] and [F7] show ${\mathbb{P}}_{{\eta }_{\epsilon }}(T_{\epsilon }\le c_{\epsilon ,\alpha })\to 0$. If they are bounded, pass to a subsequence on which $m_{{\eta }_{\epsilon }}\to m\ge u_{\ast }$ and $u_{\epsilon }\to u_{\ast }$, where $u_{\ast }>0$. Step 1.1 gives type-II limit $\mathrm{\Phi }(H_{\alpha }-m)\le \mathrm{\Phi }(H_{\alpha }-u_{\ast })$. A subsequence contradiction therefore proves the uniform upper bound ${\beta }_{\epsilon }({\mathrm{\Theta }}_2(r_{\epsilon }),{\psi }_{\epsilon ,c_{\epsilon ,\alpha }})\le \mathrm{\Phi }(H_{\alpha }-u_{\epsilon })+o(1)$.

The quartic term ${\sum }_j{\delta }_j^2{G}_j^4$ has mean $3{\sum }_j{\delta }_j^2=6u_{\epsilon }^2$ and variance $96{\sum }_j{\delta }_j^4\le 96({\max }_j{\delta }_j{)}^2{\sum }_j{\delta }_j^2\to 0$. Substituting the expansion from step 1.2 into [F3] gives $\log L_{\epsilon }=u_{\epsilon }{T}_{\epsilon }-u_{\epsilon }^2\mathrm{/}2+o_{P_0}(1)\mathrm{.}$ Together with [F2], this implies that under $P_0$ the likelihood ratio converges along any subsequence with $u_{\epsilon }\to u_{\ast }>0$ to $\exp (-u_{\ast }^2\mathrm{/}2+u_{\ast }G)$ for $G\sim N(0,1)$.

For any level-$\alpha$ test $\psi$, set $a_{\epsilon }=\exp (u_{\epsilon }{c}_{\epsilon ,\alpha }-u_{\epsilon }^2\mathrm{/}2)$. Pointwise, $L_{\epsilon }(1-\psi )\ge \min (L_{\epsilon },a_{\epsilon })-a_{\epsilon }\psi$. Hence the mixture-average type-II error is at least ${\mathbb{E}}_0\min (L_{\epsilon },a_{\epsilon })-a_{\epsilon }\alpha$. Along a subsequence with $u_{\epsilon }\to u_{\ast }>0$, step 2.1 gives $c_{\epsilon ,\alpha }\to H_{\alpha }$, and step 2.2 and [F6] yield the limit $\mathrm{\Phi }(H_{\alpha }-u_{\ast })$. Since maximal type-II error dominates the mixture average, this proves the matching lower bound for ${\beta }_{\epsilon }({\mathrm{\Theta }}_2(r_{\epsilon }),\alpha )$.

For every test $\psi$, $\psi +L_{\epsilon }(1-\psi )\ge \min (1,L_{\epsilon })$ pointwise. Thus its total error against the null and mixture is at least ${\mathbb{E}}_0\min (1,L_{\epsilon })$. By step 2.2 and [F6] this expectation tends along $u_{\epsilon }\to u_{\ast }$ to $2\mathrm{\Phi }(-u_{\ast }\mathrm{/}2)$, giving the matching minimax lower bound for ${\gamma }_{\epsilon }$. The weighted test at threshold $u_{\epsilon }\mathrm{/}2$ has type-I limit $\mathrm{\Phi }(-u_{\ast }\mathrm{/}2)$ and maximal type-II upper limit $\mathrm{\Phi }(-u_{\ast }\mathrm{/}2)$ by the same bounded-mean/unbounded-mean split as step 2.1. Continuity in $u_{\ast }$ and compactness of the allowed range of $u_{\epsilon }$ turn the subsequential bounds into the stated $o(1)$ formulas.