# Asymptotic Uncertainty of False Discovery Proportion: Supplementary
Materials

Meng Mei${}^{\text{a}}$ ${}^{\text{a}}$ Department of Statistics, Oregon State
University
${}^{\text{b}}$ Department of Statistics & Data Science, National University
of Singapore  Tao Yu${}^{\text{b}}$ ${}^{\text{a}}$ Department of Statistics,
Oregon State University
${}^{\text{b}}$ Department of Statistics & Data Science, National University
of Singapore  and Yuan Jiang${}^{\text{a}}$ Yuan Jiang is the corresponding
author. Meng Mei and Yuan Jiang’s research is supported in part by National
Institutes of Health grant R01 GM126549. Tao Yu’s research is supported in
part by the Singapore Ministry Education Academic Research Tier 1 Funds:
R-155-000-202-114. ${}^{\text{a}}$ Department of Statistics, Oregon State
University
${}^{\text{b}}$ Department of Statistics & Data Science, National University
of Singapore

##  1 Notation and Review of Technical Results in the Main Article

We need the following notation in our development. Throughout, we use
“$\lesssim$” (“$\gtrsim$”) to denote smaller (greater) than up to a universal
constant. For any $t\in[0,1]$, we denote
$\xi_{j}=\Phi(z_{t/2}+\mu_{j})+\Phi(z_{t/2}-\mu_{j})$ and
$\bar{\xi}=\frac{1}{p_{1}}\sum_{j\in\mathcal{H}_{1}}\xi_{j}$. Consider the
function

$H(\mu)=\phi(|z_{t/2}|+\mu)(|z_{t/2}|+\mu)+\phi(|z_{t/2}|-\mu)(|z_{t/2}|-\mu);$

there exists a unique root, denoted by $\mu_{t}$, of this function for
$\mu\in(|z_{t/2}|,|z_{t/2}|+1)$. Let
$C_{t}^{\max}=\sup_{\mu\in(-\mu_{t},\mu_{t})}H(\mu)$.

Furthermore, recall that in the main article, the $i$th test is given by
$t_{i}=\mathrm{1}(|Z_{i}|>|z_{t/2}|)=\mathrm{1}(P_{i}<t)$. Let

$\displaystyle
f_{\\{t_{1},\dots,t_{k}\\}}^{(i_{1},\dots,i_{k})}(a,t)=\dfrac{\partial^{\sum_{j=1}^{k}i_{j}}f_{\\{t_{1},\ldots,t_{k}\\}}(\rho_{1},\ldots,\rho_{k};a,t)}{\partial(\rho_{1})^{i_{1}}\ldots\partial(\rho_{k})^{i_{k}}}\Big{|}_{\rho_{j}=0,j=1,\ldots,k},$

where

$f_{\\{t_{1},\ldots,t_{k}\\}}(\rho_{1},\ldots,\rho_{k};a,t)=\Phi\left(\dfrac{z_{t/2}-\mu_{a}-\rho_{1}x_{t_{1}}-\ldots-\rho_{k}x_{t_{k}}}{\sqrt{1-\rho_{1}^{2}-\ldots-\rho_{k}^{2}}}\right).$

Let $\xi_{i}=E(t_{i})$ for $i\in\mathcal{H}_{1}$; clearly we have
$\xi_{i}=\Phi(z_{t/2}+\mu_{i})+\Phi(z_{t/2}-\mu_{i})$. Denote
$\bar{\xi}=\frac{1}{p_{1}}\sum_{i=1}^{p_{1}}\xi_{i}$. Then the converged limit
of $\mbox{FDP}(t)$ on the right hand side of (6) in the main article can be
written to be $\frac{p_{0}t}{p_{0}t+p_{1}\bar{\xi}}$.

The following theorem is Theorem 1 in Section 3.1 of the main article; it
establishes the asymptotic expansion of the $\mathrm{FDP}(t)$ in the framework
of the weak dependence.

######  Theorem 1.

Suppose $(Z_{1},\ldots,Z_{p})^{T}\sim
N((\mu_{1},\ldots,\mu_{p})^{T},\boldsymbol{\Sigma})$ with unit variances,
i.e., $\sigma_{jj}=1\text{ for }j=1,\ldots,p$. Assume that
$(Z_{1},\dots,Z_{p})^{T}$ are weakly dependent as defined in (5) in the main
article, that $\limsup_{p\to\infty}p_{0}t/(p_{1}\bar{\xi})<1$, and that as $p$
is sufficiently large, for a universal constant $C>0$,

$\displaystyle\sum_{i\neq
j;i,j\in\mathcal{H}_{0}}\sigma_{ij}^{2}\geq\frac{C_{t}^{\max}}{\phi(z_{t/2})|z_{t/2}|}\sum_{i\in\mathcal{H}_{1},j\in\mathcal{H}_{0},\mu_{i}\in[-\mu_{t},\mu_{t}]}\sigma_{ij}^{2},$
(1) $\displaystyle\sum_{i\neq j;i,j\in\mathcal{H}_{0}}\sigma_{ij}^{2}+p\geq
C\sum_{i\neq j;i,j\in\mathcal{H}_{1}}\sigma_{ij}^{2},$ (2)
$\displaystyle\sum_{i\neq j}\sigma_{ij}^{4}=o\left(\sum_{i\neq
j;i,j\in\mathcal{H}_{0}}\sigma_{ij}^{2}+p_{0}\right).$ (3)

We have the following asymptotic expansion of $\mathrm{FDP}(t)$:

$\mathrm{FDP}(t)=\frac{E(\bar{V})}{E(\bar{R})}+m(\bar{V},\bar{R})+r(\bar{V},\bar{R}),$
(4)

where $\bar{V}=V(t)/p$, $\bar{R}=R(t)/p$,

$m(\bar{V},\bar{R})=\frac{\bar{V}}{E(\bar{R})}-\frac{E(\bar{V})}{\\{E(\bar{R})\\}^{2}}\bar{R},$

and the remainder term $r(\bar{V},\bar{R})$ satisfies that
$E\\{r^{2}(\bar{V},\bar{R})\\}=o[\mathrm{Var}\\{m(\bar{V},\bar{R})\\}]$.

A direct outcome of this theorem is the following Corollary, which establishes
the explicit formula for the asymptotic variance of $\mathrm{FDP}(t)$. It is
presented as Corollary 1 in the main article.

######  Corollary 1.

With all the conditions in Theorem 1 effective, we have

$\lim_{p\to\infty}\dfrac{\mathrm{Var}\left\\{\mathrm{FDP}(t)\right\\}}{V_{1}(t)+V_{2}(t)}=1,$
(5)

where

$\displaystyle V_{1}(t)={}$
$\displaystyle\dfrac{p_{1}^{2}\bar{\xi}^{2}}{(p_{0}t+p_{1}\bar{\xi})^{4}}\times
p_{0}t(1-t)+\dfrac{p_{0}^{2}t^{2}}{(p_{0}t+p_{1}\bar{\xi})^{4}}\times
p_{1}\bar{\xi}(1-\bar{\xi}),$ (6) $\displaystyle V_{2}(t)={}$
$\displaystyle\dfrac{2p_{1}^{2}\bar{\xi}^{2}}{(p_{0}t+p_{1}\bar{\xi})^{4}}\sum_{\begin{subarray}{c}i<j\\\
i,j\in\mathcal{H}_{0}\end{subarray}}\mathrm{Cov}(t_{i},t_{j})-\dfrac{2p_{0}p_{1}t\bar{\xi}}{(p_{0}t+p_{1}\bar{\xi})^{4}}\sum_{\begin{subarray}{c}i\in\mathcal{H}_{0}\\\
j\in\mathcal{H}_{1}\end{subarray}}\mathrm{Cov}(t_{i},t_{j})$
$\displaystyle+\dfrac{2p_{0}^{2}t^{2}}{(p_{0}t+p_{1}\bar{\xi})^{4}}\sum_{\begin{subarray}{c}i<j\\\
i,j\in\mathcal{H}_{1}\end{subarray}}\mathrm{Cov}(t_{i},t_{j}),$ (7)

with

$\displaystyle\mathrm{Cov}(t_{i},t_{j})={}$
$\displaystyle\int_{|z_{i}|>|z_{t/2}|}\int_{|z_{j}|>|z_{t/2}|}\phi(z_{i},z_{j};\mu_{i},\mu_{j},1,1,\sigma_{ij})dz_{i}dz_{j}$
$\displaystyle-[\Phi(z_{t/2}+\mu_{i})+\Phi(z_{t/2}-\mu_{i})][\Phi(z_{t/2}+\mu_{j})+\Phi(z_{t/2}-\mu_{j})],$
(8)

in which
$\phi(z_{i},z_{j};\mu_{i},\mu_{j},\sigma_{i}^{2},\sigma_{j}^{2},\sigma_{ij})$
is the probability density function of the bivariate normal distribution with
mean $(\mu_{i},\mu_{j})^{T}$ and covariance matrix
$\left(\begin{matrix}\sigma_{i}^{2}&\sigma_{ij}\\\
\sigma_{ij}&\sigma_{j}^{2}\end{matrix}\right)$.

The following theorem, which is Theorem 2 in the main article, discusses how
the signs of the $\mathrm{Cov}(t_{i},t_{j})$ is affected by the dependence
structure among the test statistics.

######  Theorem 2.

Suppose $(Z_{1},\ldots,Z_{p})^{T}\sim
N((\mu_{1},\ldots,\mu_{p})^{T},\boldsymbol{\Sigma})$ with unit variances,
i.e., $\sigma_{jj}=1\text{ for }j=1,\ldots,p$. Assume $\sigma_{ij}\neq 0$ for
the index pair $(i,j),i\neq j$. Then we have:

  * $(a)$

for $i,j\in\mathcal{H}_{0}$, $\mathrm{Cov}(t_{i},t_{j})>0$;

  * $(b)$

for $i\in\mathcal{H}_{0}$, $j\in\mathcal{H}_{1}$,
$\mathrm{Cov}(t_{i},t_{j})<0$ if $|\mu_{j}|>2|z_{t/2}|$ and
$t<2\\{1-\Phi^{-1}(\sqrt{\log(3)/2})\\}$;

  * $(c)$

for $i,j\in\mathcal{H}_{1}$, $\mathrm{Cov}(t_{i},t_{j})>0$ if
$|\sigma_{ij}|\leq|\mu|_{\min}/(|\mu|_{\max}+z_{t/2})$ and
$\mathrm{sign}(\sigma_{ij})=\mathrm{sign}(\mu_{i}\mu_{j})$, where
$|\mu|_{\min}=\min(|\mu_{i}|,|\mu_{j}|)$ and
$|\mu|_{\max}=\max(|\mu_{i}|,|\mu_{j}|)$.

These results immediately result in the theorem blow, which gives one set of
sufficient conditions that lead to the inflation of the FDP variance; it is
Theorem 3 in the main article.

######  Theorem 3.

Suppose $(Z_{1},\ldots,Z_{p})^{T}\sim
N((\mu_{1},\ldots,\mu_{p})^{T},\boldsymbol{\Sigma})$ with $\sigma_{jj}=1\text{
for }j=1,\ldots,p$. For any $i,j\in\mathcal{H}_{1},i\neq j$, assume that
$|\mu_{i}|\geq 2|z_{t/2}|>2\sqrt{\log(3)/2}$,
$|\sigma_{ij}|\leq|\mu|_{\min}/(|\mu|_{\max}+z_{t/2})$, and
$\mathrm{sign}(\sigma_{ij})=\mathrm{sign}(\mu_{i}\mu_{j})$, where
$|\mu|_{\min}=\min(|\mu_{i}|,|\mu_{j}|)$ and
$|\mu|_{\max}=\max(|\mu_{i}|,|\mu_{j}|)$. Then, $V_{2}(t)>0$.

##  2 Lemmas

We first establish the following lemmas; they play fundamental roles in our
proofs of the theoretical results given above. In particular, Lemmas 1–5 are
to support our proof for Theorem 1; Lemma 6 is used for Theorem 2.

######  Lemma 1.

The partial derivatives $f_{t_{1},t_{2},t_{3}}^{i_{1},i_{2},i_{3}}(a,t)$ can
be written to be

$f_{t_{1},t_{2},t_{3}}^{i_{1},i_{2},i_{3}}(a,t)=C^{a}_{t/2}(i_{1}+i_{2}+i_{3})g_{i_{1}}(x_{t_{1}})g_{i_{2}}(x_{t_{2}})g_{i_{3}}(x_{t_{3}}),$

where

$\displaystyle g_{0}(x)=1,\qquad g_{1}(x)=x,\qquad g_{2}(x)=x^{2}-1,\qquad
g_{3}(x)=x^{3}-3x,$ $\displaystyle g_{4}(x)=x^{4}-6x^{2}+3,\qquad
g_{5}(x)=x^{5}-10x^{3}+15x$ $\displaystyle
g_{6}(x)=x^{6}-15x^{4}+45x^{2}-15,\qquad g_{7}(x)=x^{7}-21x^{5}+105x^{3}-105x$
$\displaystyle g_{8}(x)=x^{8}-28x^{6}+210x^{4}-420x^{2}+105,$

$C^{a}_{t/2}(0)=\Phi(z_{t/2}-\mu_{a})$, and
$C^{a}_{t/2}(i)=-\phi(z_{t/2}-\mu_{a})g_{i-1}(z_{t/2}-\mu_{a})$ for $i\geq 1$.
Furthermore, for $X\sim N(0,1)$,

$\displaystyle E\\{g_{i}(X)\\}=0,$ $\displaystyle\mbox{ for }i=1,\ldots,8,$
$\displaystyle\mbox{and }\quad E\\{g_{i_{1}}(X)g_{i_{2}}(X)\\}=0,$
$\displaystyle\mbox{ for any }i_{1},i_{2}\in\\{0,\ldots,8\\},i_{1}\neq i_{2};$

and for $\mu_{a}=0$,

$\displaystyle C^{a}_{1-t/2}(i)=\begin{cases}C^{a}_{t/2}(i),&\text{ if i is
odd}\\\ -C^{a}_{t/2}(i),&\text{ if i is even}.\\\ \end{cases}$ (9)

