#  K-instability of hyperplane sections of Segre Varieties

Shunsuke Saito  Department of Mathematics, Faculty of Science, Tokyo
University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
<EMAIL_ADDRESS>
###### Abstract.

We prove that a normal hyperplane section of the Segre variety $\Sigma_{m,n}$
is K-unstable with respect to any polarization if $m\neq n$ or it is not
smooth.

###### Key words and phrases:

K-stability, hyperplane section, Segre variety

###### 2020 Mathematics Subject Classification:

Primary 14J45; Secondary 32Q20, 32Q26

We work over an algebraically closed field of characteristic zero unless
otherwise stated.

##  1\. Introduction

The existence of a constant scalar curvature Kähler metric (cscK metric for
short) in a given Kähler class has been a central problem in Kähler geometry.
Unfortunately, not all compact Kähler manifolds admit cscK metrics. A theorem
of Matsushima [17] and Lichnerowicz [15] states that the automorphism group of
a compact Kähler manifold admitting a cscK metric must be reductive. Hano [12]
proved that a smooth hyperplane section of the Segre variety $\Sigma_{m,n}$,
the image of the Segre embedding
$\mathbf{P}^{m}\times\mathbf{P}^{n}\hookrightarrow\mathbf{P}^{mn+m+n}$, has
non-reductive automorphism group if $m\neq n$. Thus such a hyperplane section
does not admit cscK metrics in any Kähler class.

The celebrated Yau-Tian-Donaldson conjecture asserts that a polarized manifold
admits a cscK metric if and only if it is K-polystable. The “only if” part was
proved by Stoppa [19], Mabuchi [16], Berman-Darvas-Lu [3], Dyrefelt [7],
Hisamoto [13], and others for various stabilities. The reverse direction is
considered to be wide open in general. In light of the conjecture, a smooth
hyperplane section of $\Sigma_{m,n}$ should be K-unstable when $m\neq n$.
However we can not conclude this assertion due to the lack of an algebro-
geometric counterpart to the Lichnerowicz theorem, specifically the
reductivity of the automorphism group of a K-polystable polarized manifold
(except for the anticanonically polarized Fano case [1], which corresponds to
the Matsushima theorem).

In this paper, we directly prove the K-instability of certain hyperplane
sections of $\Sigma_{m,n}$, which is our main theorem.

######  Theorem A.

Let $X$ be a normal hyperplane section of the Segre variety $\Sigma_{m,n}$. If
$m\neq n$ or $X$ is not smooth, then $X$ is K-unstable with respect to any
polarization.

This is an algebro-geometric analogue of Hano’s result. Our formulation has an
advantage that hyperplane sections are allowed to be singular.

To show Theorem A, we need a formula for the Donaldson-Futaki invariant of a
product test configuration. For notational simplicity, hyperplane sections are
treated as hypersurfaces in the product of projective spaces.

######  Theorem B.

Let $X$ be a normal hypersurface of bidegree $(1,1)$ in
$\mathbf{P}^{m}\times\mathbf{P}^{n}$ and $f$ be its defining polynomial. Let
$\lambda$ be a one-parameter subgroup of
$\operatorname{SL}(m+1)\times\operatorname{SL}(n+1)$ which preserves $X$ and
$\alpha$ be the weight of $f$ for $\lambda$. For positive integers $d$ and
$e$, the Donaldson-Futaki invariant of the product test configuration
$(\mathcal{X}_{\lambda},\mathcal{L}_{\lambda})$ for $(X,\mathcal{O}_{X}(d,e))$
induced by $\lambda$ is given by

$\operatorname{DF}(\mathcal{X}_{\lambda},\mathcal{L}_{\lambda})=-\dfrac{2mnde}{(me+nd)^{2}}\alpha.$

Theorem A will be proved by combining of Theorem B with the construction of a
one-parameter subgroup $\lambda$ satisfying $\alpha=1$. When the field we work
over is the complex number field, this argument also shows the following
result on the Futaki invariant [8, 9] (see also [2] and [5]).

######  Corollary C.

Let $X$ be a smooth hyperplane section of $\Sigma_{m,n}$. If $m+n\geq 4$ and
$m\neq n$, then the Futaki invariant of $X$ does not vanish for any rational
Kähler class.

By [5, Theorem 4], this implies that such a hyperplane section has a chance to
admit a non-trivial extremal Kähler metric in a rational Kähler class. Note
that in the case when $m+n=3$, $X$ is isomorphic to the first Hirzebruch
surface as seen in the proof of Theorem A and admits non-trivial extremal
Kähler metrics in every Kähler class due to Guan [11] and Hwang [14].

##  2\. Donaldson-Futaki invariants and K-instability

This section reviews the definitions of Donaldson-Futaki invariants and
K-instability. For further details, we refer to [4].

Throughout this section, $(X,L)$ denotes an $n$-dimensonal polarized normal
variety.

######  Definition 2.1.

A test configuration for $(X,L)$ is a quadruple of the following data:

  * •

a normal $\mathbf{G}_{m}$-variety $\mathcal{X}$;

  * •

a $\mathbf{G}_{m}$-equivariant flat proper morphism
$\pi\colon\mathcal{X}\to\mathbf{A}^{1}$, where $\mathbf{G}_{m}$ acts on
$\mathbf{A}^{1}$ by the standard multiplication;

  * •

a $\mathbf{G}_{m}$-linearized ample $\mathbf{Q}$-invertible sheaf
$\mathcal{L}$ on $\mathcal{X}$;

  * •

an isomorphism $(\mathcal{X}_{1},\mathcal{L}_{1})\simeq(X,L)$.

We say that $(\mathcal{X},\mathcal{L})$ is product if $\mathcal{X}\simeq
X\times\mathbf{A}^{1}$.

######  Example 2.2.

Fix a $\mathbf{G}_{m}$-action $\lambda$ on $X$. This induces the diagonal
$\mathbf{G}_{m}$-action on $X\times\mathbf{A}^{1}$ as the product of $\lambda$
and the standard action on $\mathbf{A}^{1}$. Since some power of $L$ admits a
$\mathbf{G}_{m}$-linearization, we obtain a product test configuration for
$(X,L)$. We denote by $(\mathcal{X}_{\lambda},\mathcal{L}_{\lambda})$ the test
configuration induced by $\lambda$.

Consider a test configuration $(\mathcal{X},\mathcal{L})$ for $(X,L)$. Let
$w_{k}$ denote the total weight of the dual action of $\mathbf{G}_{m}$ on
$H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{\otimes k})$, and let $N_{k}=\dim
H^{0}(\mathcal{X}_{0},\mathcal{L}_{0}^{\otimes k})$. For a sufficiently
divisible positive integer $k$, we have the following asymptotic expansions:

$\displaystyle N_{k}$ $\displaystyle=a_{0}k^{n}+a_{1}k^{n-1}+\cdots,$
$\displaystyle w_{k}$ $\displaystyle=b_{0}k^{n+1}+b_{1}k^{n}+\cdots.$

######  Definition 2.3.

The Donaldson-Futaki invariant of $(\mathcal{X},\mathcal{L})$ is defined by

$\operatorname{DF}(\mathcal{X},\mathcal{L})=2\dfrac{a_{1}b_{0}-a_{0}b_{1}}{a_{0}^{2}}.$

Note that the Donaldson-Futaki invariant remains unchanged when the
linearization on $\mathcal{L}$ is twisted.

######  Remark 2.4.

Assume that the field we work over is the complex number field. When $X$ is
smooth, the Donaldson-Futaki invariant of the product test configuration
$(\mathcal{X}_{\lambda},\mathcal{L}_{\lambda})$ induced by a
$\mathbf{G}_{m}$-action $\lambda$ on $X$ is a non-zero constant multiple of
the Futaki invariant of $X$ for $c_{1}(L)$ evaluated at the generator of
$\lambda$. We refer to [6, Proposition 2.2.2] for the proof.

######  Definition 2.5.

A polarized normal variety $(X,L)$ is called

  * •

K-semistable if $\operatorname{DF}(\mathcal{X},\mathcal{L})\geq 0$ for all
test configurations $(\mathcal{X},\mathcal{L})$ for $(X,L)$.

  * •

K-unstable if $(X,L)$ is not K-semistable.

##  3\. Proof of the main results

###  3.1. A formula for Donaldson-Futaki invariants

###### Proof of Theorem B.

Set

$S_{a,b}=H^{0}(\mathbf{P}^{m}\times\mathbf{P}^{n},\mathcal{O}(a,b)),\quad
R_{c}=H^{0}(X,\mathcal{O}_{X}(d,e)^{\otimes c})$

for non-negative integers $a$, $b$, and $c$. In this proof, $k$ always denotes
a sufficiently divisible positive integer.

