How Uniform is the Uniform Distribution on

Permutations?⇤

Michael D. Perlman†

Department of StatisticsUniversity of WashingtonTechnical Report 649

February 26, 2019

Abstract

For large q, does the (discrete) uniform distribution on the set of q!permutations of the vector (1, 2, . . . , q) closely approximate the (con-tinuous) uniform distribution on the (q�2)-sphere that contains them?These permutations comprise the vertices of the regular permutohe-dron, a (q � 1)-dimensional convex polyhedron. Surprisingly to me,the answer is emphatically no: these permutations are confined to anegligible portion of the sphere, and the regular permutohedron occu-pies a negligible portion of the ball. However, (1, 2, . . . , q) is not themost favorable configuration for spherical uniformity of permutations.Unlike the permutations of (1, 2, . . . , q), the normalized surface area ofthe largest empty spherical cap among the permutations of the mostfavorable configuration approaches 0 as q ! 1. Nonetheless, thesepermutations do not approach spherical uniformity either.

⇤Key words: Permutations, uniform distribution, spherical cap discrepancy, largestempty cap, regular configuration, regular permutohedron, most favorable configuration,maximal configuration, maximal permutohedron, normal configuration, normal permuto-hedron, majorization, spherical code, permutation code.

†mdperlma@uw.edu. Research supported in part by U.S. Department of Defense GrantH98230-10-C-0263/0000 P0004.

1

This paper is dedicated to the memory of Ingram Olkin, my teacher, men-

tor, and friend, who introduced so many of us to the joy of majorization.

1. Are permutations spherically uniform?

Column vectors denoted by Roman letters appear in bold type, their com-ponents in plain type; thus x = (x1, . . . , xq)0 2 Rq. For any nonzero x 2 Rq

(q � 2) let ⇧(x) denote the set of all q! permutations of x, that is

(1) ⇧(x) = {Px | P 2 Pq},

where Pq is the set of all q ⇥ q permutation matrices. In this paper, thefollowing general question is examined:

Question 1: For large q, do there exist nonzero vectors x 2 Rqsuch that the

(discrete) uniform distribution on ⇧(x) closely approximates the (continu-

ous) uniform distribution on the (q � 2)-sphere in which ⇧(x) is contained?

Do there exist sequences1 {xq 2 Rq} such that ⇧(xq) approaches spherical

uniformity as q ! 1?

Because ⇧(x) is invariant under permutations of x, we may always assumethat the components of x and xq are ordered, i.e., x,xq 2 Rq

, where

(2) Rq := {x 2 Rq | x1 · · · xq}.

Clearly kPxk = kxk for all P 2 Pq, so

(3) ⇧(x) ⇢ Sq�1kxk \Mq�1

x ,

where Sq�1⇢ denotes the 0-centered (q � 1)-sphere of radius ⇢ in Rq and

(4) Mq�1x := {v 2 Rq | v0eq = x0eq}

is the (q � 1)-dimensional hyperplane containing x that is orthogonal toeq := (1, . . . , 1)0. Because Mq�1

x does not contain the origin but we wish towork with 0-centered spheres, we shall translate Mq�1

x to

(5) Mq�1 ⌘ {v 2 Rq | v0eq = 0},

the (q�1)-dimensional linear subspace parallel toMq�1x and orthogonal to eq.

1 Superscripts denote indices, not exponents, unless the contrary is evident.

2

For this purpose consider the q ⇥ q Helmert orthogonal matrix

�q ⌘ (�q1 , �q2 , . . . , �

qq )

⌘

0

BBBBB@

1 1 1 · · · 11 �1 1 · · · 11 0 �2 · · · 1...

...... · · · ...

1 0 0 · · · �(q � 1)

1

CCCCCA

0

BBBBBB@

1pq 0 0 · · · 0

0 1p1·2 0 · · · 0

0 0 1p2·3 · · · 0

......

... · · · ...0 0 0 · · · 1p

(q�1)q

1

CCCCCCA

⌘ (�q1 ,�q2),

where �q1 ⌘ 1pqe

q : q ⇥ 1 is the unit vector along the direction of eq. By the

orthogonality of �q,

(�q2)

0�q1 = 0,(6)

(�q2)

0�q2 = Iq�1,(7)

�q2(�

q2)

0 = Iq � �q1(�q1)

0 =: ⌦q,(8)

where Iq denotes the q⇥ q identity matrix. Here ⌦q is the projection matrixof rank q � 1 that projects Rq onto Mq�1, so that ⌦qMq�1

x = Mq�1. Let ybe the projection of x onto Mq�1:

y = ⌦qx = x� xeq,(9)

wherex ⌘ 1

qx0eq = 1

q

Pqi=1 xi

is the average of the q components of x. Then y = 0 since y0eq = 0, so

kyk2 = kx� xeqk2 =Pq

i=1(xi � x)2 =Pq

i=1(yi � y)2,(10)

which is proportional to their sample variance. Note that y1 · · · yq, so

(11) y 2 Mq�1 := Mq�1 \ Rq

.

Because ⌦qP = P⌦q for all P 2 Pq,

(12) ⇧(y) = ⇧(⌦qx) = ⌦q(⇧(x)),

3

so ⇧(y) is a rigid translation of ⇧(x) and satisfies

(13) ⇧(y) ⇢ Sq�1kyk \ Mq�1 =: Sq�2

kyk ,

the 0-centered (q � 2)-sphere of radius kyk in Mq�1. If the (discrete) uni-form distribution on ⇧(y) is denoted by Uq�2

y and the (continuous) uniform

distribution on Sq�2kyk denoted by Uq�2

kyk , then Question 1 can be restated equiv-alently as follows:

Question 2: For large q, do there exist nonzero vectors y 2 Mq�1 such that

Uq�2y closely approximates Uq�2

kyk? Do there exist sequences {yq 2 Mq�1 } such

that the discrepancy between Uq�2yq and Uq�2

kyqk approaches zero as q ! 1?

2. Measures of spherical discrepancy.

If we abuse notation by letting Uq�2y and Uq�2

kyk also denote random vectorshaving these distributions, then the possible existence of the vectors y andsequences {yq} in Question 2 is supported by the fact that the first andsecond moments of Uq�2

y and Uq�2kyk coincide:

E(Uq�2y ) = E(Uq�2

kyk ) = 0,(14)

Cov(Uq�2y ) = Cov(Uq�2

kyk ) =kyk2q(q�1)(qIq � eq(eq)0).(15)

(In fact all odd moments agree since these are 0 by symmetry.) Three mea-sures of the discrepancy between Uq�2

y and Uq�2kyk will be considered.

For nonzero w 2 Mq�1, �1 t < 1, and ⇢ > 0 define

C(w; t) :=�v 2 Sq�2

kwk

�� v0w > kwk2t ,(16)

Cq�2⇢ :=

�C(w; t)

�� w 2 Sq�2⇢ , �1 t < 1

.(17)

Thus C(w; t) is the open spherical cap in Sq�2kwk of angular half-width cos�1(t)

centered at w, while Cq�2⇢ is the set of all such spherical caps in Sq�2

⇢ .