###### Proof.

The proof is based on straightforward but tedious evaluations on the partial
derivatives of $f_{\\{t_{1},\ldots,t_{k}\\}}(\rho_{1},\ldots,\rho_{k};a,t)$
with respect to $\rho_{1},\ldots,\rho_{k}$; the details are omitted. ∎

######  Lemma 2.

Recall the definition:
$t_{i}=\mathrm{1}(|Z_{i}|>|z_{t/2}|)=\mathrm{1}(P_{i}<t)$; we have

  * (P1).

when $i\neq j$, and $i,j\in\mathcal{H}_{1}$,

$\displaystyle E(t_{i}t_{j})-E(t_{i})E(t_{j})$ $\displaystyle=$
$\displaystyle\left\\{C^{i}_{t/2}(1)-C^{i}_{1-t/2}(1)\right\\}\left\\{C^{j}_{t/2}(1)-C^{j}_{1-t/2}(1)\right\\}\sigma_{ij}$
$\displaystyle+\frac{1}{2}\left\\{C^{i}_{t/2}(2)-C^{i}_{1-t/2}(2)\right\\}\left\\{C^{j}_{t/2}(2)-C^{j}_{1-t/2}(2)\right\\}\sigma_{ij}^{2}$
$\displaystyle+\frac{E\\{g_{3}^{2}(X_{1})\\}}{(3!)^{2}}\left\\{C_{t/2}^{i}(3)-C_{1-t/2}^{i}(3)\right\\}\left\\{C_{t/2}^{j}(3)-C_{1-t/2}^{j}(3)\right\\}\sigma_{ij}^{3}+O(\sigma_{ij}^{4});$

  * (P2).

when $i\in\mathcal{H}_{1},j\in\mathcal{H}_{0}$, for any $t$, there exists a
unique root $\mu_{t}\in(|z_{t/2}|,|z_{t/2}|+1)$ of
$C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)=0$ in $\mu_{j}$, such that when
$\mu_{j}\leq\mu_{t}$,

$\displaystyle
O(\sigma_{ij}^{4})<E(t_{i}t_{j})-E(t_{i})E(t_{j})=-\sigma_{ij}^{2}\phi(z_{t/2})z_{t/2}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}+O(\sigma_{ij}^{4})$
$\displaystyle\leq$
$\displaystyle\phi(z_{t/2})|z_{t/2}|C_{t}^{\max}\sigma_{ij}^{2}+O(\sigma_{ij}^{4}),$

where

$\displaystyle
C_{t}^{\max}=\sup_{\mu\in(-\mu_{t},\mu_{t})}\left\\{\phi(|z_{t/2}|+\mu)(|z_{t/2}|+\mu)+\phi(|z_{t/2}|-\mu)(|z_{t/2}|-\mu)\right\\};$

and when $|\mu_{j}|\geq\mu_{t}$,

$\displaystyle E(t_{i}t_{j})-E(t_{i})E(t_{j})<O(\sigma_{ij}^{4}).$

  * (P3).

when $i\neq j$, and $i,j\in\mathcal{H}_{0}$,

$\displaystyle
E(t_{i}t_{j})-E(t_{i})E(t_{j})=2\phi^{2}(z_{t/2})z_{t/2}^{2}\sigma_{ij}^{2}+O(\sigma_{ij}^{4}).$

###### Proof.

Note that we can write

$\displaystyle
E(t_{i}t_{j})-E(t_{i})E(t_{j})=\sum_{a_{1},a_{2}\in\\{1,-1\\}}a_{1}a_{2}h_{i,j}(a_{1},a_{2}),$

where

$\displaystyle
h_{i,j}(a_{1},a_{2})=P(Z_{i}<a_{1}z_{t/2},Z_{j}<a_{2}z_{t/2})-P(Z_{i}<a_{1}z_{t/2})P(Z_{j}<a_{1}z_{t/2}).$

We derive $h_{i,j}(a_{1},a_{2})$ for $a_{1}=a_{2}=1$; the derivations for
other values of $a_{1}$ and $a_{2}$ are similar. Let $X_{1},X_{2},X_{3}$ be
i.i.d. standard normal random variables, we can write

$\displaystyle
Z_{i}=\mu_{i}+\sqrt{|\sigma_{ij}|}X_{1}+\sqrt{1-|\sigma_{ij}|}X_{2}$
$\displaystyle
Z_{j}=\mu_{j}+\sqrt{|\sigma_{ij}|}X_{1}+\sqrt{1-|\sigma_{ij}|}X_{3}$
$\displaystyle\mbox{if }\sigma_{ij}\geq 0,$

and

$\displaystyle
Z_{j}=\mu_{j}-\sqrt{|\sigma_{ij}|}X_{1}+\sqrt{1-|\sigma_{ij}|}X_{3}\quad\mbox{if
}\sigma_{ij}<0.$

Hereafer, we assume that $\sigma_{ij}\geq 0$; the case for $\sigma_{ij}<0$ can
be derived the same. The first term of $h_{i,j}(a_{1},a_{2})$ is given by

$\begin{split}&P(Z_{i}<z_{t/2},Z_{j}<z_{t/2})\\\
=&\int\Phi\left(\dfrac{z_{t/2}-\mu_{i}-\sqrt{\sigma_{ij}}x_{1}}{\sqrt{1-\sigma_{ij}}}\right)\Phi\left(\dfrac{z_{t/2}-\mu_{j}-\sqrt{\sigma_{ij}}x_{1}}{\sqrt{1-\sigma_{ij}}}\right)\phi(x_{1})dx_{1}.\\\
\end{split}$ (10)

Note that by definition
$\Phi\left(\dfrac{z_{t/2}-\mu_{a}-\sqrt{\sigma_{ij}}x_{1}}{\sqrt{1-\sigma_{ij}}}\right)=f_{\\{1\\}}(\sqrt{\sigma_{ij}};a,t)$;
applying Taylor expansion and Lemma 1, we immediately have

$\displaystyle\Phi\left(\dfrac{z_{t/2}-\mu_{a}-\sqrt{\sigma_{ij}}x_{1}}{\sqrt{1-\sigma_{ij}}}\right)=\sum_{i_{1}\leq
7}C^{a}_{t/2}(i_{1})g_{i_{1}}(x_{1})(\sqrt{\sigma_{ij}})^{i_{1}}/(i_{1}!)+R(\sigma_{ij}),$
(11)

where $R(\sigma_{ij})$ is the Lagrange residual term in the Taylor’s
expansion, satisfying $|R(\sigma_{ij})|\lesssim|f(x_{1})|\sigma_{ij}^{4}$ up
to a univeral constant not depending on $x_{1}$, with $f(x_{1})$ being a
finite order polynomial function of $x_{1}$. Combining (10) and (11) leads to

$\begin{split}&P(Z_{i}<z_{t/2},Z_{j}<z_{t/2})\\\ =&\sum_{i_{1}+i_{2}\leq
7}C^{i}_{t/2}(i_{1})C^{j}_{t/2}(i_{2})(\sqrt{\sigma_{ij}})^{i_{1}+i_{2}}E(g_{i_{1}}(X_{1})g_{i_{2}}(X_{1}))/(i_{1}!i_{2}!)+O(\sigma_{ij}^{4})\\\
=&\sum_{i_{1}=0}^{3}C^{i}_{t/2}(i_{1})C^{j}_{t/2}(i_{1})\sigma_{ij}^{i_{1}}E(g_{i_{1}}^{2}(X_{1}))/(i_{1}!)^{2}+O(\sigma_{ij}^{4}),\\\
\end{split}$ (12)

where the second “=” is because $E(g_{i_{1}}(X_{1})g_{i_{2}}(X_{1}))=0$ for
$i_{1}\neq i_{2}$ based on Lemma 1; $X_{1}$ is a standard normal random
variable. Furthermore, it is straightforward to check that for $i_{1}=0$,

$\displaystyle
C^{i}_{t/2}(i_{1})C^{j}_{t/2}(i_{1})\sigma_{ij}^{i_{1}}E(g_{i_{1}}^{2}(X_{1}))/(i_{1}!)^{2}=P(Z_{i}<z_{t/2})P(Z_{j}<z_{t/2}),$

which together with (12) leads to

$\displaystyle h_{i,j}(1,1)$ $\displaystyle=$ $\displaystyle
P(Z_{i}<z_{t/2},Z_{j}<z_{t/2})-P(Z_{i}<z_{t/2})P(Z_{j}<z_{t/2})$
$\displaystyle=$
$\displaystyle\sum_{i_{1}=1}^{3}C^{i}_{t/2}(i_{1})C^{j}_{t/2}(i_{1})\sigma_{ij}^{i_{1}}E(g_{i_{1}}^{2}(X_{1}))/(i_{1}!)^{2}+O(\sigma_{ij}^{4}).$

Similarly, we can derive $h_{i,j}(1,-1)$, $h_{i,j}(-1,1)$, and
$h_{i,j}(-1,-1)$. As consequence, we have

$\begin{split}&E(t_{i}t_{j})-E(t_{i})E(t_{j})=\sum_{a_{1},a_{2}\in\\{1,-1\\}}a_{1}a_{2}h_{i,j}(a_{1},a_{2})\\\
=&\sum_{i_{1}=1}^{3}C^{i}_{t/2}(i_{1})C^{j}_{t/2}(i_{1})\sigma_{ij}^{i_{1}}E\\{g_{i_{1}}^{2}(X_{1})\\}/(i_{1}!)^{2}\\\
-&\sum_{i_{1}=1}^{3}C^{i}_{1-t/2}(i_{1})C^{j}_{t/2}(i_{1})\sigma_{ij}^{i_{1}}E\\{g_{i_{1}}^{2}(X_{1})\\}/(i_{1}!)^{2}\\\
-&\sum_{i_{1}=1}^{3}C^{i}_{t/2}(i_{1})C^{j}_{1-t/2}(i_{1})\sigma_{ij}^{i_{1}}E\\{g_{i_{1}}^{2}(X_{1})\\}/(i_{1}!)^{2}\\\
+&\sum_{i_{1}=1}^{3}C^{i}_{1-t/2}(i_{1})C^{j}_{1-t/2}(i_{1})\sigma_{ij}^{i_{1}}E\\{g_{i_{1}}^{2}(X_{1})\\}/(i_{1}!)^{2}+O(\sigma_{ij}^{4})\\\
=&\sum_{i_{1}=1}^{3}\frac{\sigma_{ij}^{i_{1}}E\\{g_{i_{1}}^{2}(X_{1})\\}}{(i_{1}!)^{2}}\left\\{C^{i}_{t/2}(i_{1})-C^{i}_{1-t/2}(i_{1})\right\\}\left\\{C^{j}_{t/2}(i_{1})-C^{j}_{1-t/2}(i_{1})\right\\}+O(\sigma_{ij}^{4}).\end{split}$

We are able to check:

  * •

when $\mu_{i}=0$, for $i_{1}=1$ or $3$,

$\displaystyle C^{i}_{t/2}(i_{1})-C^{i}_{1-t/2}(i_{1})=0,$

and for $i_{1}=2$

$\displaystyle
C^{i}_{t/2}(i_{1})-C^{i}_{1-t/2}(i_{1})=-2\phi(z_{t/2})z_{t/2};$

  * •

when $\mu_{i}\neq 0$, for $i_{1}=1$,

$\displaystyle
C^{i}_{t/2}(i_{1})-C^{i}_{1-t/2}(i_{1})=-\phi(|z_{t/2}|+\mu_{i})+\phi(|z_{t/2}|-\mu_{i});$

for $i_{1}=2$,

$\displaystyle C^{i}_{t/2}(2)-C^{i}_{1-t/2}(2)$ $\displaystyle=$
$\displaystyle-\phi(z_{t/2}-\mu_{i})(z_{t/2}-\mu_{i})-\phi(z_{t/2}+\mu_{i})(z_{t/2}+\mu_{i})$
$\displaystyle=$
$\displaystyle\phi(|z_{t/2}|+\mu_{i})(|z_{t/2}|+\mu_{i})+\phi(|z_{t/2}|-\mu_{i})(|z_{t/2}|-\mu_{i}).$

We can verify that for any given $t\in(0,1)$, there exists a unique
$\mu_{t}\in(z_{t/2},z_{t/2}+1)$, which is the root of
$C^{i}_{t/2}(2)-C^{i}_{1-t/2}(2)=0$ for $\mu_{i}\in(|z_{t/2}|,|z_{t/2}|+1)$,
and $C^{i}_{t/2}(2)-C^{i}_{1-t/2}(2)>0$ if and only if
$\mu_{i}\in(-\mu_{t},\mu_{t})$. This follows from the following facts and that
$\phi(|z_{t/2}|+\mu_{i})(|z_{t/2}|+\mu_{i})+\phi(|z_{t/2}|-\mu_{i})(|z_{t/2}|-\mu_{i})$
as a function of $\mu_{i}$ is symmetric about $0$.

    * (1)

When $\mu_{i}\in[-|z_{t/2}|,|z_{t/2}|]$, we have both $|z_{t/2}|+\mu_{i}>0$
and $|z_{t/2}|-\mu_{i}>0$; therefore $C^{i}_{t/2}(2)-C^{i}_{1-t/2}(2)>0$.

    * (2)

When $\mu_{i}\geq|z_{t/2}|+1$, we have

$\displaystyle
C^{i}_{t/2}(2)-C^{i}_{1-t/2}(2)=\phi(u_{1})(u_{1})-\phi(u_{2})(u_{2})<0,$

where $u_{1}=\mu_{i}+|z_{t/2}|$, $u_{2}=\mu_{i}-|z_{t/2}|$, and
$u_{1}>u_{2}>1$; and we can verify $\phi(u)u$ is straightly decreasing when
$u>1$.