We have an exact sequence

$\displaystyle 0\longrightarrow S_{dk-1,ek-1}\overset{\cdot
f}{\longrightarrow}S_{dk,ek}\longrightarrow R_{k}\longrightarrow 0,$

where the second morphism is multiplication by $f$ and the third is the
restriction onto $X$. Using this, we obtain

$\displaystyle N_{k}$ $\displaystyle=\dim R_{k}=\dim S_{dk,ek}-\dim
S_{dk-1,ek-1}$
$\displaystyle=\binom{m+dk}{m}\binom{n+ek}{n}-\binom{m+dk-1}{m}\binom{n+ek-1}{n}$
$\displaystyle=\dfrac{d^{m-1}e^{n-1}}{m!n!}(me+nd)k^{m+n-1}$
$\displaystyle\qquad+\dfrac{d^{m-2}e^{n-2}}{2m!n!}(m^{2}(m-1)e^{2}+mn(m+n)de+n^{2}(n-1)d^{2})k^{m+n-2}+\cdots$

and

$a_{0}=\dfrac{d^{m-1}e^{n-1}}{m!n!}(me+nd),\quad
a_{1}=\dfrac{d^{m-2}e^{n-2}}{2m!n!}(m^{2}(m-1)e^{2}+mn(m+n)de+n^{2}(n-1)d^{2}).$

Next we compute the total weight $w_{k}$ of the $\mathbf{G}_{m}$-action on
$R_{k}$ induced by $\lambda$. Since $\lambda$ is a one-parameter subgroup of
$\operatorname{SL}(m+1)\times\operatorname{SL}(n+1)$, the total weights of the
actions on $S_{dk-1,ek-1}$ and $S_{dk,ek}$ are both zero. Combining this with
the exact sequence above, we obtain

$\displaystyle w_{k}$ $\displaystyle=-\alpha\dim
S_{dk-1,ek-1}=-\alpha\binom{m+dk-1}{m}\binom{n+ek-1}{n}$
$\displaystyle=-\dfrac{d^{m}e^{n}}{m!n!}\alpha
k^{m+n}-\dfrac{d^{m-1}e^{n-1}}{2m!n!}(m(m-1)e+n(n-1)d)\alpha k^{m+n-1}+\cdots$

and

$b_{0}=-\dfrac{d^{m}e^{n}}{m!n!}\alpha,\quad
b_{1}=-\dfrac{d^{m-1}e^{n-1}}{2m!n!}(m(m-1)e+n(n-1)d)\alpha.$

The formula for
$\operatorname{DF}(\mathcal{X}_{\lambda},\mathcal{L}_{\lambda})$ now follows
from a straightforward computation. ∎

###  3.2. K-instability of hyperplane sections

Let $X$ be a hyperplane section of the Segre variety $\Sigma_{m,n}$. In what
follows, we treat $X$ as a hypersurface of bidegree $(1,1)$ in
$\mathbf{P}^{m}\times\mathbf{P}^{n}$. We choose homogeneous coordinates on
$\mathbf{P}^{m}$ and $\mathbf{P}^{n}$ so that the defining polynomial $f$ of
$X$ is written as

$f(x_{0},\ldots,x_{m},y_{0},\ldots,y_{n})=\sum_{i=0}^{r}x_{i}y_{i},$

where $0\leq r\leq\min\\{m,n\\}$. The Jacobian criterion for smoothness then
provides the following.

######  Lemma 3.1.

  1. (1)

$X$ is smooth if and only if $r=\min\\{m,n\\}$.

  2. (2)

$X$ is normal if and only if $r\geq 1$.

###### Proof of Theorem A.

Assume that $m\leq n$. Suppose that $m\neq n$ or $X$ is not smooth. Then
$m+n\geq 3$ by Lemma 3.1.

If $m+n=3$, then $r=1$ again by Lemma 3.1. $X$ is then isomorphic to the first
Hirzebruch surface, which was proved to be slope unstable and hence K-unstable
with respect to any polarization by Ross-Thomas [18, Example 5.27].

Assume that $m+n\geq 4$. Pick an ample invertible sheaf $L$ on $X$. By the
Grothendieck-Lefschetz hyperplane theorem [10, Exposé XII, Corollaire 3.6],
the restriction
$\operatorname{Pic}(\mathbf{P}^{m}\times\mathbf{P}^{n})\to\operatorname{Pic}(X)$
is an isomorphism in this case. We thus have $L\simeq\mathcal{O}_{X}(d,e)$ for
some positive integers $d$ and $e$. Under our assumption, we have $r<n$ by
Lemma 3.1 and can define a one-parameter subgroup $\lambda$ of
$\operatorname{SL}(m+1)\times\operatorname{SL}(n+1)$ by

$\lambda(t)=(\operatorname{diag}(1,\dots,1),\operatorname{diag}(\underbrace{t^{-1},\dots,t^{-1}}_{r+1},t^{r+1},1,\dots,1)).$

One can easily see that the weight of $f$ with respect to $\lambda$ is 1. We
thus obtain

$\operatorname{DF}(\mathcal{X}_{\lambda},\mathcal{L}_{\lambda})=-\frac{2mnde}{(me+nd)^{2}}<0$

by Theorem B and the K-instability of $(X,L)$. ∎

######  Acknowledgments.

This work was partially supported by JSPS KAKENHI Grant Number JP20K14321.

## References

  * [1] J. Alper, H. Blum, D. Halpern-Leistner, and C. Xu, Reductivity of the automorphism group of K-polystable Fano varieties. _Invent. Math._ 222 (2020), 995–1032.
  * [2] S. Bando, An obstruction for Chern class forms to be harmonic. _Kodai Math. J._ 29 (2006), 337–345.
  * [3] R. J. Berman, T. Darvas, and C. H. Lu, Regularity of weak minimizers of the K-energy and applications to properness and K-stability. _Ann. Sci. Éc. Norm. Supér._ 53 (2020), 267–289.
  * [4] S. Boucksom, T. Hisamoto, and M. Jonsson, Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs. _Ann. Inst. Fourier (Grenoble)_ 67 (2017), 743–841.
  * [5] E. Calabi, Extremal Kähler metrics II. _Differential geometry and complex analysis_ , Springer-Verlag, Berlin, 1985, pp.95–114.
  * [6] S. K. Donaldson, Scalar curvature and stability of toric varieties. _J. Differential Geom._ 62 (2002), 289–349.
  * [7] Z. S. Dyrefelt, On K-polystability of cscK manifolds with transcendental cohomology class (with an appendix by R. Dervan). _Int. Math. Res. Not. IMRN_ 9 (2020), 2769–2817.
  * [8] A. Futaki, An obstruction to the existence of Einstein Kähler metrics. _Invent. Math._ 73 (1983), 437–443.
  * [9] A. Futaki, On compact Kähler manifolds of constant scalar curvatures. _Proc. Japan Acad. Ser. A Math. Sci._ 59 (1983), 401–402.
  * [10] A. Grothendieck, _Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2)_. Documents Mathématiques (Paris) 4 Société Mathématique de France, Paris, 2005.
  * [11] D. Guan, Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends. _Trans. Amer. Math. Soc._ 347 (1995), 2255–2262.
  * [12] J. Hano, Examples of projective manifolds not admitting Kähler metric with constant scalar curvature. _Osaka J. Math._ 20 (1983), 787–791.
  * [13] T. Hisamoto, Stability and coercivity for toric polarizations. arXiv:1610.07998.
  * [14] A. D. Hwang, On existence of Kähler metrics with constant scalar curvature. _Osaka J. Math._ 31 (1994), 561–595.
  * [15] A. Lichnerowicz, Sur les transformations analytiques des variétés kählériennes compactes. _C. R. Acad. Sci. Paris_ 244 (1957), 3011–3013.
  * [16] T. Mabuchi, A stronger concept of K-stability. arXiv:0910.4617.
  * [17] Y. Matsushima, Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne. _Nagoya Math. J._ 11 (1957), 145–150.
  * [18] J. Ross and R. Thomas, An obstruction to the existence of constant scalar curvature Kähler metrics. _J. Differential Geom._ 88 (2011), 109–159.
  * [19] J. Stoppa, K-stability of constant scalar curvature Kähler manifolds. _Adv. Math._ 221 (2009), 1397–1408.


unimodal, where ${\eta}\sim\mathcal{N}(0,I_{K})$, $\rho$ is a non-random
scalar, ${b}$ is a non-random vector, and $A$ is a non-random symmetric
positive definite matrix.

###### of Lemma 5.

By Lemma 3, it suffices to show that $\eta\mid\eta^{\mathrm{\scriptscriptstyle
T}}A\eta\leq a$ is symmetric unimodal. The density function of
$\eta\mid\eta^{\mathrm{\scriptscriptstyle T}}A\eta\leq a$ is

$\displaystyle g({x})=\frac{1\left\\{x^{{\mathrm{\scriptscriptstyle T}}}Ax\leq
a\right\\}}{\textnormal{pr}\left(\eta^{\mathrm{\scriptscriptstyle T}}A\eta\leq
a\right)}(2\pi)^{-K/2}\exp\left(-x^{{\mathrm{\scriptscriptstyle
T}}}x/2\right),$

so

$\displaystyle\log
g(x)=\begin{cases}-\log\left\\{\textnormal{pr}\left(\eta^{\mathrm{\scriptscriptstyle
T}}A\eta\leq a\right)\right\\}-(K/2)\log(2\pi)-x^{\mathrm{\scriptscriptstyle
T}}x/2,&\text{ if }x^{\mathrm{\scriptscriptstyle T}}Ax\leq a,\\\
-\infty,&\text{ otherwise. }\end{cases}$

It is straightforward to show that $\log g(x)$ is concave. By Lemma 4,
$\eta\mid\eta^{\mathrm{\scriptscriptstyle T}}A\eta\leq a$ is symmetric
unimodal. ∎

Now we can prove the first part of Proposition 2.

###### of Proposition 2.

We only give the proof for cluster rerandomization with individual-level
covariates, as the proof for cluster rerandomization with cluster-level
covariates is similar. By Theorem 2, the asymptotic distribution of
$M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}-\tau)\mid
M{\hat{\tau}_{\scalebox{0.6}{{haj,${x}$}}}}^{{\mathrm{\scriptscriptstyle
T}}}A_{x}{\hat{\tau}_{\scalebox{0.6}{{haj,${x}$}}}}\leq a$ is $A+B$ with

$A=V_{\textnormal{haj},\tau\tau}^{1/2}\left(1-R_{x}^{2}\right)^{1/2}\epsilon,\quad
B=V_{\textnormal{haj},\tau\tau}^{1/2}{R}_{x}{\mu}_{x}^{{\mathrm{\scriptscriptstyle
T}}}{\eta}\mid{\eta}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{haj},xx}^{1/2}A_{x}V_{\textnormal{haj},xx}^{1/2}{\eta}\leq
a,$

where $\epsilon$, $\eta_{k}\ (k=1,\ldots,K)$ are independent standard normal
random variables and
$\eta=(\eta_{1},\ldots,\eta_{K})^{\mathrm{\scriptscriptstyle T}}$. $A$ and $B$
are independent and symmetric unimodal where the symmetric unimodality of $B$
follows directly from Lemma 5. Thus, by Lemma 2, $A+B$ is symmetric unimodal.

Next, we prove the second part of Proposition 2. We need the following two
Lemmas. The first one is from Das Gupta et al., (1972, Theorem 2.1).