If U is uniformly distributed over the unit (q�2)-sphere in Rq�1 then forany unit vector u : (q � 1)⇥ 1,

(18) (u0U)2d= Beta

�12 ,

q�22

�.

4

Thus the normalized (q � 2)-dimensional surface area of the spherical capC(w; t) ⇢ Sq�2

kwk is given by

Uq�2kwk(C(w; t)) = 1

2 Pr⇥t2 Beta

�12 ,

q�22

� 1

⇤(19)

=�( q�1

2 )

2�( 12 )�(q�22 )

R 1

t2 w� 1

2 (1� w)q2�2dw(20)

=�( q�1

2 )

�( 12 )�(q�22 )

R 1

t (1� v2)q2�2dv(21)

=: �q�2(t)(22)

a strictly decreasing smooth function of t.

The following two bounds for �q�2(t) will be useful. First, from (20),

�q�2(t) <�( q�1

2 )

2�( 12 )�(q�22 )

· 1t

R 1�t2

0 uq2�2du

=�( q�1

2 )

�( 12 )�(q�22 )

· (1�t2)q2�1

t(q�2)

q

q�22⇡ · (1�t2)

q2�1

t(q�2)(23)

= (1�t2)q2�1

tp

2⇡(q�2).(24)

The inequality used to obtain (23) appears in Wendel [W]. Second, from (21)and Wendell’s inequality,

12 � �q�2(t) = 1

2 Pr⇥0 Beta

�12 ,

q�22

� t2

⇤(25)

=�( q�1

2 )

�( 12 )�(q�22 )

R t

0 (1� v2)q2�2dv

�( q�12 )

�( 12 )�(q�22 )

R t

0 e� v2(q�4)

2 dv

=�( q�1

2 )p2

�( q�22 )

pq�4)

· 1p2⇡

R tpq�4

0 e�z2

2 dz

q

q�2q�4 ·

h�⇣tpq � 4

⌘� 1

2

i,(26)

where � denotes the standard normal cumulative distribution function.

Lemma 2.1. Let {tq} be a sequence in [0, 1) such that limq!1 tqpq ⌘ �

exists, 0 � 1. Then

(27) 1� �(�) lim infq!1

�q�2(tq) lim supq!1

�q�2(tq) min(�(�)� , 12),

5

where � is the standard normal density function. Thus,

(28)

8<

:

limq!1

�q�2(tq) = 12 , � = 0;

limq!1

�q�2(tq) = 0, � = 1.

Proof. Apply (24), (26), and the bound �q�2(t) 12 for t 2 [0, 1). ⇤

For nonzero y 2 Mq�1 and any nonempty finite subset N ⇢ Sq�2

kyk , let

Uq�2N denote the (discrete) uniform distribution on N ; thus Uq�2

y = Uq�2⇧(y).

Definition 2.2. The normalized spherical cap discrepancy (NSCD) of N inSq�2kyk is defined as2

Dq�2(N) := supn��Uq�2

N (C)� Uq�2kyk (C)

����� C 2 Cq�2

kyk

o(29)

= supn�� |N\C|

|N | � Uq�2kyk (C)

����� C 2 Cq�2

kyk

o,(30)

where |N \ C| and |N | are the cardinalities of N \ C and N . The largest

empty cap discrepancy (LECD) of N in Sq�2kyk is defined as3

Lq�2(N) := supnUq�2

kyk (C)��� C 2 Cq�2

kyk , N \ C = ;o

= supn�q�2(t)

��� 9w 2 Sq�2kyk , C(w; t) 2 Cq�2

kyk , N \ C(w; t) = ;o. ⇤(31)

Obviously

(32) 0 Lq�2(N) Dq�2(N) 1.

Note that the suprema in (29)-(31) must be maxima, i.e., must be attained.This follows by applying the Blashke Selection Theorem to

�co(C)

�� C 2 Cq�2kyk

,

a collection of closed convex subsets of the closed ball B bounded by Sq�2kyk ,

where co(C) denotes the closed convex hull in B of the spherical cap C. Itfollows from this that

Lq�2(N) = �q�2�t(N)

�,(33)

2 See Leopardi [L1] Def. 2.11.5, Leopardi [L2] §1, Alishahi and Zamani [AZ] §1.2. Unlike[AZ] we divide |N \ C| by |N | to be able to compare NSCD’s of di↵ering dimensions.

3 See [AZ] §1.2.

6

where

(34) t(N) = minnt��� 9w 2 Sq�2

kyk , C(w; t) 2 Cq�2kyk , N \ C(w; t) = ;

o.

Define the unit vectors zqk 2 Mq�1 , k = 1, . . . , q � 1 as follows:

(35) zqk :=q

1q

⇣�q

q�kk , . . . ,�

qq�kk| {z }

k

,q

kq�k , . . . ,

qk

q�k| {z }

q�k

⌘0.

For 1 k < l q � 1, the inner product between zqk and zql is found to be

(36) (zqk)0zql =

qk(q�l)(q�k)l > 0.

Lemma 2.3. For nonzero y 2 Mq�1 ,

t(⇧(y)) = 1kyk min

1kq�1y0zqk,(37)

Lq�2�⇧(y)

� 1

2 .(38)

Proof. For (37), it follows from (16) that if w 2 Sq�2kyk then

⇧(y) \ C(w; t) = ; () maxP2Pq

(Py)0w kyk2t.(39)

Thus from (34) and the Rearrangement Inequality,

t(⇧(y)) = 1kyk2 min

w2Sq�2kyk

maxP2Pq

(Py)0w(40)

= 1kyk2 min

w2Mq�1 , kwk=kyk

y0w(41)

The set Mq�1 is a pointed convex simplicial cone4 whose q� 1 extreme rays

are spanned by zq1, . . . , zqq�1, so Mq�1

is their nonnegative span. Thus for

w 2 Mq�1 with kwk = kyk,

w = kyk · �1zq1+···+�q�1z

qq�1

k�1zq1+···+�q�1z

qq�1k

,

4 The geometric properties of the polyhedral cone Mq�1 that we use here stem from

its role as a fundamental region of the finite reflection group (Coxeter group) of all q ⇥ qpermutation matrices acting e↵ectively on Mq�1. A readable reference is Grove andBenson [GB]; also see Eaton and Perlman [EP].

7

for some �1 � 0, . . . ,�q�1 � 0 with �1 + · · ·+ �q�1 = 1. Therefore

y0w � kyk min1kq�1

y0zqk,

since y 2 Mq�1 ) y0zqk � 0 by (36) and k�1zq1 + · · ·+ �q�1z

qq�1k 1, hence

(42) 1kyk2 min

w2Mq�1 , kwk=kyk

y0w � 1kyk min

1kq�1y0zqk.

However equality must hold in (42) because wk := kyk zqk 2 Mq�1 and

wk = kyk. This confirms (37).