    * (3)

When $\mu_{i}\in(|z_{t/2}|,|z_{t/2}|+1)$, we have

$\displaystyle\frac{\partial\left\\{C^{i}_{t/2}(2)-C^{i}_{1-t/2}(2)\right\\}}{\partial\mu_{i}}$
$\displaystyle=$
$\displaystyle\phi(|z_{t/2}|+\mu_{i})\left\\{1-(|z_{t/2}|+\mu_{i})^{2}\right\\}-\phi(|z_{t/2}|-\mu_{i})\left\\{1-(|z_{t/2}|-\mu_{i})^{2}\right\\}$
$\displaystyle\leq$
$\displaystyle\phi(|z_{t/2}|+\mu_{i})\left\\{(|z_{t/2}|-\mu_{i})^{2}-(|z_{t/2}|+\mu_{i})^{2}\right\\}$
$\displaystyle=$ $\displaystyle-4\phi(|z_{t/2}|+\mu_{i})|z_{t/2}|\mu_{i}<0,$

where “$\leq$” is based on the facts that when
$\mu_{i}\in(|z_{t/2}|,|z_{t/2}|+1)$, $1-(|z_{t/2}-\mu_{i})^{2}>0$ and
$\phi(|z_{t/2}|+\mu_{i})<\phi(|z_{t/2}|-\mu_{i})$.

Furthermore, set

$\displaystyle
C_{t}^{\max}=\sup_{\mu\in(-\mu_{t},\mu_{t})}\left\\{\phi(|z_{t/2}|+\mu)(|z_{t/2}|+\mu)+\phi(|z_{t/2}|-\mu)(|z_{t/2}|-\mu)\right\\},$

then we have $C^{i}_{t/2}(2)-C^{i}_{1-t/2}(2)\leq C_{t}^{\max}$.

and for $i_{1}=3$,

$\displaystyle C^{i}_{t/2}(3)-C^{i}_{1-t/2}(3)$ $\displaystyle=$
$\displaystyle-\phi(z_{t/2}-\mu_{i})\left\\{(z_{t/2}-\mu_{i})^{2}-1\right\\}+\phi(z_{t/2}+\mu_{i})\left\\{(z_{t/2}+\mu_{i})^{2}-1\right\\}$
$\displaystyle=$
$\displaystyle-\phi(|z_{t/2}|+\mu_{i})\left\\{(|z_{t/2}|+\mu_{i})^{2}-1\right\\}+\phi(|z_{t/2}|-\mu_{i})\left\\{(|z_{t/2}|-\mu_{i})^{2}-1\right\\}$

We can conclude:

  * •

when $\mu_{i}\neq 0$ and $\mu_{j}\neq 0$,

$\displaystyle E(t_{i}t_{j})-E(t_{i})E(t_{j})$ $\displaystyle=$
$\displaystyle\left\\{C^{i}_{t/2}(1)-C^{i}_{1-t/2}(1)\right\\}\left\\{C^{j}_{t/2}(1)-C^{j}_{1-t/2}(1)\right\\}\sigma_{ij}$
$\displaystyle+\frac{1}{2}\left\\{C^{i}_{t/2}(2)-C^{i}_{1-t/2}(2)\right\\}\left\\{C^{j}_{t/2}(2)-C^{j}_{1-t/2}(2)\right\\}\sigma_{ij}^{2}$
$\displaystyle+\frac{E\\{g_{3}^{2}(X_{1})\\}}{(3!)^{2}}\left\\{C_{t/2}^{i}(3)-C_{1-t/2}^{i}(3)\right\\}\left\\{C_{t/2}^{j}(3)-C_{1-t/2}^{j}(3)\right\\}\sigma_{ij}^{3}+O(\sigma_{ij}^{4});$

which verifies (P1).

  * •

when $\mu_{i}\neq 0$ but $\mu_{j}=0$,

$\displaystyle
E(t_{i}t_{j})-E(t_{i})E(t_{j})=-\sigma_{ij}^{2}\phi(z_{t/2})z_{t/2}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}+O(\sigma_{ij}^{4})$
$\displaystyle\leq$
$\displaystyle\phi(z_{t/2})|z_{t/2}|C_{t}^{\max}\sigma_{ij}^{2}+O(\sigma_{ij}^{4}),$

when $\mu_{i}\in[-\mu_{t},\mu_{t}]$, and

$\displaystyle E(t_{i}t_{j})-E(t_{i})E(t_{j})<O(\sigma_{ij}^{4}),$

otherwise. Therefore, (P2) is proved.

  * •

when $\mu_{i}=0$ and $\mu_{j}=0$,

$\displaystyle
E(t_{i}t_{j})-E(t_{i})E(t_{j})=2\phi^{2}(z_{t/2})z_{t/2}^{2}\sigma_{ij}^{2}+O(\sigma_{ij}^{4}),$

which is our (P3) and we complete the proof of this lemma.

∎

######  Lemma 3.

Recall the definition:
$t_{i}=\mathrm{1}(|Z_{i}|>|z_{t/2}|)=\mathrm{1}(P_{i}<t)$; we have

$\begin{split}&\sum_{i\neq j\neq k\neq
l}\Big{\\{}E(t_{i}t_{j}t_{k}t_{l})-E(t_{i}t_{j}t_{k})E(t_{l})-E(t_{i}t_{j}t_{l})E(t_{k})-E(t_{i}t_{k}t_{l})E(t_{j})-E(t_{j}t_{k}t_{l})E(t_{i})\\\
+&E(t_{i}t_{j})E(t_{k})E(t_{l})+E(t_{i}t_{k})E(t_{j})E(t_{l})+E(t_{i}t_{l})E(t_{j})E(t_{k})+E(t_{j}t_{k})E(t_{i})E(t_{l})\\\
+&E(t_{j}t_{l})E(t_{i})E(t_{k})+E(t_{k}t_{l})E(t_{i})E(t_{j})-3E(t_{i})E(t_{j})E(t_{k})E(t_{l})\Big{\\}}\\\
=&3\left[\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(1)-C_{1-t/2}^{i}(1)\right\\}\left\\{C_{t/2}^{j}(1)-C_{1-t/2}^{j}(1)\right\\}\sigma_{ij}\right]^{2}\\\
&+3\left[\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(1)-C_{1-t/2}^{i}(1)\right\\}\left\\{C_{t/2}^{j}(1)-C_{1-t/2}^{j}(1)\right\\}\sigma_{ij}\right]\\\
&\times\left[\sum_{i\neq
j}\left\\{C_{t/2}^{i}(2)-C_{1-t/2}^{i}(2)\right\\}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}\sigma_{ij}^{2}\right]\\\
&+O\left(\sum_{i\in\mathcal{H}_{1}}\left(\sum_{j\neq
i,j\in\mathcal{H}_{1}}|\sigma_{ij}|\right)^{3}\right)+O\left(\sum_{i\neq
j}|\sigma_{ij}|\sum_{k\neq i,k\in\mathcal{H}_{1}}|\sigma_{ik}|\sum_{l\neq
j,l\in\mathcal{H}_{1}}|\sigma_{jl}|\right)+O\left(p^{2}\sum_{i\neq
j}\sigma_{ij}^{4}\right).\\\ \end{split}$

###### Proof.

We can write

$\begin{split}&E(t_{i}t_{j}t_{k}t_{l})-E(t_{i}t_{j}t_{k})E(t_{l})-E(t_{i}t_{j}t_{l})E(t_{k})-E(t_{i}t_{k}t_{l})E(t_{j})-E(t_{j}t_{k}t_{l})E(t_{i})\\\
+&E(t_{i}t_{j})E(t_{k})E(t_{l})+E(t_{i}t_{k})E(t_{j})E(t_{l})+E(t_{i}t_{l})E(t_{j})E(t_{k})+E(t_{j}t_{k})E(t_{i})E(t_{l})\\\
+&E(t_{j}t_{l})E(t_{i})E(t_{k})+E(t_{k}t_{l})E(t_{i})E(t_{j})-3E(t_{i})E(t_{j})E(t_{k})E(t_{l})\\\
=&\sum_{a_{1},a_{2},a_{3},a_{4}=\\{1,-1\\}}a_{1}a_{2}a_{3}a_{4}h_{i,j,k,l;t}(a_{1},a_{2},a_{3},a_{4}),\end{split}$
(13)

where
$h_{i,j,k,l}(a_{1},a_{2},a_{3},a_{4})=P(Z_{i}<a_{1}z_{t/2},Z_{j}<a_{2}z_{t/2},Z_{k}<a_{3}z_{t/2},Z_{l}<a_{4}z_{t/2})-P(Z_{i}<a_{1}z_{t/2},Z_{j}<a_{2}z_{t/2},Z_{k}<a_{3}z_{t/2})P(Z_{l}<a_{4}z_{t/2})-P(Z_{i}<a_{1}z_{t/2},Z_{j}<a_{2}z_{t/2},Z_{l}<a_{4}z_{t/2})P(Z_{k}<a_{3}z_{t/2})-P(Z_{i}<a_{1}z_{t/2},Z_{k}<a_{3}z_{t/2},Z_{l}<a_{4}z_{t/2})P(Z_{j}<a_{2}z_{t/2})-P(Z_{j}<a_{2}z_{t/2},Z_{k}<a_{3}z_{t/2},Z_{l}<a_{4}z_{t/2})P(Z_{i}<a_{1}z_{t/2})+P(Z_{i}<a_{1}z_{t/2},Z_{j}<a_{2}z_{t/2})P(Z_{k}<a_{3}z_{t/2})P(Z_{l}<a_{4}z_{t/2})+P(Z_{i}<a_{1}z_{t/2},Z_{k}<a_{3}z_{t/2})P(Z_{j}<a_{2}z_{t/2})P(Z_{l}<a_{4}z_{t/2})+P(Z_{i}<a_{1}z_{t/2},Z_{l}<a_{4}z_{t/2})P(Z_{j}<a_{2}z_{t/2})P(Z_{k}<a_{3}z_{t/2})+P(Z_{j}<a_{2}z_{t/2},Z_{k}<a_{3}z_{t/2})P(Z_{i}<a_{1}z_{t/2})P(Z_{l}<a_{4}z_{t/2})+P(Z_{j}<a_{2}z_{t/2},Z_{l}<a_{4}z_{t/2})P(Z_{i}<a_{1}z_{t/2})P(Z_{k}<a_{3}z_{t/2})+P(Z_{k}<a_{3}z_{t/2},Z_{l}<a_{4}z_{t/2})P(Z_{i}<a_{1}z_{t/2})P(Z_{j}<a_{2}z_{t/2})-3P(Z_{i}<a_{1}z_{t/2})P(Z_{j}<a_{2}z_{t/2})P(Z_{k}<a_{3}z_{t/2})P(Z_{l}<a_{4}z_{t/2})$.

Next, we consider the terms in $h_{i,j,k,l}(1,1,1,1)$; those for other values
of $a_{1},\ldots,a_{4}$ can be similarly derived. Without loss of generality,
we assume $\sigma_{ij}\geq 0$ for $i=1,\ldots,p;j=1,\ldots,p$. Note that
$h_{i,j,k,l}(1,1,1,1)$ contains 16 terms with the last three terms identical,
and its first term is given by

$\begin{split}&P(Z_{i}<z_{t/2},Z_{j}<z_{t/2},Z_{k}<z_{t/2},Z_{l}<z_{t/2})\\\
=&\int\Phi\left(\dfrac{z_{t/2}-\mu_{i}-\sqrt{\sigma_{ij}}x_{1}-\sqrt{\sigma_{ik}}x_{2}-\sqrt{\sigma_{il}}x_{3}}{\sqrt{1-\sigma_{ij}-\sigma_{ik}-\sigma_{il}}}\right)\Phi\left(\dfrac{z_{t/2}-\mu_{j}-\sqrt{\sigma_{ij}}x_{1}-\sqrt{\sigma_{jk}}x_{4}-\sqrt{\sigma_{jl}}x_{5}}{\sqrt{1-\sigma_{ij}-\sigma_{jk}-\sigma_{jl}}}\right)\\\
&\Phi\left(\dfrac{z_{t/2}-\mu_{k}-\sqrt{\sigma_{ik}}x_{2}-\sqrt{\sigma_{jk}}x_{4}-\sqrt{\sigma_{kl}}x_{6}}{\sqrt{1-\sigma_{ik}-\sigma_{jk}-\sigma_{kl}}}\right)\Phi\left(\dfrac{z_{t/2}-\mu_{l}-\sqrt{\sigma_{il}}x_{3}-\sqrt{\sigma_{jl}}x_{5}-\sqrt{\sigma_{kl}}x_{6}}{\sqrt{1-\sigma_{il}-\sigma_{jl}-\sigma_{kl}}}\right)\\\
&\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\phi(x_{5})\phi(x_{6})dx_{1}dx_{2}dx_{3}dx_{4}dx_{5}dx_{6}.\\\
\end{split}$ (14)

Applying Taylor expansion and Lemma 1, we have

$\begin{split}&\Phi\left(\dfrac{z_{t/2}-\mu_{i}-\sqrt{\sigma_{ij}}x_{1}-\sqrt{\sigma_{ik}}x_{2}-\sqrt{\sigma_{il}}x_{3}}{\sqrt{1-\sigma_{ij}-\sigma_{ik}-\sigma_{il}}}\right)\\\
=&\sum_{i_{1}+i_{2}+i_{3}\leq
7}C^{i}_{t/2}(i_{1}+i_{2}+i_{3})g_{i_{1}}(x_{1})g_{i_{2}}(x_{2})g_{i_{3}}(x_{3})(\sqrt{\sigma_{ij}})^{i_{1}}(\sqrt{\sigma_{ik}})^{i_{2}}(\sqrt{\sigma_{il}})^{i_{3}}/(i_{1}+i_{2}+i_{3})!+R(\vec{\rho}),\end{split}$

where $R(\vec{\rho})$ is the Lagrange residual term in the Taylor’s expansion,
and
$|R(\vec{\rho})|\lesssim|f(x_{1},x_{2},x_{3})|(|\sigma_{ij}|^{4}+|\sigma_{ik}|^{4}+|\sigma_{il}|^{4})$
up to a universal constant not depending on $x_{1},x_{2},x_{3}$, where
$f(x_{1},x_{2},x_{3})$ is a finite order polynomial function of
$\\{x_{1},x_{2},x_{3}\\}$. Similarly, we can apply Taylor expansion to other
$\Phi(\cdot)$ terms in (14); we obtain