######  Lemma 6.

Let

$\Sigma=\left[\begin{array}[]{ll}\Sigma_{11}&\Sigma_{12}\\\
\Sigma_{21}&\sigma_{pp}\end{array}\right]$

be a $p\times p$ positive definite matrix with $\Sigma_{11}$ being a
$(p-1)\times(p-1)$ matrix. Let ${x}=$
$\left(x_{1},\cdots,x_{p}\right)^{{\mathrm{\scriptscriptstyle T}}}$ be a
random vector with density function
$\left\lvert\Sigma_{\lambda}\right\rvert^{-1/2}f\left({x}\Sigma_{\lambda}^{-1}{x}^{{\mathrm{\scriptscriptstyle
T}}}\right),$ where

$\Sigma_{\lambda}=\left[\begin{array}[]{ll}\Sigma_{11}&\lambda\Sigma_{12}\\\
\lambda\Sigma_{21}&\sigma_{pp}\end{array}\right],\quad 0\leq\lambda\leq 1.$

If $E$ is a convex symmetric set in $R^{p-1}$, then
${\operatorname{pr}}\big{\\{}\left(x_{1},\cdots,x_{p-1}\right)\in
E,\left\lvert x_{p}\right\rvert\leq h\big{\\}}$ is non-decreasing in
$\lambda.$

The second one is new.

######  Lemma 7.

${\operatorname{pr}}\bigl{\\{}\lvert(1-\rho^{2})^{1/2}\epsilon+\rho{b}^{{\mathrm{\scriptscriptstyle
T}}}{\eta}\rvert<c\mid{\eta}^{{\mathrm{\scriptscriptstyle T}}}A{\eta}\leq
a\bigr{\\}}$ is a non-decreasing function of $\rho$, where
$\epsilon\sim\mathcal{N}(0,1)$ and ${\eta}\sim\mathcal{N}(0,I_{K})$ are
independent, and ${b}$ is a non-random vector satisfying
${b}^{{\mathrm{\scriptscriptstyle T}}}{b}=1$.

###### of Lemma 7.

Simple calculation gives

$\begin{pmatrix}(1-\rho^{2})^{1/2}\epsilon+\rho{b}^{{\mathrm{\scriptscriptstyle
T}}}{\eta}\\\
A^{1/2}{\eta}\end{pmatrix}\sim\mathcal{N}\left(0,\begin{bmatrix}1&\rho
b^{\mathrm{\scriptscriptstyle T}}A^{1/2}\\\ \rho
A^{1/2}b&A\end{bmatrix}\right).$

By Lemma 6,
${\operatorname{pr}}\big{\\{}\lvert(1-\rho^{2})^{1/2}\epsilon+\rho{b}^{{\mathrm{\scriptscriptstyle
T}}}{\eta}\rvert<c,{\eta}^{{\mathrm{\scriptscriptstyle T}}}A{\eta}\leq
a\big{\\}}$ is a non-decreasing function of $\rho$. The same is true for the
conditional probability
${\operatorname{pr}}\big{\\{}\lvert(1-\rho^{2})^{1/2}\epsilon+\rho{b}^{{\mathrm{\scriptscriptstyle
T}}}{\eta}\rvert<c\mid{\eta}^{{\mathrm{\scriptscriptstyle T}}}A{\eta}\leq
a\big{\\}}$. ∎

The second half of Proposition 2 follows directly from Lemma 7.

∎

###  F.8 Proof of Theorem 3

Before proving Theorem 3, we introduce two useful lemmas.

######  Lemma 8.

Let $s_{K}={2\pi^{K/2}}/{\Gamma(K/2)}$ be the surface of a $K$-dimensional
unit sphere and $B(a)=\left\\{{x}=(x_{1},\ldots,x_{K})\mid\|{x}\|_{2}^{2}\leq
a^{2}\right\\}$. Then

$\int_{B(a)}x_{1}^{2}~{}\textup{d}x_{1}\cdots\textup{d}x_{K}=\frac{1}{(K+2)K}a^{K+2}s_{K}.$

###### of Lemma 8.

By symmetry,

$\int_{B(a)}x_{1}^{2}~{}\textup{d}x_{1}\cdots\textup{d}x_{K}=K^{-1}\int_{B(a)}(x_{1}^{2}+\cdots+x_{K}^{2})~{}\textup{d}x_{1}\cdots\textup{d}x_{K}.$

Because the integrand on the right-hand side is the radius, the above formula
reduces to

$\frac{s_{K}}{K}\int_{0}^{a}r^{2}r^{K-1}\textup{d}r=\frac{1}{(K+2)K}a^{K+2}s_{K}.$

∎

######  Lemma 9.

Under Conditions 1, 3, and 5, the threshold $a$ in the cluster rerandomization
scheme $\mathcal{D}_{x}(A_{x})$ satisfies

$\displaystyle a$
$\displaystyle=(2\pi)\det(V_{\textnormal{haj},xx})^{1/K}\det(A_{x})^{1/K}\left(\frac{s_{K}}{K}\right)^{-2/K}\alpha^{2/K}+o(\alpha^{2/K}).$

Under Conditions 1, 2, and 4, the threshold $a$ in the cluster rerandomization
scheme $\mathcal{D}_{c}(A_{c})$ satisfies

$\displaystyle a$
$\displaystyle=(2\pi)\det(V_{\textnormal{ht},cc})^{1/K}\det(A_{c})^{1/K}\left(\frac{s_{K}}{K}\right)^{-2/K}\alpha^{2/K}+o(\alpha^{2/K}).$

###### of Lemma 9.

We only give the proof for cluster rerandomization with individual-level
covariates, as the proof for cluster rerandomization with cluster-level
covariates is similar. Denote the asymptotic distribution of
$(M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}-\tau),M^{1/2}{\hat{\tau}_{\scalebox{0.6}{{haj,${x}$}}}}^{\mathrm{\scriptscriptstyle
T}})$ as $\left({T}_{\infty},{{T}}_{x,\infty}^{\mathrm{\scriptscriptstyle
T}}\right)$, where ${{T}}_{x,\infty}$ is a $K$-dimensional normal random
vector. By the property of normal random vector, we have

${{T}}_{x,\infty}^{{\mathrm{\scriptscriptstyle
T}}}{A}_{x}{{T}}_{x,\infty}\sim\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2},$

where $\lambda_{1},\dots,\lambda_{K}$ are the eigenvalues of
$V_{\textnormal{haj},xx}^{1/2}{A}_{x}V_{\textnormal{haj},xx}^{1/2}$, and
$\eta_{1},\dots,\eta_{K}$ are independent standard normal random variables.
Let $\phi(x)$ be the density function of a standard normal random variable,
then

$\displaystyle{\operatorname{pr}}\Big{(}\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a\Big{)}$ $\displaystyle=\int_{\sum_{k=1}^{K}\lambda_{k}x_{k}^{2}\leq
a}~{}\prod_{k=1}^{K}\phi(x_{k})~{}\textup{d}x_{1}\cdots\textup{d}x_{K}$
$\displaystyle=(2\pi)^{-K/2}\Big{(}\prod_{k=1}^{K}\lambda_{k}\Big{)}^{-1/2}\int_{B(a^{1/2})}\exp\Big{(}2^{-1}{\sum_{k=1}^{K}x_{k}^{2}/\lambda_{k}}\Big{)}\textup{d}x_{1}\cdots\textup{d}x_{K}$
$\displaystyle=(2\pi)^{-K/2}\Big{(}\prod_{k=1}^{K}\lambda_{k}\Big{)}^{-1/2}\int_{B(a^{1/2})}\\{1+O(r^{2})\\}\textup{d}x_{1}\cdots\textup{d}x_{K}.$

The last line follows from $\min_{k=1,\dots,K}\lambda_{k}>0$ and the Taylor
expansion,

$\exp\Big{(}2^{-1}\sum_{k=1}^{K}x_{k}^{2}/\lambda_{k}\Big{)}=1+O\Big{(}2^{-1}\sum_{k=1}^{K}x_{k}^{2}/\lambda_{k}\Big{)}=1+O(r^{2}),\quad
r^{2}=\sum_{k=1}^{K}x_{k}^{2}.$

By the volume formula of a $K$-dimensional ball,

$\alpha={\operatorname{pr}}\Big{(}\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a\Big{)}=(2\pi)^{-K/2}\Big{(}\prod_{k=1}^{K}\lambda_{k}\Big{)}^{-1/2}\frac{s_{K}}{K}a^{K/2}+O(a^{K/2+1}).$
(8)

As $\lambda_{k},~{}k=1,\ldots,K$, are eigenvalues of
$V_{\textnormal{haj},xx}^{1/2}{A}_{x}V_{\textnormal{haj},xx}^{1/2}$, we have

$\displaystyle\prod_{k=1}^{K}\lambda_{k}=\det\big{(}V_{\textnormal{haj},xx}^{1/2}{A}_{x}V_{\textnormal{haj},xx}^{1/2}\big{)}=\det(V_{\textnormal{haj},xx})\det({A}_{x}).$

Substitute into (8) and reorganize the terms, we get the desired result.

∎

###### of Theorem 3.

We only give the proof for cluster rerandomization with individual-level
covariates, as the proof for cluster rerandomization with cluster-level
covariates is similar. As
$\left({T}_{\infty},{{T}}_{x,\infty}^{\mathrm{\scriptscriptstyle T}}\right)$
is the asymptotic distribution of
$(M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}-\tau),M^{1/2}{\hat{\tau}_{\scalebox{0.6}{{haj,${x}$}}}}^{\mathrm{\scriptscriptstyle
T}})$, we have

$\textnormal{var}_{\textnormal{a}}\\{M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}-\tau)\mid
M{\hat{\tau}_{\scalebox{0.6}{{haj,${x}$}}}}^{{\mathrm{\scriptscriptstyle
T}}}A_{x}{\hat{\tau}_{\scalebox{0.6}{{haj,${x}$}}}}\leq
a\\}=\operatorname{var}({T}_{\infty}\mid{{T}}_{x,\infty}^{\mathrm{\scriptscriptstyle
T}}{A}_{x}{{T}}_{x,\infty}).$

Referring to the proof of Theorem 2,

${T}_{\infty}\mid{{T}}_{x,\infty}^{\mathrm{\scriptscriptstyle
T}}{A}_{x}{{T}}_{x,\infty}\leq a\sim
V_{\textnormal{haj},\tau\tau}^{1/2}\big{\\{}(1-R_{x}^{2})^{1/2}\epsilon+R_{x}\mu_{x}^{{\mathrm{\scriptscriptstyle
T}}}{\eta}\mid{\eta}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{haj},xx}^{1/2}A_{x}V^{1/2}_{\textnormal{haj},xx}{\eta}\leq
a\big{\\}},$

where

${\mu}_{x}^{{\mathrm{\scriptscriptstyle
T}}}=(\mu_{1},\ldots,\mu_{K})=V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1/2}/(V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1}V_{\textnormal{haj},x\tau})^{1/2},\quad{\eta}=(\eta_{1},\ldots,\eta_{K})^{{\mathrm{\scriptscriptstyle
T}}},$

and $\epsilon,\eta_{1},\ldots,\eta_{K}$ are independent standard normal random
variables.