For (38), suppose that Lq�2�⇧(y)

�> 1

2 . Then ⇧(y) must be contained

in the complement of some closed hemisphere in Sq�2kyk , hence there is some

v0 2 Sq�2kyk such that 0 > w0Py for all P 2 Pq. Sum over P to obtain

0 > w0(eq(eq)0)y = w0eq((eq)0y) = 0, hence a contradiction. ⇤

It is noted in [L1] Lemma 2.11.6 and [L2] §1 that if {Nn} is a sequence offinite sets in Sq�2

kyk (q fixed), then the uniform distribution on Nn converges

weakly to Uq�2kyk as n ! 1 i↵ limn!1 Dq�2(Nn) = 0. This motivates the

following definition.

Definition 2.4. A sequence of nonzero vectors {yq 2 Mq�1 } (q varying) is

asymptotically permutation-uniform (APU) if

(43) limq!1

Dq�2(⇧(yq)) = 0;

it is asymptotically permutation-full (APF) if

(44) limq!1

Lq�2(⇧(yq)) = 0. ⇤

By (32), APU ) APF.

We also require a definition of asymptotic emptiness for a sequence ofnonzero vectors {yq 2 Mq�1

}. Because ⇧(yq) is a finite subset of the sphere

Sq�2kyqk, it always holds that Sq�2

kyqk \⇧(yq) is an infinite union of very smallempty spherical caps, so a more stringent definition of emptiness is required.

Definition 2.5. A sequence of nonzero vectors {yq 2 Mq�1 } (q varying) is

asymptotically permutation-empty (APE) if 9 ✏ > 0 and, for each q, 9 a finite

8

collection {Cqi | i = 1, . . . , nq} of (possibly overlapping) empty spherical caps

in Sq�2kyqk\⇧(yq) such that each Uq�2

kyqk(Cqi ) � ✏ and

(45) limq!1 Uq�2kyqk([nq

i=1Cqi ) = 1. ⇤

If {yq 2 Mq�1 } is APE then ⇧(yq) is asymptotically small in the sense

that ⇧(yq) ✓ ([nq

i=1Cqi )

c with Uq�2kyqk

�([nq

i=1Cqi )

c�! 0 as q ! 1. That is,

⇧(yq) occupies only an increasingly negligible portion of the sphere Sq�2kyqk.

Clearly APE ) not APF ) not APU.

Now modify the definitions of LECD and APF as follows:

Definition 2.6. The largest empty cap angular discrepancy (LECAD) of Nin Sq�2

kyk is defined to be

Aq�2(N) := supncos�1(t)

��� 9w 2 Sq�2kyk , C(w; t) 2 Cq�2

kyk , N \ C(w; t) = ;o

= cos�1(t(N)),(46)

where t(N) is defined in (34). A sequence of nonzero vectors {yq 2 Mq�1 }

(q varying) is asymptotically permutation-dense (APD) if

(47) limq!1

Aq�2(⇧(yq)) = 0. ⇤

Note that (33) and (46) yield the relation

(48) Lq�2(N) = �q�2(cos(Aq�2(N))).

If we set tq = t(⇧(yq)), it follows from (46) and (28) with � = 1 that

{yq} APD () limq!1

cos�1(tq) = 0 () tq ! 1 =) �q�2(tq) ! 0,

hence APD ) APF. However the converse need not hold: it will be shown in§4 that the sequence {yq} of maximal configurations defined in (79) is APFbut not APD.

Remark 2.7. Consider a sequence of spherical caps C(wq; tq) ✓ Mq�1 suchthat tq ! 0 while tq

pq ! 1. Then cos�1(tq) ! ⇡

2 , while �q�2(tq) ! 0 by

(28) with � = 1, that is, the spherical caps approach hemispheres in terms

9

of their angular measure but their surface areas approach 0. An examplecan be seen in §4 by taking C(wq; tq) to be the largest empty spherical capfor the set ⇧(yq), see (138) and (146). This phenomenon is yet anothermanifestation of the “curse of dimensionality”. ⇤

Question 2 now can be refined further as follows:

Question 3: For which y 2 Mq�1 , if any, are Dq�2(⇧(y)), Lq�2(⇧(y)),

and/or Aq�2(⇧(y)) small? Which sequences {yq 2 Mq�1 }, if any, are APU?

APF? APD? APE?

Some answers to these questions will be derived in §3-§5 and summarizedin §6; for example, no APD sequence exists (Proposition 6.1). Some resultsabout the volumes of the corresponding permutohedra with vertices ⇧(yq)are presented in §7. Several open questions are stated in §6-8.

Example 2.8. Despite the agreement of the first and second moments ofUq�2

y and Uq�2kyk (cf. (14), (15)), Lq�2(⇧(y)) need not be small. For example,

take y = f qq where, for i = 1, . . . , q, f qi 2 Mq�1 is the unit column vector

(49) f qi = 1pq(q�1)

(�1, . . . ,�1| {z }i�1

, q � 1, �1 . . . ,�1| {z }q�i

)0.

Here ⇧(f qq ) = {f q1 , . . . , f qq }, so |⇧(f qq )| = q not q!. From (37), (33), and (28)with tq = 1

q�1 and � = 0,

t(⇧(f qq )) =1

q�1 ,(50)

Lq�2(⇧(f qq )) = �q�2�

1q�1

�! 1

2(51)

as q ! 1. Thus the sequence {f qq } is not APF, hence not APU. ⇤

Remark 2.9. For later use, we note that for i = 1, . . . , q,

(52) f qi = 1pq(q�1)

(qeqi � eq) =q

qq�1⌦qe

qi ,

where eqi ⌘ (0, . . . , 0, 1, 0, . . . , 0)0 denotes the ith coordinate vector in Rq and⌦qe

qi is the projection of eqi onto Mq�1. Thus f q1 , . . . , f

qq form the vertices of

a standard simplex in Mq�1: an equilateral triangle when q = 3, a regulartetrahedron when q = 4, etc. ⇤

10

For any nonzero yq 2 Mq�1 ,

pq�1

kyqk Uq�2kyqk is uniformly distributed on the

sphere of radiuspq � 1 in Mq�1. It is well known (e.g. Eaton [E] Proposition

7.5) that the marginal distributions from this uniform distribution convergeto the standard normal distribution N(0, 1) as q ! 1. More precisely, forany sequence of unit vectors {uq} in Mq�1,

(uq)0⇣p

q�1kyqk U

q�2kyqk

⌘=

pq�1

kyqk (uq)0Uq�2

kyqkd! N(0, 1).

as q ! 1. If we take uq = f qi =q

qq�1⌦qe

qi (see (52)) for any fixed i, where

eqi ⌘ (0, . . . , 0, 1, 0, . . . , 0)0 is the ith coordinate vector in Rq, then

(53)pq�1

kyqk (uq)0Uq�2

kyqk =pq

kyqk(eqi )

0Uq�2kyqk ⌘

pq

kyqk(Uq�2kyqk)i

d! N(0, 1)

as q ! 1, where (Uq�2kyqk)i denotes the ith component of Uq�2

kyqk.