$\begin{split}&P(Z_{i}<z_{t/2},Z_{j}<z_{t/2},Z_{k}<z_{t/2},Z_{l}<z_{t/2})\\\
=&\int\sum_{\sum_{j=1}^{12}i_{j}\leq
7}C^{i}_{t/2}(i_{1}+i_{2}+i_{3})C^{j}_{t/2}(i_{7}+i_{4}+i_{5})C^{k}_{t/2}(i_{8}+i_{10}+i_{6})C^{l}_{t/2}(i_{9}+i_{11}+i_{12})\\\
&g_{i_{1}}(x_{1})g_{i_{7}}(x_{1})g_{i_{2}}(x_{2})g_{i_{8}}(x_{2})g_{i_{3}}(x_{3})g_{i_{9}}(x_{3})g_{i_{4}}(x_{4})g_{i_{10}}(x_{4})g_{i_{5}}(x_{5})g_{i_{11}}(x_{5})g_{i_{6}}(x_{6})g_{i_{12}}(x_{6})\\\
&(\sqrt{\sigma_{ij}})^{i_{1}+i_{7}}(\sqrt{\sigma_{ik}})^{i_{2}+i_{8}}(\sqrt{\sigma_{il}})^{i_{3}+i_{9}}(\sqrt{\sigma_{jk}})^{i_{4}+i_{10}}(\sqrt{\sigma_{jl}})^{i_{5}+i_{11}}(\sqrt{\sigma_{kl}})^{i_{6}+i_{12}}\\\
&/((i_{1}+i_{2}+i_{3})!(i_{7}+i_{4}+i_{5})!(i_{8}+i_{10}+i_{6})!(i_{9}+i_{11}+i_{12})!)\\\
&\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\phi(x_{5})\phi(x_{6})dx_{1}dx_{2}dx_{3}dx_{4}dx_{5}dx_{6}\\\
&+O\left(|\sigma_{ij}|^{4}+|\sigma_{ik}|^{4}+|\sigma_{il}|^{4}+|\sigma_{jk}|^{4}+|\sigma_{jl}|^{4}+|\sigma_{kl}|^{4}\right).\end{split}$
(15)

Applying Lemma 1, (15) can be further simplified to be

$\displaystyle P(Z_{i}<z_{t/2},Z_{j}<z_{t/2},Z_{k}<z_{t/2},Z_{l}<z_{t/2})$
(16) $\displaystyle=$
$\displaystyle\sum_{\sum_{j=1}^{6}i_{j}\in\\{0,1,2,3\\}}C^{i}_{t/2}(i_{1}+i_{2}+i_{3})C^{j}_{t/2}(i_{1}+i_{4}+i_{5})C^{k}_{t/2}(i_{2}+i_{4}+i_{6})C^{l}_{t/2}(i_{3}+i_{5}+i_{6})$
$\displaystyle\hskip
28.90755pt\times\frac{\prod_{j=1}^{6}E\left\\{g_{i_{j}}^{2}(X_{1})\right\\}\sigma_{ij}^{i_{1}}\sigma_{ik}^{i_{2}}\sigma_{il}^{i_{3}}\sigma_{jk}^{i_{4}}\sigma_{jl}^{i_{5}}\sigma_{kl}^{i_{6}}}{(i_{1}+i_{2}+i_{3})!(i_{1}+i_{4}+i_{5})!(i_{2}+i_{5}+i_{6})!(i_{3}+i_{5}+i_{6})!}$
$\displaystyle+O\left(|\sigma_{ij}|^{4}+|\sigma_{ik}|^{4}+|\sigma_{il}|^{4}+|\sigma_{jk}|^{4}+|\sigma_{jl}|^{4}+|\sigma_{kl}|^{4}\right).$

where $X_{1}$ is a standard normal random variable. Recall that (16) is the
first term of $h_{i,j,k,l}(1,1,1,1)$; we next discuss how each subitem under
the summation of the main part on the right hand side of (16) associates with
other terms contributes in the evaluation of (13). In particular, we shall
discuss the possible values of $i_{1},\ldots,i_{6}$; clearly if one of them is
zero, its corresponding “$\sigma$” term in (16) does not appear in the
expression.

  * (A1)

If the subscripts of the set of $\sigma$’s whose corresponding powers are
nonzero contain and only contain three letters, and therefore one letter does
not appear in these subscripts, the subitem in (16) is identical to the
corresponding subitem in one and only one of the second to fifth item in
$h_{i,j,k,l}(1,1,1,1)$, whose coefficients are all “$-$”. As a consequence,
they cancel each other.

For example $i_{3}=i_{5}=i_{6}=0$ but at least two of
$\\{i_{1},i_{2},i_{4}\\}$ are nonzero, so that the set of $\sigma$ whose
corresponding powers are nonzero is from
$\\{\sigma_{ij},\sigma_{ik},\sigma_{jk}\\}$, and $\\{i,j,k\\}$ all appear in
the subscript but $l$ does not. The subitem in (16) is given by

$\displaystyle
C^{i}_{t/2}(i_{1}+i_{2})C^{j}_{t/2}(i_{1}+i_{4})C^{k}_{t/2}(i_{2}+i_{4})C^{l}_{t/2}(0)$
$\displaystyle\hskip
28.90755pt\times\frac{E\left\\{g_{i_{1}}^{2}(X_{1})\right\\}E\left\\{g_{i_{2}}^{2}(X_{1})\right\\}E\left\\{g_{i_{4}}^{2}(X_{1})\right\\}\sigma_{ij}^{i_{1}}\sigma_{ik}^{i_{2}}\sigma_{jk}^{i_{4}}}{(i_{1}+i_{2})!(i_{1}+i_{4})!(i_{2}+i_{4})!},$

which is identical and only identical to a corresponding subitem in the Taylor
expansion for the second item in $h_{i,j,k,l}(1,1,1,1)$ by noting that
$C^{l}_{t/2}(0)=P(Z_{l}<z_{t/2})$; in particular, similarly to the development
of (16), we are able to derive:

$\displaystyle P(Z_{i}<z_{t/2},Z_{j}<z_{t/2},Z_{k}<z_{t/2})$ (17)
$\displaystyle=$
$\displaystyle\sum_{i_{1}+i_{2}+i_{4}\in\\{0,1,2,3\\}}C^{i}_{t/2}(i_{1}+i_{2})C^{j}_{t/2}(i_{1}+i_{4})C^{k}_{t/2}(i_{2}+i_{4})$
$\displaystyle\hskip
28.90755pt\times\frac{E\left\\{g_{i_{1}}^{2}(X_{1})\right\\}E\left\\{g_{i_{2}}^{2}(X_{1})\right\\}E\left\\{g_{i_{4}}^{2}(X_{1})\right\\}\sigma_{ij}^{i_{1}}\sigma_{ik}^{i_{2}}\sigma_{jk}^{i_{4}}}{(i_{1}+i_{2})!(i_{1}+i_{4})!(i_{2}+i_{4})!}$
$\displaystyle+O\left(|\sigma_{ij}|^{4}+|\sigma_{ik}|^{4}+|\sigma_{jk}|^{4}\right).$

  * (A2)

If the subscripts of the set $\sigma$’s whose corresponding powers are nonzero
contain only two letter in their subscripts, indicating that one and one and
only of $i_{j},j=1,\ldots,6$ is nonzero, we can find two of the second to
fifth terms and one of the sixth to thirteenth terms in $h_{i,j,k,l}(1,1,1,1)$
contain the same corresponding subitem in (16). Note that the coefficients of
the second to fifth terms in $h_{i,j,k,l}(1,1,1,1)$ are “-”, while those of
the sixth to thirteenth terms in $h_{i,j,k,l}(1,1,1,1)$ are ”$+$”, therefore
they are cancelled.

For example, if we have only $i_{1}>0$, The subitem in (16) is given by

$\displaystyle
C^{k}_{t/2}(0)C^{l}_{t/2}(0)C^{i}_{t/2}(i_{1})C^{j}_{t/2}(i_{1})\frac{E\left\\{g_{i_{1}}^{2}(X_{1})\right\\}\sigma_{ij}^{i_{1}}}{(i_{1}!)^{2}},$

which is identical to the corresponding terms in
$P(Z_{i}<z_{t/2},Z_{j}<z_{t/2},Z_{k}<Z_{t/2})P(Z_{l}<z_{t/2})$ by observing
its Taylor expansion given by (17),
$P(Z_{i}<z_{t/2},Z_{j}<z_{t/2},Z_{l}<Z_{t/2})P(Z_{k}<z_{t/2})$ (details are
omitted), and
$P(Z_{i}<z_{t/2},Z_{j}<z_{t/2})P(Z_{k}<z_{t/2})P(Z_{l}<Z_{t/2})P(Z_{k}<z_{t/2})$,
since

$\displaystyle
P(Z_{i}<z_{t/2},Z_{j}<z_{t/2})=\sum_{i_{1}=0}^{3}\frac{C^{i}_{t/2}(i_{1})C^{j}_{t/2}(i_{1})E\left\\{g_{i_{1}}^{2}(X_{1})\right\\}\sigma_{ij}^{i_{1}}}{(i_{1}!)^{2}}+O\left(|\sigma_{ij}|^{4}\right).$

  * (A3)

If $i_{j}=0$ for $j=1,\ldots,6$, it can be checked that all the terms in
$h_{i,j,k,l}(1,1,1,1)$ contain the subitem
$C^{k}_{t/2}(0)C^{l}_{t/2}(0)C^{i}_{t/2}(0)C^{j}_{t/2}(0)$, where seven have
the coefficient “$+$”, and the other seven have the coefficient “$-$”,
therefore they cancel each other.

We observe that all the subitems in the second to the sixteenth items in
$h_{i,j,k,l}(1,1,1,1)$ have been considered and therefore cancelled in Cases
(A1)–(A3). Only the subitems with the set of $\sigma$’s whose corresponding
powers are nonzeros contain all four letters in their subscripts. We consider
the following possibilities.

  * (B1)

Only two $i_{j}$’s are nonzero; therefore only two $\sigma$’s appear in the
subitem in (16). The corresponding $\sigma$ components in it have three
possibilities: $\sigma_{ij}^{i_{1}}\sigma_{kl}^{i_{6}}$,
$\sigma_{ik}^{i_{2}}\sigma_{jl}^{i_{5}}$, and
$\sigma_{il}^{i_{3}}\sigma_{jk}^{i_{4}}$. These three cases can be considered
similarly; as an example, we consider that it is
$\sigma_{ij}^{i_{1}}\sigma_{kl}^{i_{6}}$, with $i_{1}>0$ and $i_{6}>0$.
Furthermore since $0<i_{1}+i_{6}\leq 3$, we must have “$i_{1}=1,i_{6}=1$” or
“$i_{1}=1,i_{6}=2$” or “$i_{1}=2,i_{6}=1$”.

    * –

If $i_{1}=1$ and $i_{6}=1$, the subitem in (16) is give by

$\displaystyle
C^{i}_{t/2}(1)C^{j}_{t/2}(1)C^{k}_{t/2}(1)C^{l}_{t/2}(1)\sigma_{ij}\sigma_{kl},$

and we can check that the corresponding subitem in
$h_{i,j,k,l}(a_{1},a_{2},a_{3},a_{4})$ is given by

$\displaystyle\left\\{C^{i}_{t/2}(1)\right\\}^{\frac{1+a_{1}}{2}}\left\\{C^{i}_{1-t/2}(1)\right\\}^{\frac{1-a_{1}}{2}}\left\\{C^{j}_{t/2}(1)\right\\}^{\frac{1+a_{2}}{2}}\left\\{C^{j}_{1-t/2}(1)\right\\}^{\frac{1-a_{2}}{2}}$
$\displaystyle\times$
$\displaystyle\left\\{C^{k}_{t/2}(1)\right\\}^{\frac{1+a_{3}}{2}}\left\\{C^{k}_{1-t/2}(1)\right\\}^{\frac{1-a_{3}}{2}}\left\\{C^{l}_{t/2}(1)\right\\}^{\frac{1+a_{4}}{2}}\left\\{C^{l}_{1-t/2}(1)\right\\}^{\frac{1-a_{4}}{2}}\sigma_{ij}\sigma_{kl}.$

Using the fact that $C_{t/2}^{a}(1)=C_{1-t/2}^{a}(1)$ when $\mu_{a}=0$ given
by (9), we can check that when one of $\\{\mu_{i},\mu_{j},\mu_{k},\mu_{l}\\}$
is equal to 0, all these subtems will cancel each other across
$h_{i,j,k,l}(a_{1},a_{2},a_{3},a_{4})$ when evaluating (13). Therefore this
subitem is remained only when $\\{\mu_{i},\mu_{j},\mu_{k},\mu_{l}\\}$ are all
nonzero, and their contribution in (13) is given by

$\displaystyle\left\\{C_{t/2}^{i}(1)-C_{1-t/2}^{i}(1)\right\\}\left\\{C_{t/2}^{j}(1)-C_{1-t/2}^{j}(1)\right\\}$
$\displaystyle\times\left\\{C_{t/2}^{k}(1)-C_{1-t/2}^{k}(1)\right\\}\left\\{C_{t/2}^{l}(1)-C_{1-t/2}^{l}(1)\right\\}\sigma_{ij}\sigma_{kl}.$

    * –

If $i_{1}=1$, $i_{6}=2$, the subitem in (16) is given by

$\displaystyle\frac{1}{2}C^{i}_{t/2}(1)C^{j}_{t/2}(1)C^{k}_{t/2}(2)C^{l}_{t/2}(2)\sigma_{ij}\sigma_{kl}^{2},$

and the corresponding subitem in $h_{i,j,k,l}(a_{1},a_{2},a_{3},a_{4})$ is
given by

$\displaystyle\left\\{C^{i}_{t/2}(1)\right\\}^{\frac{1+a_{1}}{2}}\left\\{C^{i}_{1-t/2}(1)\right\\}^{\frac{1-a_{1}}{2}}\left\\{C^{j}_{t/2}(1)\right\\}^{\frac{1+a_{2}}{2}}\left\\{C^{j}_{1-t/2}(1)\right\\}^{\frac{1-a_{2}}{2}}$
$\displaystyle\times$
$\displaystyle\left\\{C^{k}_{t/2}(2)\right\\}^{\frac{1+a_{3}}{2}}\left\\{C^{k}_{1-t/2}(2)\right\\}^{\frac{1-a_{3}}{2}}\left\\{C^{l}_{t/2}(2)\right\\}^{\frac{1+a_{4}}{2}}\left\\{C^{l}_{1-t/2}(2)\right\\}^{\frac{1-a_{4}}{2}}\sigma_{ij}\sigma_{kl}^{2}.$