We perform the eigenvalue decomposition as follows:
$V_{\textnormal{haj},xx}^{1/2}A_{x}V^{1/2}_{\textnormal{haj},xx}=P_{x}{\Lambda}_{x}P_{x}^{{\mathrm{\scriptscriptstyle
T}}}$, where
${\Lambda}_{x}=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{K})$. Then
$\mu_{x}^{{\mathrm{\scriptscriptstyle
T}}}{\eta}\mid{\eta}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{haj},xx}^{1/2}A_{x}V^{1/2}_{\textnormal{haj},xx}{\eta}\leq
a$ has the same distribution as $\mu_{x}^{{\mathrm{\scriptscriptstyle
T}}}P_{x}{\eta}\mid{\eta}^{{\mathrm{\scriptscriptstyle
T}}}{\Lambda}_{x}{\eta}\leq a$. Let
$(\zeta_{1},\ldots,\zeta_{K})=\mu_{x}^{{\mathrm{\scriptscriptstyle T}}}P_{x}$.

First, we derive the asymptotic variance of
$\sum_{k=1}^{K}\zeta_{k}\eta_{k}\mid\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a$ with $\min_{1\leq k\leq K}\lambda_{k}>0$. Note that

$\operatorname{var}\Big{(}\eta_{1}\mid\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a\Big{)}=\alpha^{-1}\int_{\sum_{k=1}^{K}\lambda_{k}x_{k}^{2}\leq
a}x_{1}^{2}\prod_{k=1}^{K}\phi(x_{k})~{}\textup{d}x_{1}\cdots\textup{d}x_{K},~{}\text{where}~{}\alpha={\operatorname{pr}}\Big{(}\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a\Big{)},$

and

$\begin{split}\int_{\sum_{k=1}^{K}\lambda_{k}x_{k}^{2}\leq
a}x_{1}^{2}\prod_{k=1}^{K}\phi(x_{k})~{}dx_{1}\cdots\textup{d}x_{K}&=(2\pi)^{-K/2}\Big{(}\prod_{k=1}^{K}\lambda_{k}\Big{)}^{-1/2}\lambda_{1}^{-1}\int_{B({a}^{1/2})}\\{x_{1}^{2}+O(r^{4})\\}~{}\textup{d}x_{1}\cdots\textup{d}x_{K}.\end{split}$

The above approximation follows from the Taylor expansion,

$x_{1}^{2}\exp\Big{(}2^{-1}\sum_{k=1}^{K}x_{k}^{2}/\lambda_{k}\Big{)}=x_{1}^{2}+O\Big{(}2^{-1}x_{1}^{2}\sum_{k=1}^{K}x_{k}^{2}/\lambda_{k}\Big{)}=x_{1}^{2}+O(r^{4}),\quad\text{where
}~{}r^{2}=\sum_{k=1}^{K}x_{k}^{2}.$

By Lemma 8,

$\int_{\sum_{k=1}^{K}\lambda_{k}x_{k}^{2}\leq
a}x_{1}^{2}\prod_{k=1}^{K}\phi(x_{k})~{}\textup{d}x_{1}\cdots\textup{d}x_{K}=(2\pi)^{-K/2}\Big{(}\prod_{k=1}^{K}\lambda_{k}\Big{)}^{-1/2}\lambda_{1}^{-1}\frac{1}{(K+2)K}a^{K/2+1}s_{K}+O(a^{K/2+2}).$

Substitute $a$ with the result of Lemma 9, we have

$\operatorname{var}\Bigl{(}\eta_{1}\,\Bigm{|}\,\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a\Bigr{)}=p_{K}\lambda_{1}^{-1}\Bigl{(}\prod_{k=1}^{K}\lambda_{k}\Bigr{)}^{1/K}\alpha^{2/K}+o(\alpha^{2/K}).$

Similarly, for $k=1,\dots,K$,

$\operatorname{var}\Bigl{(}\eta_{k}\,\Bigm{|}\,\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a\Bigr{)}=p_{K}\lambda_{k}^{-1}\Bigl{(}\prod_{k=1}^{K}\lambda_{k}\Bigr{)}^{1/K}\alpha^{2/K}+o(\alpha^{2/K}).$

Moreover, for $m\neq n$, $\eta_{m}$ and $\eta_{n}$ are conditional
uncorrelated, then

$\displaystyle
E\Bigl{(}\eta_{m}\eta_{n}\,\Bigm{|}\,\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a\Bigr{)}$
$\displaystyle=E\biggl{\\{}\eta_{m}E\Bigl{(}\eta_{n}\,\Bigm{|}\,\eta_{m},\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a\Bigr{)}\biggm{|}\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq a\biggr{\\}}$
$\displaystyle=E\Bigl{(}\eta_{m}\times
0\Bigm{|}\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq a\Bigr{)}=0.$

The last line follows from the fact that given $\eta_{m}$ and a symmetric
ellipsoidal acceptance region, the conditional distribution of $\eta_{n}$ is
symmetric. Therefore,

$\displaystyle\operatorname{var}\Bigl{(}\sum_{k=1}^{K}\zeta_{k}\eta_{k}\Bigm{|}\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a\Bigr{)}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\zeta_{k}^{2}\operatorname{var}\Bigl{(}\eta_{k}\,\Bigm{|}\,\sum_{k=1}^{K}\lambda_{k}\eta_{k}^{2}\leq
a\Bigr{)}$ $\displaystyle=$ $\displaystyle
p_{K}\Bigl{(}\sum_{k=1}^{K}\frac{\zeta_{k}^{2}}{\lambda_{k}}\Bigr{)}\Bigl{(}\prod_{k=1}^{K}\lambda_{k}\Bigr{)}^{1/K}\alpha^{2/K}+o(\alpha^{2/K}).$

Note that

$\displaystyle\Big{(}\prod_{k=1}^{K}\lambda_{k}\Big{)}^{1/K}=\det(V_{\textnormal{haj},xx})^{1/K}\det(A_{x})^{1/K},\quad\operatorname{diag}(\lambda_{1},\ldots,\lambda_{K})={P}_{x}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{haj},xx}^{1/2}{A}_{x}{V}^{1/2}_{xx}{P}_{x},$
$\displaystyle\quad(\zeta_{1},\ldots,\zeta_{K})=V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1/2}P_{x}/(V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1}V_{\textnormal{haj},x\tau})^{1/2}.$

Thus,

$\displaystyle\sum_{k=1}^{K}\frac{\zeta_{k}^{2}}{\lambda_{k}}$
$\displaystyle=V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1/2}{P}_{x}({P}_{x}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{haj},xx}^{1/2}{A}_{x}V_{\textnormal{haj},xx}^{1/2}{P}_{x})^{-1}{P}_{x}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{haj},xx}^{-1/2}V_{\textnormal{haj},x\tau}/V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1}V_{\textnormal{haj},x\tau}$
$\displaystyle=V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1}{A}_{x}^{-1}V_{\textnormal{haj},xx}^{-1}V_{\textnormal{haj},x\tau}/(V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1}V_{\textnormal{haj},x\tau}).$

Therefore,

$\displaystyle\operatorname{var}({T}_{\infty}|{{T}}_{x,\infty}^{\mathrm{\scriptscriptstyle
T}}{A}_{x}{{T}}_{x,\infty}\leq a)=$ $\displaystyle
V_{\textnormal{haj},\tau\tau}\big{\\{}(1-R^{2}_{x})+R^{2}_{x}\operatorname{var}({\mu}_{x}^{{\mathrm{\scriptscriptstyle
T}}}P_{x}{\eta}\mid{\eta}^{{\mathrm{\scriptscriptstyle
T}}}{\Lambda}_{x}{\eta}\leq a)\big{\\}}$ $\displaystyle=$ $\displaystyle
V_{\textnormal{haj},\tau\tau}\\{(1-R^{2}_{x})+{R}_{x}^{2}p_{K}{\nu}_{x}(A_{x})\alpha^{2/K}+o(\alpha^{2/K})\\}.$

∎

###  F.9 Proof of Theorem 4

###### of Theorem 4.

It is convenient to fix $\prod_{k=1}^{K}w_{k}=1$ and minimize the term that is
related to $w_{k}$ for $k=1,\ldots,K$. Denote
$(b_{1},\ldots,b_{K})^{{\mathrm{\scriptscriptstyle
T}}}=V_{\textnormal{haj},xx}^{-1}V_{\textnormal{haj},x\tau}$. We only need to
minimize

$V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1}{A}_{x}^{-1}V_{\textnormal{haj},xx}^{-1}V_{\textnormal{haj},x\tau}=\sum_{k=1}^{K}b^{2}_{k}w_{k}^{-1}.$

Using the inequality of arithmetic and geometric means, we have

$\sum_{k=1}^{K}b^{2}_{k}w_{k}^{-1}\geq
K\Bigl{(}\prod_{k=1}^{K}b^{2}_{k}w_{k}^{-1}\Bigr{)}^{1/K}=K\Bigl{(}\prod_{k=1}^{K}b^{2}_{k}\Bigr{)}^{1/K}.$

The equality holds if and only if

$\frac{b_{1}^{2}}{w_{1}}=\cdots=\frac{b_{K}^{2}}{w_{K}},$

which implies $w_{k}=c_{0}b_{k}^{2}=c_{0}(V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1}{\xi}_{k})^{2}$ for some constant $c_{0}>0$.
Then within all positive diagonal matrix, ${\nu}_{x}(A_{x})$ reach its minimum
at $A_{x}^{\textnormal{opt}}=\operatorname{diag}(w_{1},\ldots,w_{K})$ with
$w_{k}=c_{0}(V_{\textnormal{haj},\tau
x}V_{\textnormal{haj},xx}^{-1}{\xi}_{k})^{2}$ for some constant $c_{0}>0$. The
proof for ${\nu}_{c}(A_{c})$ is similar.