Proposition 2.10. A necessary condition that a sequence {yq 2 Mq�1 } of

nonzero vectors be APU is that for each fixed i � 1,

(54)pq

kyqk(Uq�2yq )i

d! N(0, 1)

as q ! 1, where (Uq�2yq )i denotes the ith component of Uq�2

yq .

Proof. From (29) and (16)-(17),

Dq�2(⇧(yq))

= sup���Uq�2

yq (C)� Uq�2kyqk(C)

�� �� C 2 Cq�2kyk

� sup�1t<1

��Uq�2yq (C(kyqkf qi ; t))� Uq�2

kyqk(C(kyqkf qi ; t))��

= sup�1t<1

��Pr⇥(f qi )

0Uq�2yq > kyqk t

⇤� Pr

⇥(f qi )

0Uq�2kyqk > kyqk t

⇤��

= sup�1t<1

��Pr⇥q

qq�1(e

qi )

0Uq�2yq > kyqk t

⇤� Pr

⇥qq

q�1(eqi )

0Uq�2kyqk > kyqk t

⇤��

= sup�1t<1

��Pr⇥ p

qkyqk(U

q�2yq )i >

pq � 1 t

⇤� Pr

⇥ pq

kyqk(Uq�2kyqk)i >

pq � 1 t

⇤��.

Because Dq�2(⇧(yq)) ! 0 if {yq} is APU, this and (53) yield (54). ⇤

11

3. The regular configurations xq and yq are not spherically uniform.

It is seen from (38) and (51) that {f qq } fails to be APF (hence fails to be APUand APD) to the greatest possible extent. Clearly this is due to the fact thatthe components of f qq comprise only two distinct values �1 and q � 1. Thissuggests that the APU, APF, and APD properties are more likely to hold forvectors yq ⌘ ⌦qxq 2 Mq�1

whose components are distinct, so that |⇧(yq)|,equivalently |⇧(xq)|, attains its maximum value q!.

At this point, it seems reasonable to conjecture that the APU, APD, andAPF properties are most likely to hold for vectors whose components areevenly spaced, that is, for the vectors

xq = (1, 2, . . . , q)0,(55)

yq = ⌦qxq = (� q�1

2 ,� q�32 , . . . , q�3

2 , q�12 )0,(56)

We call xq and yq the regular configurations in Rq and Mq�1

respectively.

This conjecture is supported by the case q = 2 with y2 = (�12 ,

12)

0,where the two permutations (�1

2 ,12)

0 and (12 , �12)

0 trivially are uniformly

distributed on S0ky2k, and by the case q = 3 with y3 = (1, 2, 3)0, where the

3!=6 permutations of y3 comprise the vertices of a regular hexagon, themost uniform among all configurations of 6 points on the circle S1

ky3k. When

q = 4, however, the 4!=24 permutations of y(4) ⌘ (�32 ,�

12 ,

12 ,

32) comprise the

vertices of the regular permutohedron R4 (see §7), a truncated octahedronwhose 14 faces consist of 8 regular hexagons and 6 squares, hence is not aregular solid.

In this section we present two arguments that show this asymptotic spher-ical uniformity conjecture is invalid for the regular configurations. The firstargument (Propositions 3.1 and 3.2) examines the APF and APE proper-ties for {xq} and {yq}, the second argument (Proposition 3.4) compares theunivariate marginal distributions of Uq�2

yq and Uq�2kyqk. A third comparison of

Uq�2yq and Uq�2

kyqk will be presented in §7.

Proposition 3.1. The sequences of regular configurations {xq 2 Rq} and

{yq 2 Mq�1 } are not APF, hence not APU and not APD.

12

Proof. It su�ces to consider {yq}. Beginning with the relations

kyqk =q

112q(q

2 � 1),(57)

(yq)0zqk = (xq)0zqk =12

pqk(q � k),(58)

it follows from (33) and (37) that the LECD of ⇧(yq) is given by

Lq�2�⇧(yq)

�= �q�2

⇣1

kyqk min1kq�1

(yq)0zqk

⌘(59)

= �q�2⇣q

3q+1

⌘,(60)

where the minimum is attained for k = 1 and k = q � 1. From (27) with

tq =q

3q+1 and � =

p3,

lim infq!1

�q�2�q

3q+1

�� 1� �(

p3) ⇡ .0416 > 0,(61)

so {yq} is not APF. ⇤

In fact, {xq} and {yq} fail asymptotic uniformity in a stronger sense:

Proposition 3.2. The sequences of regular configurations {xq 2 Rq} and

{yq 2 Mq�1 } are APE.

Proof. Again it su�ces to consider {yq}. Define zqk = kyqkzqk, where zqk isthe unit vector in (35). Because the minimum in (59) is attained for k = 1

and q � 1, i.e., for zq1 and zqq�1, both C�zq1;

q3

q+1

�and C

�zqq�1;

q3

q+1

�are

(overlapping) largest empty spherical caps for ⇧(yq) in Sq�2kyqk. (Note that z

q1 =

�f q1 and zqq�1 = f qq .) Because P⇧(yq) = ⇧(yq) for all P 2 Pq, C�P zq1;

q3

q+1

�

and C�P zqq�1;

q3

q+1

�also are (overlapping) largest empty spherical caps for

⇧(yq); there are 2q! such caps, all congruent. However

{P zq1 | P 2 Pq} = {�f q1 , . . . ,�f qq },{P zqq�1 | P 2 Pq} = {f q1 , . . . , f qq },

where f qi = kyqkf qi , so these 2q! empty caps reduce to 2q, namelynC�� f qi ;

q3

q+1

� ��� i = 1, . . . qo[nC�f qi ;

q3

q+1

� ��� i = 1, . . . qo.

13

By (59)-(61), each of these congruent empty caps remains nonnegligible asq ! 1, so {yq} is APE if

limq!1

Uq�2kyqk(⌥

q) = 1,

where

⌥q =Sq

i=1

hC�� f qi ;

q3

q+1

�[ C

�f qi ;

q3

q+1

�i.

Therefore, because

Sq�2kyqk

T�(⌥q)c

�= Sq�2

kyqkT�

\qi=1 S

qi

�,

where Sqi is the closed symmetric slab

Sqi =

nv 2 Mq�1

�� |v0f qi | kyqk2q

3q+1

o,

to show that {yq} is APE it su�ces to show that

(62) limq!1

Uq�2kyqk

�(⌥q)

c�⌘ lim

q!1Uq�2

kyqk�\qi=1 S

qi

�= 0.

If Sq1 , . . . , S

qq were mutually geometrically orthogonal, i.e., if f q1 , . . . , f

qq

were orthonormal, then the Sqi would be subindependent under Uq�2

kyqk (cf.Ball and Perissinaki [BP]), that is,

Uq�2kyqk(\

qi=1S

qi )

Yq

i=1Uq�2

kyqk(Sqi ),

which would readily yield (62). However, (f qi )0f qj = � 1

q�1 6= 0 if i 6= j so this

approach fails.5 Instead we can apply the cruder one-sided bound

Uq�2kyqk(\

qi=1S

qi ) Uq�2

kyqk(\qi=2H

qi ),(63)

where Hqi is the halfspace

Hqi :=

�v 2 Mq�1

�� v0f qi kyqk2q

3q+1

.