When $\mu_{i}=0$ or $\mu_{j}=0$, all these subitems will cancel each other
across $h_{i,j,k,l}(a_{1},a_{2},a_{3},a_{4})$ when evaluating (13). Therefore
this subitem is remained only when $\mu_{i}\neq 0$ and $\mu_{j}\neq 0$, and
their contribution in (13) is given by

$\displaystyle\frac{1}{2}\left\\{C_{t/2}^{i}(1)-C_{1-t/2}^{i}(1)\right\\}\left\\{C_{t/2}^{j}(1)-C_{1-t/2}^{j}(1)\right\\}$
$\displaystyle\times\left\\{C_{t/2}^{k}(2)-C_{1-t/2}^{k}(2)\right\\}\left\\{C_{t/2}^{l}(2)-C_{1-t/2}^{l}(2)\right\\}\sigma_{ij}\sigma_{kl}^{2}.$

    * –

If $i_{1}=2$ and $i_{6}=1$, we can similarly conclude that the corresponding
subitem in $h_{i,j,k,l}(a_{1},a_{2},a_{3},a_{4})$ can be remained only when
$\mu_{k}\neq 0$ and $\mu_{l}\neq 0$, and their contribution in (13) is given
by

$\displaystyle\frac{1}{2}\left\\{C_{t/2}^{i}(2)-C_{1-t/2}^{i}(2)\right\\}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}$
$\displaystyle\times\left\\{C_{t/2}^{k}(1)-C_{1-t/2}^{k}(1)\right\\}\left\\{C_{t/2}^{l}(1)-C_{1-t/2}^{l}(1)\right\\}\sigma_{ij}^{2}\sigma_{kl}.$

In summary, for subitems considered in (B1), when summed over $i,j,k,l$ for
$i\neq j\neq k\neq l$, they are equal to

$\displaystyle 3\left[\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(1)-C_{1-t/2}^{i}(1)\right\\}\left\\{C_{t/2}^{j}(1)-C_{1-t/2}^{j}(1)\right\\}\sigma_{ij}\right]^{2}$
$\displaystyle+3\left[\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(1)-C_{1-t/2}^{i}(1)\right\\}\left\\{C_{t/2}^{j}(1)-C_{1-t/2}^{j}(1)\right\\}\sigma_{ij}\right]$
$\displaystyle\times\left[\sum_{i\neq
j}\left\\{C_{t/2}^{i}(2)-C_{1-t/2}^{i}(2)\right\\}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}\sigma_{ij}^{2}\right].$

  * (B2)

There are three $i_{j}$’s are nonzero, and therefore each is equal to 1. Note
that the corresponding subscripts of $\sigma$ have six letters; and each of
$i,j,k,l$ must appear at least once. There are two possibilities

    * –

One letter appears three times, but each of the other letters appear once. For
example, letter $i$ appears three times, but $j,k,l$ appear once, namely
$i_{1}=i_{2}=i_{3}=1$, which corresponds to the subitem in (16):

$\displaystyle
C_{t/2}^{i}(3)C_{t/2}^{j}(1)C_{t/2}^{k}(1)C_{t/2}^{l}(1)\frac{\sigma_{ij}\sigma_{ik}\sigma_{il}}{3!}.$

Using similar arguments as those in (B1), we can conclude that this subitem is
remained in $h_{i,j,k,l}(a_{1},a_{2},a_{3},a_{4})$ only when
$\mu_{i},\mu_{j},\mu_{k},\mu_{l}$ are all nonzero, and they are in the order
of $O(\sigma_{ij}\sigma_{ik}\sigma_{il})$.

    * –

Two letters appear twice, but each of the other two letters appear once. For
example, letter $i$ and $j$ appear twice, but $k,l$ appear once, namely
$i_{1}=i_{2}=i_{5}=1$, which corresponds to the subitem in (16):

$\displaystyle
C_{t/2}^{i}(2)C_{t/2}^{j}(2)C_{t/2}^{k}(1)C_{t/2}^{l}(1)\frac{\sigma_{ij}\sigma_{ik}\sigma_{jl}}{2!2!}.$

Using similar arguments as those in (B1), we can conclude that this subitem is
remained in $h_{i,j,k,l}(a_{1},a_{2},a_{3},a_{4})$ only when $\mu_{k}\neq 0$
and $\mu_{l}\neq 0$, and they are in the order of
$O(\sigma_{ij}\sigma_{ik}\sigma_{jl})$.

In summary, the subitems considered in (B2), when summed over $i,j,k,l$ for
$i\neq j\neq k\neq l$, they are in the order of

$\displaystyle O\left(\sum_{i\in\mathcal{H}_{1}}\left(\sum_{j\neq
i,j\in\mathcal{H}_{1}}|\sigma_{ij}|\right)^{3}\right)+O\left(\sum_{i\neq
j}|\sigma_{ij}|\sum_{k\neq i,k\in\mathcal{H}_{1}}|\sigma_{ik}|\sum_{l\neq
j,l\in\mathcal{H}_{1}}|\sigma_{jl}|\right).$

∎

######  Lemma 4.

Recall the definition:
$t_{i}=\mathrm{1}(|Z_{i}|>|z_{t/2}|)=\mathrm{1}(P_{i}<t)$; we have

$\begin{split}&\sum_{i\neq j\neq
k}\Big{|}E(t_{i}t_{j}t_{k})-E(t_{i}t_{j})E(t_{k})-E(t_{i}t_{k})E(t_{j})-E(t_{j}t_{k})E(t_{i})+2E(t_{i})E(t_{j})E(t_{k})\Big{|}\\\
=&O\left(\sum_{i=1}^{p}\left(\sum_{j\neq
i,j\in\mathcal{H}_{1}}|\sigma_{ij}|\right)^{2}\right)+O\left(\sum_{i\in\mathcal{H}_{1}}\left(\sum_{j\neq
i}|\sigma_{ij}|\right)\left(\sum_{k\neq
i,k\in\mathcal{H}_{1}}|\sigma_{ik}|\right)\right)\\\ &+O\left(\sum_{i\neq
j\neq k}|\sigma_{ij}\sigma_{ik}\sigma_{kl}|\right)+O\left(p\sum_{i\neq
j}\sigma_{ij}^{4}\right).\\\ \end{split}$

###### Proof.

The proof of this lemma is similar to but simpler than that of Lemma 3. The
details are thus omitted. ∎

######  Lemma 5.

Let

$\displaystyle
m(\bar{V},\bar{R})=\frac{\bar{V}}{E(\bar{R})}-\frac{E(\bar{V})}{[E(\bar{R})]^{2}}\bar{R}.$

Assume $\limsup_{p\to\infty}p_{0}t/(p_{1}\bar{\xi})<1$, and as $p$ is
sufficiently large,

$\displaystyle\sum_{i,j}|\sigma_{ij}|=O\left(p^{2-\delta}\right),$ (18)
$\displaystyle\sum_{i\neq
j;i,j\in\mathcal{H}_{0}}\sigma_{ij}^{2}\geq\frac{C_{t}^{\max}}{\phi(z_{t/2})|z_{t/2}|}\sum_{i\in\mathcal{H}_{1},j\in\mathcal{H}_{0},\mu_{i}\in[-\mu_{t},\mu_{t}]}\sigma_{ij}^{2},$
(19) $\displaystyle\sum_{i\neq
j;i,j\in\mathcal{H}_{0}}\sigma_{ij}^{2}+p\gtrsim\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\sigma_{ij}^{2}$ (20) $\displaystyle\sum_{i\neq
j}\sigma_{ij}^{4}=o\left(\sum_{i\neq
j;i,j\in\mathcal{H}_{0}}\sigma_{ij}^{2}+p_{0}\right),$ (21)

where $\mu_{t}$ and $C_{t}^{\max}$ are defined in Lemma 2. We have

$\displaystyle E\left[\\{R-E(R)\\}^{4}\right]$ $\displaystyle=$ $\displaystyle
o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right)$ (22) $\displaystyle
E\left[\\{V-E(V)\\}^{4}\right]$ $\displaystyle=$ $\displaystyle
o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right).$ (23)

###### Proof.

We consider $\mbox{Var}\left(m(\bar{V},\bar{R})\right)$ first. Recall that
$R=V+S$; therefore $\bar{R}=\bar{V}+\bar{S}$ with $\bar{R}=R/p$, $\bar{V}=V/p$
and $\bar{S}=S/p$, and

$\displaystyle m(\bar{V},\bar{R})$ $\displaystyle=$
$\displaystyle\frac{E(\bar{S})}{\\{E(\bar{R})\\}^{2}}\bar{V}-\frac{E(\bar{V})}{\\{E(\bar{R})\\}^{2}}\bar{S}$
$\displaystyle=$ $\displaystyle
A\left(\frac{\bar{V}}{E(\bar{V})}-\frac{\bar{S}}{E(\bar{S})}\right),$

where $A=E(\bar{S})E(\bar{V})/\\{E(\bar{R})\\}^{2}$. We shall derive

$\displaystyle\mbox{Var}\left\\{m(\bar{V},\bar{R})\right\\}\gtrsim\frac{\sum_{i\neq
j;i,j\in\mathcal{H}_{0}}\sigma_{ij}^{2}+p_{0}+\mbox{Var}(S)}{p^{2}}.$ (24)

To this end, consider

$\displaystyle\frac{1}{A^{2}}\mbox{Var}\left(m(\bar{V},\bar{R})\right)$
$\displaystyle=$
$\displaystyle\mbox{Var}(\bar{V}/E(\bar{V}))+\mbox{Var}(\bar{S}/E(\bar{S}))-2\mbox{Cov}(\bar{V}/E(\bar{V}),\bar{S}/E(\bar{S})),$
(25)

and based on Lemma 2,

$\displaystyle\mbox{Cov}(\bar{V}/E(\bar{V}),\bar{S}/E(\bar{S}))=\frac{1}{E(V)E(S)}\sum_{i\in\mathcal{H}_{1},j\in\mathcal{H}_{0}}\left\\{E(t_{i}t_{j})-E(t_{i})E(t_{j})\right\\}$
(26) $\displaystyle\leq$
$\displaystyle\frac{\phi(z_{t/2})|z_{t/2}|C_{t}^{\max}}{p_{0}p_{1}t\bar{\xi}}\sum_{i\in\mathcal{H}_{1},j\in\mathcal{H}_{0};\mu_{i}\in[-\mu_{t},\mu_{t}]}\sigma_{ij}^{2}+\frac{1}{p_{0}p_{1}t\bar{\xi}}O\left(\sum_{i\in\mathcal{H}_{1},j\in\mathcal{H}_{0}}\sigma_{ij}^{4}\right).$

Combining (25) with (26), and applying Lemma 2, we have

$\displaystyle\frac{1}{A^{2}}\mbox{Var}\left(m(\bar{V},\bar{R})\right)$
$\displaystyle\geq$
$\displaystyle\frac{2\phi^{2}(z_{t/2})z_{t/2}^{2}\sum_{i\neq
j;i,j\in\mathcal{H}_{0}}\sigma_{ij}^{2}+p_{0}t(1-t)+O\left(\sum_{i\neq
j,i,j\in\mathcal{H}_{0}}\sigma_{ij}^{4}\right)}{p_{0}^{2}t^{2}}$
$\displaystyle-\frac{p_{0}t}{p_{1}\bar{\xi}}\frac{2\phi(z_{t/2})|z_{t/2}|C_{t}^{\max}}{p_{0}^{2}t^{2}}\sum_{i\in\mathcal{H}_{1},j\in\mathcal{H}_{0};\mu_{i}\in[-\mu_{t},\mu_{t}]}\sigma_{ij}^{2}-\frac{O\left(\sum_{i\in\mathcal{H}_{1},j\in\mathcal{H}_{0}}\sigma_{ij}^{4}\right)}{p_{0}p_{1}t\bar{\xi}}$
$\displaystyle+\mbox{Var}(S/E(S))$ $\displaystyle\gtrsim$
$\displaystyle\frac{\sum_{i\neq
j;i,j\in\mathcal{H}_{0}}\sigma_{ij}^{2}+p_{0}}{p_{0}^{2}}+\mbox{Var}(S/E(S)),$

where “$\gtrsim$” is based on assumptions (19) and (21); this verifies (24).