∎

###  F.10 Proof of Corollary 2

###### of Corollary 2.

It suffices to show that the optimal weights
$w_{k}=c_{0}{R}_{x_{k}}^{2}/{V}_{\textnormal{haj},x_{k}x_{k}}$,
$k=1,\ldots,K$, for some constant $c_{0}>0$. Denote
$V_{\textnormal{haj},xx}=\operatorname{diag}(V_{\textnormal{haj},x_{1}x_{1}},\ldots,V_{\textnormal{haj},x_{K}x_{K}})$.
Recall that $\sum_{k=1}^{K}{R}^{2}_{x_{k}}=R^{2}_{x}$ with

$R_{x_{k}}^{2}=V_{\textnormal{haj},\tau
x_{k}}^{2}/(V_{\textnormal{haj},\tau\tau}V_{\textnormal{haj},x_{k}x_{k}}).$

Then

$\displaystyle{V}_{\textnormal{haj},\tau x}{V}_{\textnormal{haj},xx}^{-1}$
$\displaystyle=$ $\displaystyle({V}_{\textnormal{haj},\tau
x_{1}}{V}_{\textnormal{haj},x_{1}x_{1}}^{-1},\ldots,{V}_{\textnormal{haj},\tau
x_{K}}{V}^{-1}_{\textnormal{haj},x_{K}x_{K}})$ $\displaystyle=$
$\displaystyle({R}_{x_{1}}{V}_{\textnormal{haj},\tau\tau}^{1/2}/{V}_{\textnormal{haj},x_{1}x_{1}}^{1/2},\ldots,{R}_{x_{K}}{V}_{\textnormal{haj},\tau\tau}^{1/2}/{V}_{\textnormal{haj},x_{K}x_{K}}^{1/2}).$

We minimize the term related to ${A}_{x}$ in ${\nu}_{x}(A_{x})$. Note that

$\displaystyle{V}_{\textnormal{haj},\tau
x}{V}_{\textnormal{haj},xx}^{-1}{A}_{x}^{-1}{V}_{\textnormal{haj},xx}^{-1}{V}_{\textnormal{haj},x\tau}\det(A_{x})^{1/K}\det(V_{\textnormal{haj},xx})^{1/K}$
$\displaystyle=\sum_{k=1}^{K}{R}^{2}_{x_{k}}{V}_{\textnormal{haj},\tau\tau}/({V}_{\textnormal{haj},x_{k}x_{k}}w_{k})\Bigl{(}\prod_{k=1}^{K}w_{k}{V}_{\textnormal{haj},x_{k}x_{k}}\Bigr{)}^{1/K}$
$\displaystyle\geq
K{V}_{\textnormal{haj},\tau\tau}\Bigl{(}\prod_{k=1}^{K}{R}_{x_{k}}^{2}\Bigr{)}^{1/K},$

where the last inequality follows from the inequality of arithmetic and
geometric means. Therefore,

${\nu}_{x}\big{(}A_{x}\big{)}\geq
K{V}_{\textnormal{haj},\tau\tau}\Bigl{(}\prod_{i=1}^{K}{R}_{x_{k}}^{2}\Bigr{)}^{1/K}\Bigm{/}({V}_{\textnormal{haj},\tau
x}{V}^{-1}_{\textnormal{haj},xx}{V}_{\textnormal{haj},x\tau})=K\Bigl{(}\prod_{i=1}^{K}{R}_{x_{k}}^{2}\Bigr{)}^{1/K}\Bigm{/}R^{2}_{x}.$

The equality holds if and only if

${R}^{2}_{x_{1}}/({V}_{\textnormal{haj},x_{1}x_{1}}w_{1})=\cdots={R}^{2}_{x_{K}}/({V}_{\textnormal{haj},x_{K}x_{K}}w_{K}),$

which implies $w_{k}=c_{0}{R}_{x_{k}}^{2}/{V}_{\textnormal{haj},x_{k}x_{k}}$
for some constant $c_{0}>0$. ∎

###  F.11 Proof of Corollary 3

###### of Corollary 3.

By Theorem 3 of Li et al., (2018),

$M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}-\tau)\mid\mathcal{M}_{{[1]}},\ldots,\mathcal{M}_{{[L]}}\mathrel{\xrightarrow{d}}{V}_{\textnormal{haj},\tau\tau}^{1/2}\Big{\\{}(1-R^{2}_{x})^{1/2}\epsilon+\sum_{l=1}^{L}{R}_{x_{[l]}}L_{K_{l},a_{[l]}}\Big{\\}},$

where $\epsilon$ is a standard normal random variable, and
$\epsilon,L_{K_{1},a_{[1]}},\ldots,L_{K_{l},a_{[l]}}$ are jointly independent.
Applying our Theorems 1 and 3 with $K=K_{l}$, $\alpha=\alpha_{[l]}$,
$V_{\textnormal{haj},\tau\tau}=1$, $A_{x}=V_{\textnormal{haj},xx}^{-1}$, and
$R^{2}_{x}=1$, we have

$\operatorname{var}(L_{K_{l},a_{l}})=p_{K_{l}}\alpha_{[l]}^{2/K_{l}}+o(\alpha_{[l]}^{2/K_{l}}).$

Therefore,

$\textnormal{var}_{\textnormal{a}}\bigl{\\{}M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}-\tau)\mid\mathcal{M}_{[1]},\ldots,\mathcal{M}_{[L]}\bigr{\\}}={V}_{\textnormal{haj},\tau\tau}\Bigl{\\{}(1-R^{2}_{x})+\sum_{l=1}^{L}p_{K_{l}}{R}_{x_{[l]}}^{2}\alpha_{[l]}^{2/K_{l}}+o(\alpha_{[l]}^{2/K_{l}})\Bigr{\\}}.$

The leading term in
$\textnormal{var}_{\textnormal{a}}\bigl{\\{}M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}-\tau)\mid\mathcal{M}_{[1]},\ldots,\mathcal{M}_{[L]}\bigr{\\}}$
has the following lower bound:

$\displaystyle{V}_{\textnormal{haj},\tau\tau}\bigl{\\{}(1-R^{2}_{x})+\sum_{l=1}^{L}p_{K_{l}}{R}_{x_{[l]}}^{2}\alpha_{[l]}^{2/K_{l}}\bigr{\\}}$
$\displaystyle\geq{V}_{\textnormal{haj},\tau\tau}\biggl{[}(1-R^{2}_{x})+K\biggl{\\{}\prod_{l=1}^{L}\Bigl{(}\frac{p_{K_{l}}{R}_{x_{[l]}}^{2}\alpha_{[l]}^{2/K_{l}}}{K_{l}}\Bigr{)}^{K_{l}}\biggr{\\}}^{1/K}\biggr{]}$
$\displaystyle={V}_{\textnormal{haj},\tau\tau}\biggl{[}(1-R^{2}_{x})+K\biggl{\\{}\prod_{l=1}^{L}\Bigl{(}\frac{p_{K_{l}}{R}_{x_{[l]}}^{2}}{K_{l}}\Bigr{)}^{K_{l}}\biggr{\\}}^{1/K}\alpha^{2/K}\biggr{]}.$

The equality holds if and only if

$p_{K_{1}}{R}_{x_{[1]}}^{2}\alpha_{[1]}^{2/K_{1}}/{K_{1}}=\cdots=p_{K_{L}}{R}_{x_{[L]}}^{2}\alpha_{[L]}^{2/K_{L}}/{K_{L}},$

which implies that for some constant $c_{0}>0$,

$\alpha_{[l]}=\left({c_{0}{R}_{x_{[l]}}^{2}p_{K_{l}}}/{K_{l}}\right)^{-K_{l}/2}\quad(l=1,\ldots,L).$

∎

###  F.12 Proof of Theorem 5

To prove Theorem 5, we need Lemma 10 below.

######  Lemma 10.

$p_{K}=\frac{2\pi}{K+2}\left\\{\frac{2\pi^{K/2}}{K\Gamma(K/2)}\right\\}^{-2/K}\quad(K=1,2,\ldots)$

is decreasing in $K$.

###### of Lemma 10.

First,

$\displaystyle\log p_{K}$
$\displaystyle=\log(2\pi)-\log(K+2)-2\\{\log(2/K)+(K/2)\log\pi-\log\Gamma(K/2)\\}/K$
$\displaystyle=\log(2\pi)-\log\\{\pi(K+2)\\}+(2/K)\log\Gamma(K/2+1).$

Let $x=K/2$ and we consider the function

$f(x)=-\log\\{2\pi(x+1)\\}+(1/x)\log\Gamma(x+1).$

It suffices to prove that $f(x)$ is decreasing in $x$ for $x\geq 0$. Simple
calculation gives

$f^{\prime}(x)=-1/(x+1)+\psi(x+1)/x-(1/x^{2})\log\Gamma(x+1),$

where $\psi(x)=\textup{d}\log\Gamma(x)/\textup{d}x$ is the Digamma function.
Let

$g(x)=x^{2}f^{\prime}(x)=-x^{2}/(x+1)+x\psi(x+1)-\log\Gamma(x+1),$

We only need to show that $g(x)\leq 0$ for $x\geq 0$. In fact,

$\displaystyle g^{\prime}(x)$
$\displaystyle=-x(x+2)/(x+1)^{2}+\psi(x+1)+x\psi^{\prime}(x+1)-\psi(x+1)$
$\displaystyle=-x(x+2)/(x+1)^{2}+x\psi^{\prime}(x+1).$

By Alzer, (1997, Theorem 1), $\log x-\psi(x)-1/x$ is an increasing function
for $x>0$. Thus, we have $1/x+1/x^{2}-\psi^{\prime}(x)$ is positive for $x>0$,
which implies that $\psi^{\prime}(x+1)<(x+2)/(x+1)^{2}$ for $x\geq 0$.
Therefore, $g^{\prime}(x)\leq 0$ and $g(x)\leq g(0)=0$ for $x\geq 0$. ∎