5 In fact, Theorem 2.1 of Das Gupta et al. [DEOPSS] suggests that Sq1 , . . . , S

qq may be

superdependent under Uq�2kyqk.

14

Again Hq2 , . . . , H

qq are not mutually geometrically orthogonal, but now this

works in our favor: because (f qi )0f qj < 0 if i 6= j, the extension of Slepian’s

inequality to spherically symmetric density functions ([DEOPSS], Lemma5.1) and a standard approximation argument yields

(64) Uq�2kyqk(\

qi=2H

qi ) Uq�2

kyqk(\qi=2K

qi ),

where Kqi is the halfspace

Kqi :=

�v 2 Mq�1

�� v0�qi kyqkq

3q+1

and �q2 , . . . , �qq are the last q � 1 columns of the Helmert matrix �q in §1,

which form an orthonormal basis in Mq�1 so (�qi )0�qj = 0. Now Proposition

A.1 in the Appendix and the orthogonal invariance of Uq�2kyqk imply that

Uq�2kyqk(\

qi=2K

qi )

Qqi=2 U

q�2kyqk(K

qi )(65)

=⇥Uq�2

kyqk(Kqi )⇤q�1

(66)

=h1� �q�2

⇣q3

q+1

⌘iq�1

.(67)

Therefore by (61),

(68) lim supq!1

⇥Uq�2

kyqk(\qi=2K

qi )⇤ 1

q�1 �(p3) ⇡ .9584,

hence by (62)-(64),

(69) Uq�2kyqk((⌥

q)c) Uq�2kyqk(\

qi=2K

qi ) (.96)q�1

for su�ciently large q. Thus (62) holds, in fact Uq�2kyqk((⌥

q)c) ! 0 at ageometric rate, hence {yq} is APE as asserted. ⇤

Remark 3.3. The above result can be framed in terms of statistical hypoth-esis testing. Based on one random observation Y ⌘ (Y1, . . . , Yq)0 2 Sq�2

kyqk,suppose that it is wished to test the spherical-uniformity hypothesis H0

that Yd= Uq�2

kyqk against the permutation-uniformity alternative H1 that

Yd= Uq�2

yq . Consider the test that rejects H0 in favor of H1 i↵ Y 2 (⌥q)c,that is, i↵

max1iq |Yi � Y | q�12 ,

15

where Y = 1q

Pqi=1 Yi. The size of this test is Uq�2

kyqk((⌥q)c), which by (69)

rapidly approaches 0 as q ! 1, while its power = 1 for every q because⇧(yq) ⇢ (⌥q)c. ⇤

A second argument for the invalidity of the spherical uniformity conjec-ture for the regular configuration {yq} (and {xq}) stems from Proposition2.10 and the following fact:

Proposition 3.4. For each fixed i � 1, as q ! 1,q

12q2�1(U

q�2yq )i

d! Uniform��

p3,p3�

(70)

as q ! 1. Thus {yq} does not satisfy (54), hence is not APU.

Proof. By (56), for each i = 1, . . . , q, (Uq�2yq )i is uniformly distributed over

the range

(71) � q�12 ,� q�3

2 , . . . , q�32 , q�1

2 ,

so its moment generating function (mgf) is

etq/2 � e�tq/2

q(et/2 � e�t/2).

(Thus the distribution of (Uq�2yq )i is the same for each i.) Therefore the mgf

ofq

12q2�1(U

q�2yq )i is

et

r3q2

q2�1 � e�t

r3q2

q2�1

q⇣etq

3q2�1 � e

�tq

3q2�1

⌘ ,

which converges to sinh(tp3)

tp3

as q ! 1, the mgf of Uniform⇥�p3,p3⇤. ⇤

4. The most favorable configuration for spherical uniformity.

It was shown in Proposition 3.1 that the regular configurations xq and yq

are not APF, hence not APU or APD, although the components of xq andyq are exactly evenly spaced. Is there is a more favorable configuration forspherical uniformity of permutations? We show now that the answer is yes.

Continuing the discussion in §2-3, we wish to find a nonzero vector y inSq�2kyqk \Rq

that minimizes the LECD Lq�2(⇧(y)) in Sq�2kyqk; equivalently, that

16

minimizes the LECAD Aq�2(⇧(y)). From (33), (37), and (48),

Lq�2(⇧(y)) = �q�2(t(⇧(y))) = �q�2⇣

1kyk min

1kq�1y0zqk

⌘,(72)

Aq�2(⇧(y)) = cos�1(t(⇧(y))) = cos�1⇣

1kyk min

1kq�1y0zqk

⌘.(73)

Thus, because �q�2(·) and cos�1(·) are decreasing and ykyk is a unit vector,

we seek a unit vector z ⌘ (z1, . . . , zq)0 2 Mq�1 that attains the maximum

(74) ⇤q := maxz2Mq�1

, kzk=1min

1kq�1z0zqk.

For 1 k q define

bqk =q

3k(q�k)q(q+1) , (bq0 = 0),(75)

aqk = bqk�1 � bqk,(76)

aq = (aq1, . . . , aqq)

0,(77)

zq ⌘ (zq1, . . . , zqq)

0 = aq

kaqk .(78)

Then aq1 + · · ·+ aqq = 0 so zq1 + · · ·+ zqq = 0, and it is straightforward to show

that aq1 < · · · < aqq, so zq1 < · · · < zqq , hence zq 2 Mq�1 . Trivially, kzqk = 1.

Proposition 4.1. The unit vector zq uniquely attains the maximum ⇤q.Thus in the original scale,

(79) yq := kyqk zq =q

q(q2�1)12

aq

kaqk

uniquely minimizes the LECD and the LECAD of ⇧(y) for y 2 Sq�2kyqk \ Rq

,and yq 6= yq when q � 4. The minimum LECD and LECAD are

Lq�2(⇧(yq)) = �q�2⇣

1kaqk

q3

q+1

⌘,(80)

Aq�2(⇧(yq)) = cos�1⇣

1kaqk

q3

q+1

⌘.(81)

Proof. For any unit vector z ⌘ (z1, . . . , zq)0 2 Mq�1 , z1 + · · · + zq = 0, so

after some algebra we find that

z0zqk =q

qk(q�k) (zk+1 + · · ·+ zq),(82)

17

hence

(83) ⇤q = maxz2Mq�1

, kzk=1min

1kq�1

qq

k(q�k) (zk+1 + · · ·+ zq).

We now show that the maximum in (83) is uniquely attained when z = zq.