We next consider $E[\\{R-E(R)\\}^{4}]$:

$\begin{split}&E[\\{R-E(R)\\}^{4}]=E(R^{4})-4E(R^{3})E(R)+6E(R^{2})E^{2}(R)-3E^{4}(R)\\\
=&E\left\\{\left(\sum_{i=1}^{p}t_{i}\right)^{4}\right\\}-4E\left\\{\left(\sum_{i=1}^{p}t_{i}\right)^{3}\right\\}E\left(\sum_{i=1}^{p}t_{i}\right)\\\
&+6E\left\\{\left(\sum_{i=1}^{p}t_{i}\right)^{2}\right\\}E^{2}\left(\sum_{i=1}^{p}t_{i}\right)-3E^{4}\left(\sum_{i=1}^{p}t_{i}\right)\\\
=&E\left(\sum_{i,j,k,l=1}^{p}t_{i}t_{j}t_{k}t_{l}\right)-4E\left(\sum_{i,j,k=1}^{p}t_{i}t_{j}t_{k}\right)E\left(\sum_{l=1}^{p}t_{l}\right)\\\
&+6E\left(\sum_{i,j=1}^{p}t_{i}t_{j}\right)E\left(\sum_{k=1}^{p}t_{k}\right)E\left(\sum_{l=1}^{p}t_{l}\right)\\\
&-3E\left(\sum_{i=1}^{p}t_{i}\right)E\left(\sum_{j=1}^{p}t_{j}\right)E\left(\sum_{k=1}^{p}t_{k}\right)E\left(\sum_{l=1}^{p}t_{l}\right).\end{split}$
(27)

We need to consider the following possibilities:

  * (C1)

Based on Lemma 4, the collection of terms in (27) where $\\{i,j,k,l\\}$ are
all different is given by

$\begin{split}&\sum_{i\neq j\neq k\neq
l}\Big{\\{}E(t_{i}t_{j}t_{k}t_{l})-E(t_{i}t_{j}t_{k})E(t_{l})-E(t_{i}t_{j}t_{l})E(t_{k})-E(t_{i}t_{k}t_{l})E(t_{j})-E(t_{j}t_{k}t_{l})E(t_{i})\\\
+&E(t_{i}t_{j})E(t_{k})E(t_{l})+E(t_{i}t_{k})E(t_{j})E(t_{l})+E(t_{i}t_{l})E(t_{j})E(t_{k})+E(t_{j}t_{k})E(t_{i})E(t_{l})\\\
+&E(t_{j}t_{l})E(t_{i})E(t_{k})+E(t_{k}t_{l})E(t_{i})E(t_{j})-3E(t_{i})E(t_{j})E(t_{k})E(t_{l})\Big{\\}}\\\
=&3\left[\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(1)-C_{1-t/2}^{i}(1)\right\\}\left\\{C_{t/2}^{j}(1)-C_{1-t/2}^{j}(1)\right\\}\sigma_{ij}\right]^{2}\\\
&+3\left[\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(1)-C_{1-t/2}^{i}(1)\right\\}\left\\{C_{t/2}^{j}(1)-C_{1-t/2}^{j}(1)\right\\}\sigma_{ij}\right]\\\
&\times\left[\sum_{i\neq
j}\left\\{C_{t/2}^{i}(2)-C_{1-t/2}^{i}(2)\right\\}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}\sigma_{ij}^{2}\right]\\\
&+O\left(\sum_{i\in\mathcal{H}_{1}}\left(\sum_{j\neq
i,j\in\mathcal{H}_{1}}|\sigma_{ij}|\right)^{3}\right)+O\left(\sum_{i\neq
j}|\sigma_{ij}|\sum_{k\neq i,k\in\mathcal{H}_{1}}|\sigma_{ik}|\sum_{l\neq
j,l\in\mathcal{H}_{1}}|\sigma_{jl}|\right)+O\left(p^{2}\sum_{i\neq
j}\sigma_{ij}^{4}\right)\\\ &\equiv
3\mathcal{I}_{1}+3\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}+\mathcal{I}_{5}.\end{split}$
(28)

We consider the above $\mathcal{I}$ terms one by one. We consider
$\mathcal{I}_{1}+\mathcal{I}_{2}$ first. Note that

$\displaystyle\mathcal{I}_{1}=\mathcal{I}_{1,1}\cdot\mathcal{I}_{1,1}$
$\displaystyle\mathcal{I}_{2}=\mathcal{I}_{1,1}\cdot\mathcal{I}_{2,2},$

with

$\displaystyle\mathcal{I}_{1,1}$ $\displaystyle=$ $\displaystyle\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(1)-C_{1-t/2}^{i}(1)\right\\}\left\\{C_{t/2}^{j}(1)-C_{1-t/2}^{j}(1)\right\\}\sigma_{ij}$
$\displaystyle\mathcal{I}_{2,2}$ $\displaystyle=$ $\displaystyle\sum_{i\neq
j}\left\\{C_{t/2}^{i}(2)-C_{1-t/2}^{i}(2)\right\\}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}\sigma_{ij}^{2}.$

As a consequence

$\displaystyle\mathcal{I}_{1}+\mathcal{I}_{2}=\mathcal{I}_{1,1}(\mathcal{I}_{1,1}+\mathcal{I}_{2,2}),$
(29)

and we need to verify

$\displaystyle\mathcal{I}_{1}+\mathcal{I}_{2}=o\left(p^{4}\mbox{Var}(\bar{V},\bar{R})\right)$
(30)

Based on the assumption (18), we immediately have

$\displaystyle\mathcal{I}_{1,1}=o(p^{2}),$ (31)

and we have the following decomposition for
$\mathcal{I}_{2,2}+\mathcal{I}_{1,1}$:

$\displaystyle\mathcal{I}_{2,2}+\mathcal{I}_{1,1}$ $\displaystyle=$
$\displaystyle\left\\{\mathcal{I}_{1,1}+\frac{1}{2}\mathcal{I}_{2,2}+\mathcal{I}_{2,3}+p_{0}t(1-t)+\sum_{i=1}^{p_{1}}\xi_{i}(1-\xi_{i})\right\\}$
(32)
$\displaystyle+\left\\{\frac{1}{2}\mathcal{I}_{2,2}-\mathcal{I}_{2,3}-p_{0}t(1-t)-\sum_{i=1}^{p_{1}}\xi_{i}(1-\xi_{i})\right\\},$

where

$\displaystyle\mathcal{I}_{2,3}=\frac{E\\{g_{3}^{2}(X_{1})\\}}{(3!)^{2}}\sum_{i\neq
j}\left\\{C_{t/2}^{i}(3)-C_{1-t/2}^{i}(3)\right\\}\left\\{C_{t/2}^{j}(3)-C_{1-t/2}^{j}(3)\right\\}\sigma_{ij}^{3}.$

Furthermore, with Lemmas 1 and 2, we are able to check

$\displaystyle\mbox{Var}(R)$ $\displaystyle=$ $\displaystyle\sum_{i\neq
j}\\{E(t_{i}t_{j})-E(t_{i})E(t_{j})\\}+\sum_{i=1}^{p}\left[E(t_{i}^{2})-\\{E(t_{i})\\}^{2}\right]$
(33) $\displaystyle=$
$\displaystyle\mathcal{I}_{1,1}+\frac{1}{2}\mathcal{I}_{2,2}+\mathcal{I}_{2,3}+p_{0}t(1-t)+\sum_{i=1}^{p_{1}}\xi_{i}(1-\xi_{i})+O\left(\sum_{i\neq
j}\sigma_{ij}^{4}\right),$

and

$\displaystyle\mbox{Var}(V)+2\mbox{Cov}(V,S)$ (34) $\displaystyle=$
$\displaystyle\sum_{i\neq
j;i,j\in\mathcal{H}_{0}}\\{E(t_{i}t_{j})-E(t_{i})E(t_{j})\\}$
$\displaystyle+\sum_{i\in\mathcal{H}_{1},j\in\mathcal{H}_{0}}\\{E(t_{i}t_{j})-E(t_{i})E(t_{j})\\}+\sum_{i=1}^{p_{0}}\left[E(t_{i}^{2})-\\{E(t_{i})\\}^{2}\right]$
$\displaystyle=$ $\displaystyle\frac{1}{2}\mathcal{I}_{2,2}+p_{0}t(1-t)$
$\displaystyle-\frac{1}{2}\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(2)-C_{1-t/2}^{i}(2)\right\\}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}\sigma_{ij}^{2}$
$\displaystyle+O\left(\sum_{i\neq j}\sigma_{ij}^{4}\right)$

Combining (32), (33), and (34) leads to

$\displaystyle\mathcal{I}_{2,2}+\mathcal{I}_{1,1}$ $\displaystyle=$
$\displaystyle\mbox{Var}(R)+\mbox{Var}(V)+2\mbox{Cov}(V,S)-2p_{0}t(1-t)-\sum_{i=1}^{p_{1}}\xi_{i}(1-\xi_{i})$
(35) $\displaystyle-I_{2,3}+\frac{1}{2}\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(2)-C_{1-t/2}^{i}(2)\right\\}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}\sigma_{ij}^{2}$
$\displaystyle+O\left(\sum_{i\neq j}\sigma_{ij}^{4}\right)$ $\displaystyle=$
$\displaystyle
2\mbox{Var}(V)+4\mbox{Cov}(V,S)+\mbox{Var}(S)-2p_{0}t(1-t)-\sum_{i=1}^{p_{1}}\xi_{i}(1-\xi_{i})$
$\displaystyle-I_{2,3}+\frac{1}{2}\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(2)-C_{1-t/2}^{i}(2)\right\\}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}\sigma_{ij}^{2}$
$\displaystyle+O\left(\sum_{i\neq j}\sigma_{ij}^{4}\right).$

Based on (24) and Cauchy-Schwartz inequality, we conclude

$2\mbox{Var}(V)+4\mbox{Cov}(V,S)+\mbox{Var}(S)-2p_{0}t(1-t)-\sum_{i=1}^{p_{1}}\xi_{i}(1-\xi_{i})=O\left(p^{2}\mbox{Var}(m(\bar{V},\bar{R})\right);$
(36)

and based on the assumptions (20) and (21), we have

$\frac{1}{2}\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\left\\{C_{t/2}^{i}(2)-C_{1-t/2}^{i}(2)\right\\}\left\\{C_{t/2}^{j}(2)-C_{1-t/2}^{j}(2)\right\\}\sigma_{ij}^{2}+O\left(\sum_{i\neq
j}\sigma_{ij}^{4}\right)=O\left(p^{2}\mbox{Var}(m(\bar{V},\bar{R})\right)$
(37)

Combining (35) (36) and (37), we have

$\displaystyle\mathcal{I}_{2,2}+\mathcal{I}_{1,1}$ $\displaystyle=$
$\displaystyle-\mathcal{I}_{2,3}+O\left(p^{2}\mbox{Var}(m(\bar{V},\bar{R})\right),$

which together with (31) leads to

$\displaystyle\mathcal{I}_{1,1}(\mathcal{I}_{2,2}+\mathcal{I}_{1,1})=-\mathcal{I}_{1,1}\mathcal{I}_{2,3}+o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right).$
(38)

For $\mathcal{I}_{1,1}\mathcal{I}_{2,3}$:

$\displaystyle|\mathcal{I}_{1,1}\mathcal{I}_{2,3}|\lesssim\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}|\sigma_{i,j}|\sum_{k\neq l}|\sigma_{k,l}|^{3}\lesssim
p^{2}\sum_{i\neq
j}\sigma_{ij}^{4}=o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right),$

which together with (38) and (29) verifies (30).

We proceed to consider $\mathcal{I}_{3}$.

$\displaystyle\sum_{i\neq j\neq k\neq
l;i,j,k,l\in\mathcal{H}_{1}}|\sigma_{ij}||\sigma_{ik}||\sigma_{il}|$
$\displaystyle\leq$ $\displaystyle\sum_{k\neq
l,k,l\in\mathcal{H}_{1}}\left(\sum_{i\neq
j,i,j\in\mathcal{H}_{1}}\sigma_{ij}^{2}\right)^{1/2}\left(\sum_{i\neq j\neq
k\neq l;i,j\in\mathcal{H}_{1}}\sigma_{ik}^{2}\sigma_{il}^{2}\right)^{1/2}$
(39) $\displaystyle=$ $\displaystyle\left(\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\sigma_{ij}^{2}\right)^{1/2}\sum_{k\neq
l;k,l\in\mathcal{H}_{1}}\left\\{(p_{1}-3)\sum_{i\neq k\neq
l;i\in\mathcal{H}_{1}}(\sigma_{ik}^{2}\sigma_{il}^{2})\right\\}^{1/2}$
$\displaystyle\leq$ $\displaystyle p_{1}(p_{1}-3)^{1/2}\left(\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\sigma_{ij}^{2}\right)^{1/2}\left\\{\sum_{k\neq l\neq
i;i,k,l\in\mathcal{H}_{1}}(\sigma_{ik}^{2}\sigma_{il}^{2})\right\\}^{1/2}$
$\displaystyle\lesssim$ $\displaystyle p_{1}^{3/2}\left(\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\sigma_{ij}^{2}\right)^{1/2}\left(p_{1}\sum_{i\neq
k;i,k\in\mathcal{H}_{1}}\sigma_{ik}^{4}+p\sum_{i\neq
l;i,l\in\mathcal{H}_{1}}\sigma_{il}^{4}\right)^{1/2}$ $\displaystyle=$
$\displaystyle p_{1}^{2}\left(\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\sigma_{ij}^{2}\right)^{1/2}\left(2\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}\sigma_{ij}^{4}\right)^{1/2},$

where the first “$\leq$” is because of the Cauchy–Schwarz inequality, the
second “$\leq$” is derived from the Jensen’s inequality by noting that
$\sqrt{x}$ is a concave function, the “$\lesssim$” is based on
$\sigma_{ik}^{2}\sigma_{il}^{2}\lesssim\sigma_{ik}^{4}+\sigma_{il}^{4}$. The
far right hand side of (39) is in the order of
$o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right)$ based on (24) and the
assumptions (20) and (21); this implies

$\displaystyle\mathcal{I}_{3}=O\left(\sum_{i\in\mathcal{H}_{1}}\left(\sum_{j\neq
i,j\in\mathcal{H}_{1}}|\sigma_{ij}|\right)^{3}\right)=o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right).$
(40)

Similarly to the development of (39), we can obtain

$\displaystyle\mathcal{I}_{4}=O\left(\sum_{i\neq j}|\sigma_{ij}|\sum_{k\neq
i,k\in\mathcal{H}_{1}}|\sigma_{ik}|\sum_{l\neq
j,l\in\mathcal{H}_{1}}|\sigma_{jl}|\right)=o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right).$
(41)

Last,

$\displaystyle\mathcal{I}_{5}=O\left(p^{2}\sum_{i\neq
j}\sigma_{ij}^{4}\right)=o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right)$
(42)

is implied by (24) and the assumption (21).

Combining (30), (40), (41), and (42), we conclude that the collection of terms
in (27) where $\\{i,j,k,l\\}$ are all different is in the order of
$o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right)$.