###### of Theorem 5.

By Lemma 10, $p_{K}$ is decreasing in $K$. Therefore,

$\displaystyle\sum_{l=1}^{L}R_{x_{[l]}}^{2}p_{K_{l}}\alpha_{[l]}^{2/K}$
$\displaystyle\geq\biggl{\\{}\prod_{l=1}^{L}\Bigl{(}\frac{p_{K_{l}}{R}_{x_{[l]}}^{2}}{K_{l}}\Bigr{)}^{K_{l}}\biggr{\\}}^{1/K}\alpha^{2/K}\geq
p_{K}\biggl{\\{}\prod_{l=1}^{L}\Bigl{(}\frac{{R}_{x_{[l]}}^{2}}{K_{l}}\Bigr{)}^{K_{l}}\biggr{\\}}^{1/K}\alpha^{2/K}$
$\displaystyle\geq
Kp_{K}\Bigl{(}\prod_{k=1}^{K}R_{x_{k}}^{2}\Bigr{)}^{1/K}\alpha^{2/K}.$

The last inequality follows from the inequality of geometric and arithmetic
means, and

${R}_{x_{[l]}}^{2}=\sum_{k\in[l]}{R}^{2}_{x_{k}},\quad\Bigl{(}\frac{{R}_{x_{[l]}}^{2}}{K_{l}}\Bigr{)}^{K_{l}}\geq\prod_{k\in[l]}{R}^{2}_{x_{k}}.$

∎

###  F.13 Proof of Theorem 7

Define

$\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}=M_{1}^{-1}{\sum_{i=1}^{M}Z_{i}\\{\tilde{Y}_{i\cdot}(1)-{v}_{i}^{{\mathrm{\scriptscriptstyle
T}}}{\beta}_{v}(1)\\}}-M_{0}^{-1}{\sum_{i=1}^{M}(1-Z_{i})\\{\tilde{Y}_{i\cdot}(0)-{v}_{i}^{{\mathrm{\scriptscriptstyle
T}}}{\beta}_{v}(0)\\}},$

and

$\check{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}=N_{1}^{-1}{\sum_{i,j}Z_{ij}\\{{Y}_{ij}(1)-{w}_{ij}^{{\mathrm{\scriptscriptstyle
T}}}{\beta}_{w}(1)\\}}-N_{0}^{-1}{\sum_{i,j}(1-Z_{i})\\{Y_{ij}(0)-{w}_{ij}^{{\mathrm{\scriptscriptstyle
T}}}{\beta}_{w}(0)\\}}.$

Lemma 11 below shows that $\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}$ and
$\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}$
have the same asymptotic distributions under cluster rerandomization. The same
is true for $\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}$ and
$\check{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}$.

######  Lemma 11.

If Conditions 1, 2, 4, 6 hold, then

$M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}})|\mathcal{D}_{{c}}(A_{c})=o_{\textup{p}}(1).$

If Condition 1, 3, 5, 7 hold, then

$M^{1/2}({\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}}-\check{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}^{\scalebox{0.6}{$\textnormal{adj}$}})|\mathcal{D}_{{x}}(A_{x})=o_{\textup{p}}(1).$

###### of Lemma 11.

By Lemma A7 of Su and Ding, (2021),
$M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}})=o_{\textup{p}}(1)$,
which implies that, for any $\delta>0$,

$\operatorname{pr}\left\\{|M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}})|>\delta\right\\}=o(1).$

Proposition 1 implies that

$\lim\limits_{M\rightarrow\infty}\operatorname{pr}\\{\mathcal{D}_{c}(A_{c})\\}>0.$

Therefore,

$\operatorname{pr}\left\\{|M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}})|>\delta\mid{D}_{c}(A_{c})\right\\}\leq\operatorname{pr}\left\\{|M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}})|>\delta\right\\}\Bigm{/}\operatorname{pr}\\{{D}_{c}(A_{c})\\}=o(1),$

which implies that

$M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}})\mathrel{|}\mathcal{D}_{c}(A_{c})=o_{\textup{p}}(1).$

Similarly, we can prove the second part of Lemma 11.

∎

###### of Theorem 7.

The asymptotic results for
$\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}$
follow from Lemma 11 and Theorems 1 and 2 with
$\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}=\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}$.
The asymptotic results for
${\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}}$
follow from Lemma 11 and Theorems 1 and 2 with
$\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}=\check{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}$.

∎

###  F.14 Proof of Corollary 4

###### of Corollary 4.

By Lemma 11,
$M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\tau)\mathrel{\
\dot{\sim}\
}M^{1/2}(\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\tau)$
under cluster rerandomization. When ${c}_{i}=B{v}_{i}$,
$M^{1/2}(\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\tau)$
is uncorrelated with $M^{1/2}{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$ }}}$.
Applying Theorem 2, the conclusion follows immediately. ∎

###  F.15 Proof of Theorem 8

###### of Theorem 8.

By Theorems 2 and 4 of Su and Ding, (2021), we have

$\displaystyle M\hat{V}_{\scalebox{0.5}{$\textnormal{LZ}$}}$
$\displaystyle\mathrel{\xrightarrow{\textup{p}}}e_{1}^{-1}\operatorname{var}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)\\}+e_{0}^{-1}\operatorname{var}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(0)\\}\geq
V_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}},$
$\displaystyle M\hat{V}_{\scalebox{0.5}{$\textnormal{HW}$}}$
$\displaystyle\mathrel{\xrightarrow{\textup{p}}}e_{1}^{-1}\operatorname{var}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(1)\\}+e_{0}^{-1}\operatorname{var}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(0)\\}\geq
V_{\textnormal{ht},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}.$

As
$\lim\limits_{M\rightarrow\infty}\operatorname{pr}\\{{D}_{\textnormal{*}}(A_{\textnormal{*}})\\}>0,\
\text{for }*=x,c,$ similar to the proof of Lemma 11, we have

$\displaystyle
M\hat{V}_{\scalebox{0.5}{$\textnormal{LZ}$}}-V_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}\mid\mathcal{D}_{x}(A_{x})\mathrel{\geq_{\textup{p}}}0,\quad
M\hat{V}_{\scalebox{0.5}{$\textnormal{HW}$}}-V_{\textnormal{ht},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}\mid\mathcal{D}_{c}(A_{c})\mathrel{\geq_{\textup{p}}}0.$

∎

###  F.16 Proof of Theorem 6

###### of Theorem 6.

When $v_{i}=c_{i}$ and $x_{ij}=w_{ij}$, we have $R^{\textnormal{adj}}_{c}=0$,
$R^{\textnormal{adj}}_{x}\geq 0$, and

$V_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}\\{1-(R^{\textnormal{adj}}_{x})^{2}\\}=V_{\textnormal{haj},\tau\tau}(1-R^{2}_{x}),\quad
V_{\textnormal{ht},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}\\{1-(R^{\textnormal{adj}}_{c})^{2}\\}=V_{\textnormal{ht},\tau\tau}(1-R^{2}_{c}).$

By Theorem 7, we have

$M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\tau)\mid\mathcal{D}_{c}(A_{c})\mathrel{\
\dot{\sim}\ }(V_{\textnormal{ht},\tau\tau})^{1/2}(1-R^{2}_{c})\epsilon,$

and

$M^{1/2}({\hat{\tau}_{\scalebox{0.6}{$\textnormal{haj}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}}-\tau)\mid\mathcal{D}_{x}(A_{x})\mathrel{\
\dot{\sim}\
}(V_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}})^{1/2}\big{[}\\{1-(R^{\textnormal{adj}}_{x})^{2}\\}^{1/2}\epsilon+R^{\textnormal{adj}}_{x}{\mu}_{x}^{{\mathrm{\scriptscriptstyle
T}}}{\eta}\mid{\eta}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{haj},xx}^{1/2}A_{x}V_{\textnormal{haj},xx}^{1/2}{\eta}\leq
a\big{]}.$

Moreover, by Theorem 8,
$M\hat{V}^{\textnormal{adj}}_{\scalebox{0.5}{$\textnormal{LZ}$}}$ and
$M\hat{V}^{\textnormal{adj}}_{\scalebox{0.5}{$\textnormal{HW}$}}$ are
conservative estimators of
$V_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}$ and
$V_{\textnormal{ht},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}$,
respectively. The conservativeness of the Wald-type confidence intervals
follows from the conservativeness of
$M\hat{V}^{\textnormal{adj}}_{\scalebox{0.5}{$\textnormal{LZ}$}}$ and
$M\hat{V}^{\textnormal{adj}}_{\scalebox{0.5}{$\textnormal{HW}$}}$ and
Proposition 2. Therefore, (i) and (ii) hold.

When ${c}_{i}=(n_{i},\tilde{{x}}_{i\cdot}^{{\mathrm{\scriptscriptstyle
T}}})^{{\mathrm{\scriptscriptstyle T}}}$, Corollary 1 implies that

$V_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}\\{1-(R^{\textnormal{adj}}_{x})^{2}\\}\geq
V_{\textnormal{ht},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}\\{1-(R^{\textnormal{adj}}_{c})^{2}\\}.$

Thus, the normal component of the asymptotic distribution of the cluster-level
regression adjustment is more concentrated than that of the individual-level
regression adjustment. Moreover, the asymptotic distribution of the
individual-level regression adjustment has a truncated normal component while
the cluster-level regression adjustment does not. Thus, (iii) holds. ∎

###  F.17 Proof of Corollary 5

###### of Corollary 5.

By Lemma 11,
$M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\tau)\mathrel{\
\dot{\sim}\
}M^{1/2}(\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\tau)$
under cluster rerandomization. By Proposition 1 of Li and Ding, (2020),

$\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\tau=\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}-\tau-\operatorname{cov}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}},{\hat{\tau}_{\textnormal{ht},v}})\\{\operatorname{cov}({\hat{\tau}_{\textnormal{ht},v}})\\}^{-1}{\hat{\tau}_{\textnormal{ht},v}}.$

Since ${v}_{i}=B{c}_{i}$, then
${\hat{\tau}_{\textnormal{ht},v}}=B{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$
}}}$. As
$M\textnormal{var}_{\textnormal{a}}({\hat{\tau}_{\scalebox{0.6}{{ht},${c}$
}}})=V_{\textnormal{ht},cc}$ and
$M\textnormal{cov}_{\textnormal{a}}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}},{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$
}}})=V_{\textnormal{ht},\tau c}$, substituting them into the above equation,
we have