Because zqk+1 + · · ·+ zqq =bqk

kaqk ,

(84)q

qk(q�k) (z

qk+1 + · · ·+ zqq) =

1kaqk

q3

q+1

for each k = 1, . . . , q � 1. Thus we must show that

(85) min1kq�1

qq

k(q�k) (zk+1 + · · ·+ zq) <1

kaqk

q3

q+1

for every z 6= zq such that z1 + · · ·+ zq = 0, kzk = 1, z1 · · · zq. Supposethat there is such a z that satisfies

(86) min1kq�1

qq

k(q�k) (zk+1 + · · ·+ zq) � 1kaqk

q3

q+1 .

Therefore if 1 k q � 1 then

zk+1 + · · ·+ zq �bqk

kaqk = zqk+1 + · · ·+ zqq ,

with equality for k = 0, so zmajorizes zq (Marshall and Olkin [MO]). Becausekzk2 is symmetric and strictly convex in (z1, . . . , zq) and z 6= zq, this impliesthat

(87) 1 = kzk2 > kzqk2 = 1,

a contradiction. Thus the maximum value ⇤q is uniquely achieved whenz = zq as asserted. It is easy to verify that aq1, . . . , a

qq are not evenly spaced

when q � 4, hence yq 6= yq. Lastly, (80) and (81) follow from (84). ⇤

The vectors yq and xq ⌘ yq+ q+12 eq are called the maximal configurations

in Mq�1 and Rq

respectively. It is now natural to ask whether the sequences{yq} and {xq} are APU or not. Because the LECD of ⇧(yq) given by (80)depends on kaqk, bounds for kaqk are needed. Because yq 6= yq, necessarilykaqk < 1 by the uniqueness of yq, but sharper bounds will be required.

18

Lemma 4.2.q

3[log(2q+1)�2]2(q+1) < kaqk <

q3[2 log(2q�1)+1]

2(q+1) .(88)

Therefore

(89) kaqk = O⇣q

log qq+1

⌘as q ! 1.

Proof. For k = 1, . . . , q set

cqk =⇥p

(k � 1(q � k + 1)�pk(q � k)

⇤2,(90)

dqk =p

k(k � 1)(q � k)(q � k + 1),

q = q+12 ,

then verify that

(91) cqk = 2⇥q2�14 � (k � q)2 � dqk

⇤.

From (75)-(76) and (90)-(91) we find that

kaqk2 = 3q(q+1)

Pqk=1 c

qk(92)

= (q � 1)� 6q(q+1)

qPk=1

dqk.

For the upper bound, use the harmonic mean-geometric mean inequality:

kaqk2 < (q � 1)� 6q(q+1)

Pqk=1

k(k�1)(q�k)(q�k+1)(k� 1

2 )(q�k+ 12 )

= (q � 1)� 6q(q+1)

Pqk=1

[(k� 12 )

2� 14 ][(q�k+ 1

2 )2� 1

4 ]

(k� 12 )(q�k+ 1

2 )

< (q � 1)� 6q(q+1)

Pqk=1

n(k � 1

2)(q � k + 12)�

k� 12

4(q�k+ 12 )

� q�k+ 12

4(k� 12 )

o

= (q � 1)� 6q(q+1)

nPqk=1(k � 1

2)(q � k + 12)�

Pqk=1

q�k+ 12

2(k� 12 )

o

= (q � 1)� 6q(q+1)

nPqk=1(k � 1

2)(q � k + 12)�

q2

Pqk=1

1k� 1

2

+ q2

o

= 3q+1

nPqk=1

1k� 1

2

� 32

o

< 3[2 log(2q�1)+1]2(q+1) ;(93)

19

the final inequality follows from (7) of Qi and Guo [QG].

Similarly, the geometric mean-arithmetic mean inequality yields the non-logarithmic lower bound 3(3q�2)

2q(q+1) . However, the asserted logarithmic lowerbound, which is sharper, can be obtained as follows. We will show that

cqk ⌘⇥p

k(q � k)�p

(k � 1)(q � k + 1)⇤2 � (k�q)2

(k� 12 )(q�k+ 1

2 ),(94)

for k = 1, . . . , q, where q = q+12 . Thus from (92),

kaqk2 � 3q(q+1)

Pqk=1

(k�q)2

(k� 12 )(q�k+ 1

2 )

= 3q2(q+1)

Pqk=1

h(k�q)2

k� 12

+ (k�q)2

q�k+ 12

i

= 3q2(q+1)

Pqk=1

h(k� 1

2 )2�2(k� 1

2 )(q2 )+( q2 )

2

k� 12

+(q�k+ 1

2 )2�2(q�k+ 1

2 )(q2 )+( q2 )

2

q�k+ 12

i

= 3q+1

hPqk=1

14

�1

k� 12

+ 1q�k+ 1

2

�� 1

i

= 3q+1

⇥Pqk=1

12k�1 � 1

⇤

> 3[log(2q+1)�2]2(q+1) ,(95)

where the inequality used in (95) also follows from (7) of [QG].

To establish (94), rewrite it in the equivalent form�p

(q + u)(q � u� 1)�p(q + u� 1)(q � u)

�2 � u2

q2�u2 ,(96)

where u ⌘ k � q 2 {� q�12 , . . . , q�1

2 } and q = q � 12 = q

2 . Now set v = uq , so

|v| q�1q < 1. Then (96) can be written in the equivalent forms

�p(q + qv)(q � qv � 1)�

p(q + qv � 1)(q � qv)

�2 � v2

1�v2 ,�p(q2 � q2v2)� (q + qv)�

p(q2 � q2v2)� (q � qv)

�2 � v2

1�v2 ,

2µ(v)� 2p

(µ(v)� qv)(µ(v) + qv) � v2

1�v2 ,

2µ(v)� 2p

µ(v)2 � q2v2 � v2

1�v2(97)

whereµ(v) = q2 � q2v2 � q = 1

4 [q2(1� v2)� 1].

It will be shown that for |v| q�1q ,

(98) 2µ(v)� v2

1�v2 � 0,

20

so (97) is equivalent to each of the following inequalities:

[2µ(v)� v2

1�v2 ]2 � 4µ(v)2 � q2v2,

q2v2 � 4µ(v) v2

1�v2 +v4

(1�v2)2 � 0,

q2v2 � [q2(1� v2)� 1] v2

1�v2 +v4

(1�v2)2 � 0,

q2v2(1� v2)� [q2(1� v2)� 1]v2 + v4

1�v2 � 0,

v2(1 + v2

1�v2 ) � 0,

which clearly is true. Thus (94) will be established once (98) is verified.