  * (C2)

If one and only one pair in $\\{i,j,k,l\\}$ are equal to each other, as an
example, we consider $i\neq j\neq k=l$, the collection of terms in (27) is
given by

$\begin{split}&\sum_{i\neq j\neq
k}\Big{\\{}E(t_{i}t_{j}t_{k})-E(t_{i}t_{k})E(t_{j})-E(t_{j}t_{k})E(t_{i})-2E(t_{i}t_{j}t_{k})E(t_{k})+E(t_{i})E(t_{j})E(t_{k})\\\
&+2E(t_{i}t_{k})E(t_{j})E(t_{k})+2E(t_{j}t_{k})E(t_{i})E(t_{k})+E(t_{i}t_{j})E^{2}(t_{k})-3E^{2}(t_{k})E(t_{i})E(t_{j})\Big{\\}}\\\
=&\sum_{i\neq j\neq
k}\Big{[}\big{\\{}E(t_{i}t_{j}t_{k})-E(t_{i}t_{k})E(t_{j})-E(t_{j}t_{k})E(t_{i})-E(t_{i}t_{j})E(t_{k})\\\
&\hskip
43.36243pt+2E(t_{i})E(t_{j})E(t_{k})\big{\\}}\\{1-2E(t_{k})\\}\Big{]}\\\
&+\sum_{i\neq j\neq
k}\big{\\{}E(t_{k})-E^{2}(t_{k})\big{\\}}\big{\\{}E(t_{i}t_{j})-E(t_{i})E(t_{j})\big{\\}}\\\
=&O\left(\sum_{i=1}^{p}\left(\sum_{j\neq
i,j\in\mathcal{H}_{1}}|\sigma_{ij}|\right)^{2}\right)+O\left(\sum_{i\in\mathcal{H}_{1}}\left(\sum_{j\neq
i}|\sigma_{ij}|\right)\left(\sum_{k\neq
i,k\in\mathcal{H}_{1}}|\sigma_{ik}|\right)\right)\\\ &+O\left(\sum_{i\neq
j\neq k}|\sigma_{ij}\sigma_{ik}\sigma_{kl}|\right)+O\left(p\sum_{i\neq
j}\sigma_{ij}^{4}\right)+O\left(p\sum_{i\neq
j}\sigma_{ij}^{2}\right)+O\left(p\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}|\sigma_{ij}|\right)\\\
=&\mathcal{I}_{6}+\mathcal{I}_{7}+\mathcal{I}_{8}+\mathcal{I}_{9}+\mathcal{I}_{10}+\mathcal{I}_{11},\end{split}$
(43)

where to achieve the second “$=$”, we have applied Lemmas 2 and 4. Next, we
consider these “$\mathcal{I}$” terms one by one. For $\mathcal{I}_{6}$, since
$|\sigma_{ij}|<1$, therefore by noting (24) and the assumption (18), we have

$\displaystyle\mathcal{I}_{6}=O\left(\sum_{i=1}^{p}\left(\sum_{j\neq
i,j\in\mathcal{H}_{1}}|\sigma_{ij}|\right)^{2}\right)\lesssim p_{1}\sum_{j\neq
i,j\in\mathcal{H}_{1}}|\sigma_{ij}|=p_{1}O(p^{2-\delta})=o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right);$
(44)

and similarly

$\displaystyle\mathcal{I}_{7}=O\left(\sum_{i\in\mathcal{H}_{1}}\left(\sum_{j\neq
i}|\sigma_{ij}|\right)\left(\sum_{k\neq
i,k\in\mathcal{H}_{1}}|\sigma_{ik}|\right)\right)\lesssim p\sum_{i\neq
k;i,k\in\mathcal{H}_{1}}|\sigma_{ik}|=o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right).$
(45)

For $\mathcal{I}_{8}$, since $|\sigma_{ij}|<1$, we have

$\displaystyle\mathcal{I}_{8}=O\left(\sum_{i\neq j\neq
k}|\sigma_{ij}\sigma_{ik}\sigma_{kl}|\right)\lesssim p\sum_{i\neq
j}|\sigma_{ij}|=o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right).$ (46)

For $\mathcal{I}_{9}$ and $\mathcal{I}_{10}$, clearly

$\displaystyle\mathcal{I}_{9}\lesssim\mathcal{I}_{10}=o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R}))\right);$
(47)

and

$\displaystyle\mathcal{I}_{11}=O\left(p\sum_{i\neq
j;i,j\in\mathcal{H}_{1}}|\sigma_{ij}|\right)=o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R}))\right)$
(48)

is obtained based on the assumption (18).

Combining (43)–(48), we conclude that the collection of terms in (27) where
one and only one pair in $\\{i,j,k,l\\}$ are equal to each other is in the
order of $o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right)$.

  * (C3)

The collection of terms in (27) where two pairs in $\\{i,j,k,l\\}$ are equal
to each other carries only $O(p^{2})$ terms; then based on (24), we conclude
that this collection of terms is in the order of
$o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right)$.

  * (C4)

The collection of terms in (27) where at least three of $\\{i,j,k,l\\}$ are
mutually equal to each other also carries $O(p^{2})$ terms, therefore this
collection of terms is in the order of
$o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R})\right)$.

In summary, the discussion of (C1)–(C4) implies

$\displaystyle
E[\\{R-E(R)\\}^{4}]=o\left(p^{4}\mbox{Var}(m(\bar{V},\bar{R}))\right),$

which complete the proof for (22).

The proof for (23) is similar but simpler. In particular, similar developments
in Lemmas 3 and 5 can be applied to derive the rates of convergence for the
terms in $E(V-E(V))^{4}$. But we observe that since $V$ is the summation of
$t_{i}$ over $i\in\mathcal{H}_{0}$, the corresponding terms involving
$\mathcal{H}_{1}$ in both Lemmas will not appear, which simplifies the
developments. Here we skip the details for this development to avoid lengthy
presentations. ∎

######  Lemma 6.

Suppose $(Z_{1},\ldots,Z_{p})^{T}\sim
N((\mu_{1},\ldots,\mu_{p})^{T},\boldsymbol{\Sigma})$ with unit variances,
i.e., $\sigma_{jj}=1\text{ for }j=1,\ldots,p$. For any pair of indices $(i,j)$
with $i\neq j$,

$\displaystyle\frac{\partial\mathrm{Cov}(t_{i},t_{j})}{\partial\sigma_{ij}}=\frac{1}{2\pi\sqrt{1-\sigma_{ij}^{2}}}$
$\displaystyle\left[\exp\left\\{-\frac{\mu_{i-}^{2}+\mu_{j-}^{2}-2\sigma_{ij}\mu_{i-}\mu_{j-}}{2(1-\sigma_{ij}^{2})}\right\\}+\exp\left\\{-\frac{\mu_{i+}^{2}+\mu_{j+}^{2}-2\sigma_{ij}\mu_{i+}\mu_{j+}}{2(1-\sigma_{ij}^{2})}\right\\}\right.$
$\displaystyle\left.-\exp\left\\{-\frac{\mu_{i-}^{2}+\mu_{j+}^{2}-2\sigma_{ij}\mu_{i-}\mu_{j+}}{2(1-\sigma_{ij}^{2})}\right\\}-\exp\left\\{-\frac{\mu_{i+}^{2}+\mu_{j-}^{2}-2\sigma_{ij}\mu_{i+}\mu_{j-}}{2(1-\sigma_{ij}^{2})}\right\\}\right],$

where $\mu_{i-}=\mu_{i}+z_{t/2}$ and $\mu_{i+}=\mu_{i}-z_{t/2}$ for
$i=1,\ldots,p$.

###### Proof.

Let $R=(-\infty,z_{t/2})\cup(-z_{t/2},\infty)$ and
$R-\mu_{i}=(-\infty,z_{t/2}-\mu_{i})\cup(-z_{t/2}-\mu_{i},\infty)$, then

$\displaystyle P(|Z_{i}|>|z_{t/2}|,|Z_{j}|>|z_{t/2}|)$ $\displaystyle={}$
$\displaystyle\int_{R}\phi(z_{i}-\mu_{i})\int_{R}\frac{1}{\sqrt{1-\sigma_{ij}^{2}}}\phi\left\\{\frac{(z_{j}-\mu_{j})-\sigma_{ij}(z_{i}-\mu_{i})}{\sqrt{1-\sigma_{ij}^{2}}}\right\\}dz_{j}dz_{i}$
$\displaystyle={}$
$\displaystyle\int_{R}\phi(z_{i}-\mu_{i})\left[\Phi\left\\{\frac{-\mu_{j+}-\sigma_{ij}(z_{i}-\mu_{i})}{\sqrt{1-\sigma_{ij}^{2}}}\right\\}+\Phi\left\\{\frac{\mu_{j-}+\sigma_{ij}(z_{i}-\mu_{i})}{\sqrt{1-\sigma_{ij}^{2}}}\right\\}\right]dz_{i}$
$\displaystyle={}$
$\displaystyle\int_{R-\mu_{i}}\phi(z_{i})\left[1-\Phi\left(\frac{\mu_{j+}+\sigma_{ij}z_{i}}{\sqrt{1-\sigma_{ij}^{2}}}\right)+\Phi\left(\frac{\mu_{j-}+\sigma_{ij}z_{i}}{\sqrt{1-\sigma_{ij}^{2}}}\right)\right]dz_{i}$

where the first equality holds because the conditional distribution of $Z_{j}$
given $Z_{i}=z_{i}$ is
$N[\mu_{j}+\sigma_{ij}(z_{i}-\mu_{i}),1-\sigma_{ij}^{2}]$. Then,

$\displaystyle\frac{\partial\mathrm{Cov}(t_{i},t_{j})}{\partial\sigma_{ij}}={}$
$\displaystyle\frac{\partial
P(|Z_{i}|>|z_{t/2}|,|Z_{j}|>|z_{t/2}|)}{\partial\sigma_{ij}}$
$\displaystyle={}$
$\displaystyle-(1-\sigma_{ij}^{2})^{-3/2}\int_{R-\mu_{i}}\phi(z_{i})\phi\left(\frac{\mu_{j+}+\sigma_{ij}z_{i}}{\sqrt{1-\sigma_{ij}^{2}}}\right)(\mu_{j+}\sigma_{ij}+z_{i})dz_{i}$
$\displaystyle+(1-\sigma_{ij}^{2})^{-3/2}\int_{R-\mu_{i}}\phi(z_{i})\phi\left(\frac{\mu_{j-}+\sigma_{ij}z_{i}}{\sqrt{1-\sigma_{ij}^{2}}}\right)(\mu_{j-}\sigma_{ij}+z_{i})dz_{i}$
$\displaystyle={}$
$\displaystyle-\frac{1}{2\pi}(1-\sigma_{ij}^{2})^{-3/2}\int_{R-\mu_{i}}\exp\left\\{-\frac{z_{i}^{2}+2\sigma_{ij}z_{i}\mu_{j+}+\mu_{j+}^{2}}{2(1-\sigma_{ij}^{2})}\right\\}(\mu_{j+}\sigma_{ij}+z_{i})dz_{i}$
$\displaystyle+\frac{1}{2\pi}(1-\sigma_{ij}^{2})^{-3/2}\int_{R-\mu_{i}}\exp\left\\{-\frac{z_{i}^{2}+2\sigma_{ij}z_{i}\mu_{j-}+\mu_{j-}^{2}}{2(1-\sigma_{ij}^{2})}\right\\}(\mu_{j-}\sigma_{ij}+z_{i})dz_{i}$
$\displaystyle={}$
$\displaystyle\frac{1}{2\pi\sqrt{1-\sigma_{ij}^{2}}}\left[\exp\left\\{-\frac{\mu_{i+}^{2}+\mu_{j+}^{2}-2\sigma_{ij}\mu_{i+}\mu_{j+}}{2(1-\sigma_{ij}^{2})}\right\\}-\exp\left\\{-\frac{\mu_{i-}^{2}+\mu_{j+}^{2}-2\sigma_{ij}\mu_{i-}\mu_{j+}}{2(1-\sigma_{ij}^{2})}\right\\}\right.$
$\displaystyle\left.-\exp\left\\{-\frac{\mu_{i+}^{2}+\mu_{j-}^{2}-2\sigma_{ij}\mu_{i+}\mu_{j-}}{2(1-\sigma_{ij}^{2})}\right\\}+\exp\left\\{-\frac{\mu_{i-}^{2}+\mu_{j-}^{2}-2\sigma_{ij}\mu_{i-}\mu_{j-}}{2(1-\sigma_{ij}^{2})}\right\\}\right].$

∎

##  3 Proof of Theorem 1 and Corollary 1

Define $\bar{V}=V(t)/p$ and $\bar{R}=R(t)/p$, then
$\mathrm{FDP}(t)=\bar{V}/\bar{R}$. Let $H(v,r)=v/r$ be a function of $(v,r)$
and apply Taylor expansion on this function at $(E(\bar{V}),E(\bar{R}))$ and
plugging in $v=\bar{V}$, $r=\bar{R}\vee c$ to the expansion, where
$\bar{R}\vee c=\max\\{\bar{R},c\\}$ and $c$ is a sufficiently small constant
with $0<c<0.5E(\bar{R})$; we have

$\displaystyle\frac{\bar{V}}{\bar{R}\vee
c}=\frac{E(\bar{V})}{E(\bar{R})}+\frac{\bar{V}-E(\bar{V})}{E(\bar{R})}-\frac{E(\bar{V})}{\\{E(\bar{R})\\}^{2}}\Big{\\{}\bar{R}\vee
c-E(\bar{R})\Big{\\}}+r^{*}(\bar{V},\bar{R}),$

where $r^{*}(\bar{V},\bar{R})$ is the remainder term in the Taylor expansion.
Since $E(\bar{R})>c$ and $\bar{R}\vee c\geq c$, and in an arbitrary order of
the partial derivatives of the function $H(v,r)$, only $r$ can appear in the
denominator, therefore, we can verify

$\displaystyle|r^{*}(\bar{V},\bar{R})|\lesssim\left\\{\bar{V}-E(\bar{V})\right\\}^{2}+\left\\{\bar{R}\vee
c-E(\bar{R})\right\\}^{2}.$ (49)

Therefore, we can have

$\displaystyle\mbox{FDP}(t)=E(\bar{V})/E(\bar{R})+m(\bar{V},\bar{R})+\mathcal{J}+r^{*}(\bar{V},\bar{R}),$
(50)

where

$\displaystyle m(\bar{V},\bar{R})$ $\displaystyle=$
$\displaystyle\frac{\bar{V}}{E(\bar{R})}-\frac{E(\bar{V})}{\\{E(\bar{R})\\}^{2}}\bar{R}$
$\displaystyle\mathcal{J}$ $\displaystyle=$
$\displaystyle\frac{\bar{V}}{\bar{R}}-\frac{\bar{V}}{\bar{R}\vee c}.$