$\displaystyle\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\tau\mathrel{\
\dot{\sim}\ }$
$\displaystyle\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}-\tau-
V_{\textnormal{ht},\tau c}B^{{\mathrm{\scriptscriptstyle
T}}}(BV_{\textnormal{ht},cc}B^{{\mathrm{\scriptscriptstyle
T}}})^{-1}B{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$ }}}$ $\displaystyle=$
$\displaystyle\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}-\tau-
V_{\textnormal{ht},\tau
c}V_{\textnormal{ht},cc}^{-1}{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$
}}}+V_{\textnormal{ht},\tau
c}V_{\textnormal{ht},cc}^{-1}{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$
}}}-V_{\textnormal{ht},\tau c}B^{{\mathrm{\scriptscriptstyle
T}}}(BV_{\textnormal{ht},cc}B^{{\mathrm{\scriptscriptstyle
T}}})^{-1}B{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$ }}}$ $\displaystyle=$
$\displaystyle\\{\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}-\tau-
V_{\textnormal{ht},\tau
c}V_{\textnormal{ht},cc}^{-1}{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$ }}}\\}$
$\displaystyle+V_{\textnormal{ht},\tau
c}V_{\textnormal{ht},cc}^{-1/2}\left\\{I_{K}-V_{\textnormal{ht},cc}^{1/2}B^{{\mathrm{\scriptscriptstyle
T}}}(BV_{\textnormal{ht},cc}B^{{\mathrm{\scriptscriptstyle
T}}})^{-1}BV_{\textnormal{ht},cc}^{1/2}\right\\}V_{\textnormal{ht},cc}^{-1/2}{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$
}}}.$

Note that

$\textnormal{var}_{\textnormal{a}}\\{M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}-\tau-
V_{\textnormal{ht},\tau
c}V_{\textnormal{ht},cc}^{-1}{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$
}}})\\}=(1-R^{2}_{c})V_{\textnormal{ht},\tau\tau},$
$\textnormal{cov}_{\textnormal{a}}\\{M^{1/2}(\hat{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}-\tau-
V_{\textnormal{ht},\tau
c}V_{\textnormal{ht},cc}^{-1}{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$
}}}),M^{1/2}{\hat{\tau}_{\scalebox{0.6}{{ht},${c}$ }}}\\}=0.$

Therefore, by Proposition 5, we have

$\displaystyle
M^{1/2}(\check{\tau}_{\scalebox{0.6}{$\textnormal{ht}$}}^{\scalebox{0.6}{$\textnormal{adj}$}}-\tau)|\mathcal{D}_{c}(A_{c})\mathrel{\
\dot{\sim}\ }V_{\textnormal{ht},\tau\tau}^{1/2}(1-R^{2}_{{c}})^{1/2}\epsilon+$
$\displaystyle\quad\quad\quad V_{\textnormal{ht},\tau
c}V_{\textnormal{ht},cc}^{-1/2}\left\\{I_{K}-V_{\textnormal{ht},cc}^{1/2}B^{{\mathrm{\scriptscriptstyle
T}}}(BV_{\textnormal{ht},cc}B^{{\mathrm{\scriptscriptstyle
T}}})^{-1}BV_{\textnormal{ht},cc}^{1/2}\right\\}{\eta}\mid{\eta}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{ht},cc}^{1/2}A_{c}V_{\textnormal{ht},cc}^{1/2}{\eta}\leq
a,$

which concludes the proof. ∎

###  F.18 Proof of Corollary 6

###### of Corollary 6.

Let ${\mu}_{c}^{\mathrm{\scriptscriptstyle
T}}=(\mu_{1},\ldots,\mu_{K})=V_{\textnormal{ht},\tau
c}V_{\textnormal{ht},cc}^{-1/2}/\\{V_{\textnormal{ht},\tau
c}V_{\textnormal{ht},cc}^{-1}V_{\textnormal{ht},c\tau}\\}^{1/2}.$ It suffices
to compare the variances of

$(V_{\textnormal{ht},\tau\tau})^{1/2}\big{\\{}(1-R^{2}_{c})^{1/2}\epsilon+R_{c}{\mu}_{c}^{{\mathrm{\scriptscriptstyle
T}}}H{\eta}\mid{\eta}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{ht},cc}^{1/2}A_{c}V_{\textnormal{ht},cc}^{1/2}{\eta}\leq
a\big{\\}}$

and

$(V_{\textnormal{ht},\tau\tau})^{1/2}\big{\\{}(1-R^{2}_{c})^{1/2}\epsilon+R_{c}{\mu}_{c}^{{\mathrm{\scriptscriptstyle
T}}}{\eta}\mid{\eta}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{ht},cc}^{1/2}A_{c}V_{\textnormal{ht},cc}^{1/2}{\eta}\leq
a\big{\\}},$

where

$H=\begin{bmatrix}0&0\\\ 0&I_{(K-J)\times(K-J)}\end{bmatrix}.$

Due to the conditional uncorrelatedness of $\eta_{m}$ and $\eta_{n}$ for
$m\not=n$, the difference between the variances of the above two distributions
is equal to

$V_{\textnormal{ht},\tau\tau}{R}_{c}^{2}\sum_{k=1}^{J}\mu^{2}_{k}\operatorname{var}(\eta_{k}\mid{\eta}^{{\mathrm{\scriptscriptstyle
T}}}V_{\textnormal{ht},cc}^{1/2}A_{c}V_{\textnormal{ht},cc}^{1/2}{\eta}\leq
a)\geq 0,$

which concludes the proof.

∎

###  F.19 Proof of Theorem 9

Let ${{\hat{\beta}}_{v}}(1)$ and ${{\hat{\beta}}_{v}}(0)$ be the coefficients
of $v$ in the least squares fit of $\tilde{Y}_{i\cdot}$ on $(1,v_{i})$ under
treatment and control arms, respectively. Let ${{\hat{\beta}}_{w}}(1)$ and
${{\hat{\beta}}_{w}}(0)$ be the coefficients of $w$ in the least squares fit
of $Y_{ij}$ on $(1,w_{ij})$ under treatment and control arms, respectively.
Lemmas 12–14 below show the consistency of the ordinary least squares
coefficients, sample means, variances, and covariances under cluster
rerandomization. Their unconditional versions can be found in Su and Ding,
(2021) and Li et al., (2020). The proof is similar to that of Lemma 11, so we
omit. Let $\bar{Y}_{z}$ be the sample mean of $Y_{ij}$ under treatment arm
$z$.

######  Lemma 12.

If Conditions 1, 2, 4, 6 hold, then

$\displaystyle{{\hat{\beta}}_{v}}(z)-{{\beta}_{v}}(z)\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}0.$

If Condition 1, 3, 5, 7 hold, then

$\displaystyle{{\hat{\beta}}_{w}}(z)-{{\beta}_{w}}(z)\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0.$

######  Lemma 13.

If Condition 1, 3, 5, 7 hold, then

$\displaystyle\bar{Y}_{z}-\bar{Y}(z)\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0.$

######  Lemma 14.

If Conditions 1, 2, 4, 6 hold, then

$\hat{\operatorname{var}}_{\textnormal{f,z}}\\{\tilde{\varepsilon}(z)\\}-\operatorname{var}_{\textnormal{f}}\\{\tilde{\varepsilon}(z)\\}\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0,\quad\hat{\operatorname{cov}}_{\textnormal{f,z}}(\tilde{\omega})-\operatorname{cov}_{\textnormal{f}}(\tilde{\omega})\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0,$
$\hat{\operatorname{cov}}_{\textnormal{f,z}}(\tilde{{X}})-\operatorname{cov}_{\textnormal{f}}(\tilde{{X}})\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0,\quad\hat{\operatorname{cov}}_{\textnormal{f,z}}(\tilde{W})-\operatorname{cov}_{\textnormal{f}}(\tilde{W})\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0,$
$\hat{\operatorname{cov}}_{\textnormal{f,z}}(\tilde{{X}},\tilde{W})-\operatorname{cov}_{\textnormal{f}}(\tilde{{X}},\tilde{W})\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0,\quad\hat{\operatorname{cov}}_{\textnormal{f,z}}\\{\tilde{{X}},\tilde{\varepsilon}(z)\\}-\operatorname{cov}_{\textnormal{f}}\\{\tilde{{X}},\tilde{\varepsilon}(z)\\}\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0,$
$\hat{\operatorname{cov}}_{\textnormal{f,z}}(\tilde{{X}},\tilde{\omega})-\operatorname{cov}_{\textnormal{f}}(\tilde{{X}},\tilde{\omega})\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0,\quad\hat{\operatorname{cov}}_{\textnormal{f,z}}\\{\tilde{W},\tilde{\varepsilon}(z)\\}-\operatorname{cov}_{\textnormal{f}}\left\\{\tilde{W},\tilde{\varepsilon}(z)\right\\}\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0,$
$\hat{\operatorname{cov}}_{\textnormal{f,z}}(\tilde{W},\tilde{\omega})-\operatorname{cov}_{\textnormal{f}}(\tilde{W},\tilde{\omega})\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0,\quad\hat{\operatorname{cov}}_{\textnormal{f,z}}\\{\tilde{\omega},\tilde{\varepsilon}(z)\\}-\operatorname{cov}_{\textnormal{f}}\left\\{\tilde{\omega},\tilde{\varepsilon}(z)\right\\}\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0.$

If Condition 1, 3, 5, 7 hold, then

$\hat{\operatorname{var}}_{\textnormal{f,z}}\\{\tilde{Y}(z)\\}-\operatorname{var}_{\textnormal{f}}\\{\tilde{Y}(z)\\}\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}0,\quad\hat{\operatorname{cov}}_{\textnormal{f,z}}(V)-\operatorname{cov}_{\textnormal{f}}(V)\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}0,$
$\hat{\operatorname{cov}}_{\textnormal{f,z}}\\{\tilde{Y}(z),V\\}-\operatorname{cov}_{\textnormal{f}}\\{\tilde{Y}(z),V\\}\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}0,\quad\hat{\operatorname{cov}}_{\textnormal{f,z}}(C)-\operatorname{cov}_{\textnormal{f}}(C)\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}0,$
$\hat{\operatorname{cov}}_{\textnormal{f,z}}(C,V)-\operatorname{cov}_{\textnormal{f}}(C,V)\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}0,\quad\hat{\operatorname{cov}}_{\textnormal{f,z}}\\{C,\tilde{Y}(z)\\}-\operatorname{cov}_{\textnormal{f}}\\{C,\tilde{Y}(z)\\}\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}0.$

By the property of projection, Lemma 15 below holds.