For this set x = v2, so (98) can be expressed equivalently as

h(x) ⌘ (1� x)[q2(1� x)� 1]� 2x � 0,

where 0 x ( q�1q )2. The quadratic function h(x) satisfies

h(0) = q2 � 1 > h⇥�

q�1q

�2⇤= 2� 2

q > 0 > h(1) = �2,

hence h(x) > 0 for 0 x ( q�1q )2, as required. ⇤

It follows from (53), (57), and (70) that for each fixed i � 1,

Wq :=12

q2�1

⇥(Uq�2

kyqk)i⇤2 d! �2

1,(99)

Wq :=12

q2�1

⇥(Uq�2

yq )i⇤2 d! 3Beta(12 , 1),(100)

as q ! 1. The bounds for kaqk in (88) yield a comparable result for themaximal configuration:

Proposition 4.5. For each fixed i � 1,

2Yq

log(2q�1)+2 <st Wq :=12

q2�1

⇥(Uq�2

yq )i⇤2

<st4Yq+1

log(2q+1)�2 ,(101)

where {Yq} is a sequence of positive random variables such that

Yqd! F1,2(102)

as q ! 1. Here <st denotes stochastic ordering and F1,2 denotes the Fdistribution with 1 and 2 degrees of freedom. Therefore

Wq = Op

⇣F1,2

log q

⌘p! 0,(103)

q12

q2�1(Uq�2yq )i = Op

�t2plog q

� p! 0,(104)

21

where t2 denotes Student’s t-distribution with 2 degrees of freedom.

Proof. From (75)-(79) and (90), Wq is uniformly distributed over the set

�12

q2�1q(q2�1)

123

kaqk2q(q+1) cqk

��� k = 1, . . . , q =� 3cqk

(q+1)kaqk2�� k = 1, . . . , q

,(105)

so by (95), Wq is stochastically smaller than the uniform distribution on

n2cqk

log(2q+1)�2

��� k = 1, . . . , qo=n

4[ q2�14 �(k�q)2�dqk]

log(2q+1)�2

��� k = 1, . . . , qo.

Now apply the harmonic mean-geometric mean inequality to dqk to obtain

q2�14 � (k � q)2 � dqk <

q2�14 � (k � q)2 � k(k�1)(q�k)(q�k+1)

(k� 12 )(q�k+ 1

2 )

= q2�14 � (k � q)2 � [(k� 1

2 )2� 1

4 ][(q�k+ 12 )

2� 14 ]

(k� 12 )(q�k+ 1

2 )

= q2�14 � (k � q)2 � (k � 1

2)(q � k + 12)

+q�k+ 1

2

4(k� 12 )

+k� 1

2

4(q�k+ 12 )

� 116

1(k� 1

2 )(q�k+ 12 )

= 14

hq2� 1

4

(k� 12 )(q�k+ 1

2 )� 3

i

= 14

hq2� 1

4q2

4 �(k�q)2� 3

i

< 1

1�4�

k�qq

�2 � 34 ,

where we have twice used the relation

(106) (k � q)2 + (k � 12)(q � k + 1

2) =q2

4 .

Therefore Wq is stochastically smaller than

4log(2q+1)�2

�1

1�4V 2q� 3

4

�⌘ 4Yq+1

log(2q+1)�2 ,

where

Vqd= Uniform

�k�qq

�� k = 1, . . . , q ,

Yq =4V 2

q

1�4V 2q.

22

Because k�qq = 2k�q�1

2q , clearly

Vqd! V

d= Uniform(�1

2 ,12),

4V 2q

d! 4V 2 d= Beta(12 , 1),

as q ! 1, from which it follows that Yqd! F1,2.

Similarly from (93), (94), (105), and (106), Wq is stochastically largerthan the uniform distribution on

n2

log(2q+1)�2

h(k�q)2

q2

4 �(k�q)2

i ��� k = 1, . . . , qo,

so Wq is stochastically larger than

2log(2q+1)�2

� 4V 2q

1�4V 2q

�⌘ 2Yq

log(2q+1)�2 ,

as asserted. ⇤

It is now easy to answer the question posed earlier:

Proposition 4.6. The sequences of maximal configurations {yq} and {xq}are not APU.

Proof. It follows from (101) and (103) that for any fixed i,

(107)pq

kyqk(Uq�2yq )i =

q12

q2�1(Uq�2yq )i = Op

�1plog q

� p! 0,

hence by Proposition 2.10 {yq} and {xq} cannot be APU. ⇤

5. The normal configuration.

The sequence {yq}, like {yq}, fails to satisfy the necessary condition (54)for APU, yet {yq} uniquely minimizes the LECD and LECAD, so it seemsreasonable to conjecture that no APU sequence exists. However, it is easyto find a sequence {yq 2 Mq�1

} that does satisfy (54). Define

(108) aq ⌘ (aq1, . . . , aqq) =

���1( 1

q+1),��1( 2

q+1), . . . ,��1( q

q+1)�0,

the kq+1 -quantiles of the N(0, 1) distribution, then in the original scale let

(109) yq = kyqk aq

kaqk .

23

Clearly aq1 < · · · < aqq while aq1 + · · · + aqq = 0 by the symmetry of N(0, 1),

hence yq 2 Mq�1 . The vector yq is called the normal configuration.

For each i = 1, . . . , q, (Uq�2aq )i

d= ��1(Uq), where

Uqd= Uniform

����1( 1

q+1), . . . ,��1( q

q+1) � d! Uniform(0, 1),

hence (Uq�2aq )i

d! N(0, 1) as q ! 1. Furthermore,

kaqk2q+1 + 1

q+1 [��1( q

q+1)⇤2 ⌘ 1

q+1

Pqk=1

⇥��1( k

q+1)⇤2

+ 1q+1 [�

�1( qq+1)

⇤2

is an approximating Riemann sum for

R 1

0 [��1(u)]2du =

R1�1 x2�(x)dx = 1,

while

(110) ��1( qq+1) =

p2 log(q + 1)(1 + o(1))

as q ! 1 (e.g. Fung and Seneta [FS] p.1092), hence

kaqk2q+1 = 1� 2 log(q+1)

q+1 + o(1),(111)

kaqk ⇠pq + 1.(112)

Therefore {yq} satisfies (54):

(113)pq

kyqk(Uq�2yq )i =

pq

kaqk(Uq�2aq )i

d! N(0, 1)

as q ! 1. However, it is now shown that the LECD of {yq}, necessarilygreater than that of {yq}, does not approach 0.

24

Proposition 5.1. {yq} is not APF, hence is not APU.

Proof. By (33)-(37) and (108)-(109),

Lq�2�⇧(yq)

�= �q�2(tq),(114)

tq : = 1kyqk min

1kq�1(yq)0zqk(115)

1kyqk(y

q)0zqq�1

= 1kaqk(a

q)0zqq�1(116)

= 1kaqk

qq

q�1��1�

qq+1

�(117)

< 1kaqk

qq

q�1

�(��1( qq+1 ))

1��(��1( qq+1 ))

= q+1p2⇡kaqk

qq

q�1e� 1

2 [��1( q

q+1 )]2

.

It follows from Fung and Seneta [FS] p.1092 that

(118) ��1�

qq+1

�=q

2 log�(q + 1)

p4⇡ log(q + 1)

�✓1 +�q

◆

where �q = O⇣

log(log(q+1))(log(q+1))2

⌘, hence

e�12 [�

�1( qq+1 )]

2

= 1q+1

1p4⇡ log(q+1)

�1

q+1

��q+�2q�

1p4⇡ log(q+1)

��q+�2q

= 1q+1o(1)(1 + o(1))(1 + o(1))

= 1q+1o(1).