Define $r(\bar{V},\bar{R})=r^{*}(\bar{V},\bar{R})+\mathcal{J}$. To complete
the proof of the theorem, it suffices to show that both $E(\mathcal{J}^{2})$
and $E\\{r^{*2}(\bar{V},\bar{R})\\}$ can be dominated by
$\mbox{Var}\\{m(\bar{V},\bar{R})\\}$. We consider
$E\\{r^{*2}(\bar{V},\bar{R})\\}$ first. Based on (3), we have

$\displaystyle|r^{*}(\bar{V},\bar{R})|$ $\displaystyle\lesssim$
$\displaystyle\left\\{\bar{V}-E(\bar{V})\right\\}^{2}+\left\\{\bar{R}\vee
c-\bar{R}\right\\}^{2}+\left\\{\bar{R}-E(\bar{R})\right\\}^{2}$
$\displaystyle=$
$\displaystyle\left\\{\bar{V}-E(\bar{V})\right\\}^{2}+\left\\{c-\bar{R}\right\\}^{2}\mathrm{1}(\bar{R}<c)+\left\\{\bar{R}-E(\bar{R})\right\\}^{2}$
$\displaystyle\leq$
$\displaystyle\left\\{\bar{V}-E(\bar{V})\right\\}^{2}+c^{2}\mathrm{1}(\bar{R}<c)+\left\\{\bar{R}-E(\bar{R})\right\\}^{2},$

which leads to

$\displaystyle E\left\\{r^{*2}(\bar{V},\bar{R})\right\\}$
$\displaystyle\lesssim$ $\displaystyle
E\left\\{\bar{V}-E(\bar{V})\right\\}^{4}+c^{4}P(\bar{R}<c)+E\left\\{\bar{R}-E(\bar{R})\right\\}^{4}$
(51) $\displaystyle\leq$ $\displaystyle
E\left\\{\bar{V}-E(\bar{V})\right\\}^{4}+2E\left\\{\bar{R}-E(\bar{R})\right\\}^{4},$

since based on the definition of $c\in(0,0.5E(\bar{R}))$, we have

$\displaystyle P(\bar{R}<c)\leq
P(|\bar{R}-E(\bar{R})|>c)\leq\frac{E\left\\{\bar{R}-E(\bar{R})\right\\}^{4}}{c^{4}}.$

Combining (51) with Lemma 5, we conclude

$\displaystyle
E\\{r^{*2}(\bar{V},\bar{R})\\}=o\left(\mbox{Var}\\{m(\bar{V},\bar{R})\\}\right).$
(52)

Next, we consider $E(\mathcal{J}^{2})$; in particular

$\displaystyle\mathcal{J}=\frac{\bar{V}}{\bar{R}}-\frac{\bar{V}}{\bar{R}\vee
c}=\frac{\bar{V}}{\bar{R}}\cdot\frac{\bar{R}\vee c-\bar{R}}{\bar{R}\vee
c}=\frac{\bar{V}}{\bar{R}}\cdot\frac{c-\bar{R}}{c}\cdot\mathrm{1}(\bar{R}<c).$

By noting $\bar{V}/\bar{R}\in(0,1)$ and $(c-\bar{R})/c\in(0,1)$ when
$\bar{R}<c$, we have

$\displaystyle E(\mathcal{J}^{2})\leq
E\\{\mathrm{1}(\bar{R}<c)\\}\leq\frac{E\left\\{\bar{R}-E(\bar{R})\right\\}^{4}}{c^{4}}=o\left(\mbox{Var}\\{m(\bar{V},\bar{R}))\right).$
(53)

Combining (52), (53) with (50), we complete the proof of Theorem 1.

We proceed to show Corollary 1. From (50), we have that

$\displaystyle\mbox{Var}\left\\{\mbox{FDP}(t)\right\\}$ $\displaystyle=$
$\displaystyle\mbox{Var}\left\\{m(\bar{V},\bar{R})+\mathcal{J}+r^{*}(\bar{V},\bar{R})\right\\}$
$\displaystyle=$
$\displaystyle\mbox{Var}\left\\{m(\bar{V},\bar{R})\right\\}+\mbox{Var}\left\\{\mathcal{J}\right\\}+\mbox{Var}\left\\{r^{*}(\bar{V},\bar{R})\right\\}$
$\displaystyle+2\mbox{Cov}\left\\{m(\bar{V},\bar{R}),\mathcal{J}\right\\}+2\mbox{Cov}\left\\{m(\bar{V},\bar{R}),r^{*}(\bar{V},\bar{R})\right\\}+2\mbox{Cov}\left\\{\mathcal{J},r^{*}(\bar{V},\bar{R})\right\\},$

which together with (52), (53), and Cauchy-Schwarz inequality immediately
leads to

$\lim_{p\to\infty}\frac{\mbox{Var}\left\\{\mbox{FDP}(t)\right\\}}{\mbox{Var}\left\\{m(\bar{V},\bar{R})\right\\}}=1.$

With straightforward evaluation, we can check that the denominator above is
identical to $V_{1}(t)+V_{2}(t)$. We complete the proof of Corollary 1.

##  4 Proof of Theorem 2

### Part (a)

When $\mu_{i}=\mu_{j}=0$, according to Lemma 6,

$\frac{\partial\mathrm{Cov}(t_{i},t_{j})}{\partial\sigma_{ij}}=\frac{1}{\pi\sqrt{1-\sigma_{ij}^{2}}}\left\\{\exp\left(-\frac{z_{t/2}^{2}}{1+\sigma_{ij}}\right)-\exp\left(-\frac{z_{t/2}^{2}}{1-\sigma_{ij}}\right)\right\\}.$

Thus, $\partial\mathrm{Cov}(t_{i},t_{j})/\partial\sigma_{ij}>0$ if
$\sigma_{ij}>0$ and $\partial\mathrm{Cov}(t_{i},t_{j})/\partial\sigma_{ij}<0$
if $\sigma_{ij}<0$. In addition, $\mathrm{Cov}(t_{i},t_{j})=0$ when
$\sigma_{ij}=0$. Then $\mathrm{Cov}(t_{i},t_{j})>0$ for $\sigma_{ij}\neq 0$.

### Part (b)

When $\mu_{i}=0$ and $\mu_{j}\neq 0$, based on Lemma 6,
$\partial\mathrm{Cov}(t_{i},t_{j})/\partial\sigma_{ij}$ is an odd function of
$\sigma_{ij}$ and an even function of $\mu_{j}$. As
$\mathrm{Cov}(t_{i},t_{j})=0$ when $\sigma_{ij}=0$, it suffices to show that
$\partial\mathrm{Cov}(t_{i},t_{j})/\partial\sigma_{ij}<0$ when $\sigma_{ij}>0$
and $\mu_{j}>0$.

According to Lemma 6, up to positive multiplicative factors,

$\displaystyle\frac{\partial\mathrm{Cov}(t_{i},t_{j})}{\partial\sigma_{ij}}\propto{}$
$\displaystyle\exp\left\\{-\frac{\mu_{j-}^{2}-2\sigma_{ij}z_{t/2}\mu_{j-}}{2(1-\sigma_{ij}^{2})}\right\\}+\exp\left\\{-\frac{\mu_{j+}^{2}+2\sigma_{ij}z_{t/2}\mu_{j+}}{2(1-\sigma_{ij}^{2})}\right\\}$
$\displaystyle-\exp\left\\{-\frac{\mu_{j+}^{2}-2\sigma_{ij}z_{t/2}\mu_{j+}}{2(1-\sigma_{ij}^{2})}\right\\}-\exp\left\\{-\frac{\mu_{j-}^{2}+2\sigma_{ij}z_{t/2}\mu_{j-}}{2(1-\sigma_{ij}^{2})}\right\\}$
$\displaystyle\propto{}$
$\displaystyle\exp\left\\{-\frac{(1-\sigma_{ij})z_{t/2}\mu_{j}-\sigma_{ij}z_{t/2}^{2}}{1-\sigma_{ij}^{2}}\right\\}+\exp\left\\{-\frac{-(1-\sigma_{ij})z_{t/2}\mu_{j}-\sigma_{ij}z_{t/2}^{2}}{1-\sigma_{ij}^{2}}\right\\}$
$\displaystyle-\exp\left\\{-\frac{-(1+\sigma_{ij})z_{t/2}\mu_{j}+\sigma_{ij}z_{t/2}^{2}}{1-\sigma_{ij}^{2}}\right\\}-\exp\left\\{-\frac{(1+\sigma_{ij})z_{t/2}\mu_{j}+\sigma_{ij}z_{t/2}^{2}}{1-\sigma_{ij}^{2}}\right\\}$
$\displaystyle\leq{}$
$\displaystyle\exp\left\\{-\frac{2z_{t/2}\mu_{j}-\sigma_{ij}z_{t/2}\mu_{j}}{2(1-\sigma_{ij}^{2})}\right\\}+\exp\left\\{-\frac{-2z_{t/2}\mu_{j}+3\sigma_{ij}z_{t/2}\mu_{j}}{2(1-\sigma_{ij}^{2})}\right\\}$
$\displaystyle-\exp\left\\{-\frac{-2z_{t/2}\mu_{j}-3\sigma_{ij}z_{t/2}\mu_{j}}{2(1-\sigma_{ij}^{2})}\right\\}-\exp\left\\{-\frac{2z_{t/2}\mu_{j}+\sigma_{ij}z_{t/2}\mu_{j}}{2(1-\sigma_{ij}^{2})}\right\\},$
(54)

where the last inequality holds as $\mu_{j}>2|z_{t/2}|$. Let

$a=\exp[-\sigma_{ij}z_{t/2}\mu_{j}/\\{2(1-\sigma_{ij}^{2})\\}]\quad\text{and}\quad
b=\exp\\{-z_{t/2}\mu_{j}/(1-\sigma_{ij}^{2})\\},$

then the expression in (54) becomes

$\displaystyle a^{-1}b+a^{3}b^{-1}-a^{-3}b^{-1}-ab$
$\displaystyle=a^{-3}b^{-1}(a^{2}b^{2}+a^{6}-a^{4}b^{2}-1)$
$\displaystyle=a^{-3}b^{-1}(a^{2}-1)(a^{4}+a^{2}+1-a^{2}b^{2}).$

Since $a>1$, $\partial\mathrm{Cov}(t_{i},t_{j})/\partial\sigma_{ij}<0$ is
implied by $a^{4}+a^{2}+1-a^{2}b^{2}<0$, which is further implied by
$3a^{2}<b^{2}$. It is straightforward to show that
$t<2\\{1-\Phi^{-1}(\sqrt{\log(3)/2})\\}$ and $\mu_{j}>2|z_{t/2}|$ are
sufficient conditions for $3a^{2}<b^{2}$.

### Part (c)

As $\mathrm{sign}(\sigma_{ij})=\mathrm{sign}(\mu_{i}\mu_{j})$, we can assume
$\mu_{i}>0$, $\mu_{j}>0$, and $\sigma_{ij}>0$ in the proof, because for the
other possible cases, we can add a negative sign to $Z_{i}$ and/or $Z_{j}$,
which does not actually change $\mathrm{Cov}(t_{i},t_{j})$ and thus does not
affect the conclusion. Similar to Part (b), It suffices to show that
$\partial\mathrm{Cov}(t_{i},t_{j})/\partial\sigma_{ij}>0$ when $\mu_{i}>0$,
$\mu_{j}>0$, and $\sigma_{ij}>0$. Further, we assume $\mu_{i}<\mu_{j}$ without
loss of generality.

According to Lemma 6, up to a positive multiplicative factor,

$\displaystyle\frac{\partial\mathrm{Cov}(t_{i},t_{j})}{\partial\sigma_{ij}}\propto{}$
$\displaystyle\exp\left(\frac{z_{t/2}^{2}\sigma_{ij}}{1-\sigma_{ij}^{2}}\right)\left[\exp\left\\{-\frac{z_{t/2}(\mu_{i}+\mu_{j})(1-\sigma_{ij})}{1-\sigma_{ij}^{2}}\right\\}+\exp\left\\{\frac{z_{t/2}(\mu_{i}+\mu_{j})(1-\sigma_{ij})}{1-\sigma_{ij}^{2}}\right\\}\right]$
$\displaystyle-\exp\left(-\frac{z_{t/2}^{2}\sigma_{ij}}{1-\sigma_{ij}^{2}}\right)\left[\exp\left\\{-\frac{z_{t/2}(\mu_{i}-\mu_{j})(1+\sigma_{ij})}{1-\sigma_{ij}^{2}}\right\\}+\exp\left\\{\frac{z_{t/2}(\mu_{i}-\mu_{j})(1+\sigma_{ij})}{1-\sigma_{ij}^{2}}\right\\}\right]$
(55)

Introducing the following notation

$a=\exp\left(\frac{z_{t/2}^{2}\sigma_{ij}}{1-\sigma_{ij}^{2}}\right),\
x=\exp\left\\{-\frac{z_{t/2}(\mu_{i}+\mu_{j})(1-\sigma_{ij})}{1-\sigma_{ij}^{2}}\right\\},\
y=\exp\left\\{\frac{z_{t/2}(\mu_{i}-\mu_{j})(1+\sigma_{ij})}{1-\sigma_{ij}^{2}}\right\\},$

where $a>1$, $x>1$, and $y>1$. Then, the expression in (55) becomes
$a(x+x^{-1})-a^{-1}(y+y^{-1})$.

Next, we show that $ax>a^{-1}y$ implies that $a(x+x^{-1})>a^{-1}(y+y^{-1})$.
When $x>y$, it is easy to see that $a(x+x^{-1})>a^{-1}(y+y^{-1})$. When $x\leq
y$ and $ax>a^{-1}y$, $a(x+x^{-1})>a^{-1}y+ay^{-1}\geq a^{-1}(y+y^{-1})$.

Therefore, a sufficient condition for
$\partial\mathrm{Cov}(t_{i},t_{j})/\partial\sigma_{ij}>0$ is $ax>a^{-1}y$,
which is equivalent to $\sigma_{ij}<\mu_{i}/(\mu_{j}+z_{t/2})$.