######  Lemma 15.

$\displaystyle{\operatorname{cov}_{\textnormal{f}}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0),\tilde{G}\\}\\{\operatorname{cov}_{\textnormal{f}}(\tilde{G})\\}^{-1}{\operatorname{cov}_{\textnormal{f}}}\\{\tilde{G},\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0)\\}$
$\displaystyle\leq\operatorname{var}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0)\\},$
$\displaystyle{\operatorname{cov}_{\textnormal{f}}}\\{\tilde{Y}^{\textnormal{adj}}(1)-\tilde{Y}^{\textnormal{adj}}(0),C\\}\\{\operatorname{cov}_{\textnormal{f}}(C)\\}^{-1}{\operatorname{cov}_{\textnormal{f}}}\\{C,\tilde{Y}^{\textnormal{adj}}(1)-\tilde{Y}^{\textnormal{adj}}(0)\\}$
$\displaystyle\leq\operatorname{var}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(1)-\tilde{Y}^{\textnormal{adj}}(0)\\}.$

###### of Theorem 9.

Recall that

$\displaystyle(\hat{R}^{\textnormal{adj}}_{c})^{2}$
$\displaystyle=\big{[}e_{1}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(1)\mid{C}\\}+e_{0}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(0)\mid{C}\\}-\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(1)-\tilde{Y}^{\textnormal{adj}}(0)\mid{C}\\}\big{]}\bigm{/}{\hat{V}_{\textnormal{ht},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}},$
$\displaystyle(\hat{R}^{\textnormal{adj}}_{x})^{2}$
$\displaystyle=\big{[}{e_{1}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)\mid\tilde{{X}}\\}+e_{0}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(0)\mid\tilde{{X}}\\}-\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0)\mid\tilde{{X}}\\}}\big{]}\bigm{/}{\hat{V}_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}}.$

First, we prove that the denominators,
$\hat{V}_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}$ and
$\hat{V}_{\textnormal{ht},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}$, are
conservative. Recall that

$\displaystyle\hat{V}_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}$
$\displaystyle={e_{1}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)\\}+e_{0}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(0)\\}-\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0)\mid\tilde{G}\\}},$
$\displaystyle\hat{V}_{\textnormal{ht},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}$
$\displaystyle=e_{1}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(1)\\}+e_{0}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(0)\\}-\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(1)-\tilde{Y}^{\textnormal{adj}}(0)\mid
C\\}.$

Note that

$\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(z)\\}=\hat{\operatorname{var}}_{\textnormal{f,z}}\\{\tilde{Y}(z)-{{\hat{\beta}}_{v}}^{\mathrm{\scriptscriptstyle
T}}(z)V\\},$
$\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(z)\\}=\hat{\operatorname{var}}_{\textnormal{f,z}}\\{\tilde{Y}(z)-\tilde{\omega}\bar{Y}_{z}-{{\hat{\beta}}_{v}}^{\mathrm{\scriptscriptstyle
T}}(z)\tilde{W}\\}=\hat{\operatorname{var}}_{\textnormal{f,z}}\bigl{[}\tilde{\varepsilon}(z)-\tilde{\omega}\\{\bar{Y}_{z}-\bar{Y}(z)\\}-{{\hat{\beta}}_{v}}^{\mathrm{\scriptscriptstyle
T}}(z)\tilde{W}\bigr{]}.$

Therefore, by Lemmas 12–14, we have

$\displaystyle\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(z)\\}\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}\operatorname{var}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(z)\\},\quad\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(z)\\}\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}\operatorname{var}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(z)\\}.$

Recall that

$\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0)\mid\tilde{{G}}\\}=\\{\hat{\operatorname{cov}}_{\textnormal{f,1}}(\tilde{U},\tilde{{G}})-\hat{\operatorname{cov}}_{\textnormal{f,0}}(\tilde{U},\tilde{{G}})\\}\\{\operatorname{cov}_{\textnormal{f}}(\tilde{{G}})\\}^{-1}\\{\hat{\operatorname{cov}}_{\textnormal{f,1}}(\tilde{{G}},\tilde{U})-\hat{\operatorname{cov}}_{\textnormal{f,0}}(\tilde{{G}},\tilde{U})\\}.\\\
$

As $\tilde{{G}}$ is the union of $\tilde{X}$ and $\tilde{W}$,

$\hat{\operatorname{cov}}_{\textnormal{f,z}}(\tilde{U},\tilde{{G}})\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}\operatorname{cov}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(z),\tilde{{G}}\\},$

which implies that

$\displaystyle\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0)\mid\tilde{{G}}\\}\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}{\operatorname{cov}_{\textnormal{f}}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0),\tilde{{G}}\\}\\{\operatorname{cov}_{\textnormal{f}}(\tilde{{G}})\\}^{-1}{\operatorname{cov}_{\textnormal{f}}}\\{\tilde{{G}},\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0)\\}.$

By Lemma 15, the right-hand side is smaller than or equal to
$\operatorname{var}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0)\\}$.
Similarly, the probability limit of
$\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(1)-\tilde{Y}^{\textnormal{adj}}(0)\mid
C\\}$ is smaller than or equal to
${\operatorname{var}_{\textnormal{f}}}\\{\tilde{Y}^{\textnormal{adj}}(1)-\tilde{Y}^{\textnormal{adj}}(0)\\}$.
Therefore,
$\hat{V}_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}$ is
a conservative estimator of
$V_{\textnormal{haj},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}$, and
$\hat{V}_{\textnormal{ht},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}$ is a
conservative estimator of
$V_{\textnormal{ht},\tau\tau}^{\scalebox{0.6}{$\textnormal{adj}$}}$.

Second, we prove that the estimated numerators for the multiple correlation
coefficients are consistent. Note that

$\hat{\operatorname{cov}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(z),C\\}=\hat{\operatorname{cov}}_{\textnormal{f,z}}\\{\tilde{Y}(z)-{{\hat{\beta}}_{v}}^{\mathrm{\scriptscriptstyle
T}}(z)V,C\\},$
$\hat{\operatorname{cov}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(z),\tilde{X}\\}=\hat{\operatorname{cov}}_{\textnormal{f,z}}\\{\tilde{Y}(z)-\tilde{\omega}\bar{Y}_{z}-{{\hat{\beta}}_{v}}^{\mathrm{\scriptscriptstyle
T}}(z)\tilde{W},\tilde{X}\\}=\hat{\operatorname{cov}}_{\textnormal{f,z}}\left[\tilde{\varepsilon}(z)-\tilde{\omega}\\{\bar{Y}_{z}-\bar{Y}(z)\\}-{{\hat{\beta}}_{v}}^{\mathrm{\scriptscriptstyle
T}}(z)\tilde{W},\tilde{X}\right].$

Therefore, by Lemmas 12–14, we have

$\displaystyle\hat{\operatorname{cov}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(z),C\\}\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}\operatorname{cov}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(z),C\\},\quad\hat{\operatorname{cov}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(z),\tilde{X}\\}\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}\operatorname{cov}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(z),\tilde{X}\\}.$

Together with the following property by Lemma 14,

$\displaystyle\hat{\operatorname{cov}}_{\textnormal{f,z}}(\tilde{{X}})-\operatorname{cov}_{\textnormal{f}}(\tilde{{X}})\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}0,$
$\displaystyle\quad\hat{\operatorname{cov}}_{\textnormal{f,z}}(C)-\operatorname{cov}_{\textnormal{f}}(C)\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}0,$

we have

$\displaystyle
e_{1}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)\mid\tilde{{X}}\\}+e_{0}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(0)\mid\tilde{{X}}\\}-\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0)\mid\tilde{{X}}\\}\mid\mathcal{D}_{x}(A_{x})\xrightarrow{\textnormal{p}}$
$\displaystyle
e_{1}^{-1}\operatorname{var}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)\mid\tilde{{X}}\\}+e_{0}^{-1}\operatorname{var}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(0)\mid\tilde{{X}}\\}-\operatorname{var}_{\textnormal{f}}\\{\tilde{\varepsilon}^{\textnormal{adj}}(1)-\tilde{\varepsilon}^{\textnormal{adj}}(0)\mid\tilde{{X}}\\},$
$\displaystyle
e_{1}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(1)\mid{C}\\}+e_{0}^{-1}\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(0)\mid{C}\\}-\hat{\operatorname{var}}_{\textnormal{f}}\\{\tilde{Y}^{\textnormal{adj}}(1)-\tilde{Y}^{\textnormal{adj}}(0)\mid{C}\\}\mid\mathcal{D}_{c}(A_{c})\xrightarrow{\textnormal{p}}$
$\displaystyle
e_{1}^{-1}{\operatorname{var}_{\textnormal{f}}}\\{\tilde{Y}^{\textnormal{adj}}(1)\mid{C}\\}+e_{0}^{-1}{\operatorname{var}_{\textnormal{f}}}\\{\tilde{Y}^{\textnormal{adj}}(0)\mid{C}\\}-{\operatorname{var}_{\textnormal{f}}}\\{\tilde{Y}^{\textnormal{adj}}(1)-\tilde{Y}^{\textnormal{adj}}(0)\mid{C}\\}.$

Thus, the estimated numerators for the multiple correlation coefficients are
consistent. Similarly, we can prove the consistency of $\hat{\mu}_{x}$ and
$\hat{\mu}_{c}$. The conclusion follows immediately.

∎

###  F.20 Proof of Corollaries 7 and 8, and Theorem 10

Cluster rerandomization procedures with individual-level covariates and
cluster-level covariates have the same form of asymptotic distributions. The
proof is just a slight modification of the proof of Corollaries 2, 3, and
Theorem 5.