Therefore by (112),

tqpq < 1p

2⇡kaqkqpq�1

o(1) = o(1),(119)

hence limq!1 tqpq = 0, so

(120) limq!1

Lq�2�⇧(yq)

�= 1

2

by Lemma 2.1 with � = 0. This completes the proof. ⇤

Remark 5.2. It should be noted that the convergences in (107) and (120)occur at very slow sublogarithmic rates. ⇤

25

Proposition 5.3. {yq} is APE.

Proof. The proof is similar to that of Proposition 3.2. Again define zqk =kyqkzqk, where zqk is the unit vector in (35), and define

sq = 1kaqk(a

q)0zqq�1,

cf. (116). As in (116)-(119),

sqpq < 1p

2⇡kaqkqpq�1

o(1) = o(1),(121)

hence from (19)-(22) and Lemma 2.1 with � = 0,

limq!1

Uq�2kyqk

�C�zqq�1; sq

��= lim

q!1�q�2(sq) =

12 .(122)

Furthermore, from (39) and the Rearrangement Inequality,

⇧(yq) \ C�zqq�1; sq

�= ; () max

P2Pq(P yq)0zqq�1 kyqk2sq

() (yq)0zqq�1 kyqk2sq() 1

kaqk(aq)0zqq�1 sq,

hence C�zqq�1; sq

�is an empty spherical cap for ⇧(yq).

Because P⇧(yq) = ⇧(yq) for all P 2 Pq, each C�P zqq�1; sq

�is an empty

spherical cap for ⇧(yq) in Sq�2kyqk; there are q! such congruent caps. However

{P zqq�1 | P 2 Pq} = {f q1 , . . . , f qq },

where f qi = kyqkf qi , so these q! empty caps reduce to q congruent ones, namely�C�f qi ; sq

� �� i = 1, . . . q .

By (122) each of these congruent caps remains nonnegligible as q ! 1, soto show that {yq} is APE it su�ces to show that

(123) limq!1

Uq�2kyqk

�⌥q�= 1,

where

⌥q =Sq

i=1

hC�f qi ; sq

�i.

26

Clearly

(124) Sq�2kyqk

T�(⌥q)c

�✓ Sq�2

kyqkT�

\qi=2 H

qi

�,

where Hqi is the halfspace

Hqi :=

�v 2 Mq�1

�� v0f qi kyqk2 sq .

As in the proof of Proposition 3.2, (f qi )0f qj = � 1

q�1 < 0 if i 6= j so

(125) Uq�2kyqk

�\q

i=2 Hqi

� Uq�2

kyqk�\q

i=2 Kqi

�,

where Kqi is the halfspace

Kqi :=

�v 2 Mq�1

�� v0�qi kyqk sq .

Again apply Proposition A.1 in the Appendix and the orthogonal invarianceof Uq�2

kyqk to obtain

Uq�2kyqk

�\q

i=2 Kqi

�Qq

i=2 Uq�2kyqk(K

qi )(126)

=⇥Uq�2

kyqk(Kqi )⇤q�1

=⇥1� �q�2

�sq�⇤q�1

.(127)

Therefore by (122),

(128) lim supq!1

⇥Uq�2

kyqk�\q

i=2 Kqi

�⇤ 1q�1 1

2 ,

hence by (124)-(128),

(129) Uq�2kyqk

�(⌥q)c

� Uq�2

kyqk�\q

i=2 Kqi

� (.51)q�1

for su�ciently large q. Thus (123) holds, in fact Uq�2kyqk((⌥

q)c) ! 0 at ageometric rate, hence {yq} is APE as asserted. ⇤

6. Comparisons among the distributions.

Based on the results in §3-5, comparisons among the three uniform distribu-tions Uq�2

yq , Uq�2yq , Uq�2

yq on permutations and the uniform distribution Uq�2kyqk

on the sphere Sq�2kyqk are collected here.

27

The LECDs of ⇧(yq), ⇧(yq), and ⇧(yq) are as follows:

Lq�2(⇧(yq)) = �q�2⇣q

3q+1

⌘;(130)

Lq�2(⇧(yq)) = �q�2⇣

1kaqk

q3

q+1

⌘;(131)

Lq�2�⇧(yq)

�= �q�2(tq).(132)

Here kaqk is given by (77) and approximated in (88), while tq is given by (115)and bounded above by (117) together with (111). Some explicit bounds andasymptotic comparisons among these LECDs are now developed.

First, from (24) and (26),

12 �

qq�2q�4

h�⇣q

3(q�4)q+1

⌘� 1

2

i< Lq�2(⇧(yq)) <

hq�2q+1

i q�22q

q+16⇡(q�2) .(133)

Also, if we set tq =q

3q+1 then tq

pq =

q3qq+1 !

p3, so asymptotically

.0416 = 1� �(p3) lim inf

q!1Lq�2(⇧(yq))(134)

lim supq!1

Lq�2(⇧(yq)) = �(p3)p3

= .0514(135)

by Lemma 2.1 with � =p3.

Second, from (88),

�q�2⇣q

2log(2q+1)�2

⌘< Lq�2(⇧(yq)) < �q�2

⇣q2

2 log(2q�1)+1

⌘,

which, combined with (24) and (26), yields the explicit bounds

12 �

qq�2q�4

h�⇣q

2(q�4)log(2q+1)�2

⌘� 1

2

i< Lq�2(⇧(yq))(136)

<h2 log(2q�1)�12 log(2q�1)+1

i q�22q

2 log(2q�1)+14⇡(q�2) .(137)

Also, if we set tq = 1kaqk

q3

q+1 then (88) yields

qq

2 log(2q�1)+1 < tqpq <

q2q

log(2q+1)�2 ,

28

so from Lemma 2.1 with � = 1 we obtain the asymptotic limit

(138) limq!1

Lq�2(⇧(yq)) = 0.

Third, from (26) and (117),

12 �

qq�2q�4

h�⇣

1kaqk

qq(q�4)q�1 �

�1�

qq+1

�⌘� 1

2

i< Lq�2(⇧(yq)),

while asymptotically (repeating (120)),

(139) limq!1

Lq�2�⇧(yq)

�= 1

2 .

The LECADs of ⇧(yq), ⇧(yq), and ⇧(yq) are as follows:

Aq�2(⇧(yq)) = cos�1⇣q

3q+1

⌘;(140)

Aq�2(⇧(yq)) = cos�1⇣

1kaqk

q3

q+1

⌘;(141)

Aq�2�⇧(yq)

�= cos�1(tq).(142)

These yield som explicit expressions and bounds for the LECADs:

Aq�2(⇧(yq)) = cos�1⇣q

3q+1

⌘;(143)

cos�1⇣q

2log(2q+1)�2

⌘<Aq�2(⇧(yq)) < cos�1

⇣q2

2 log(2q�1)+1

⌘;(144)

cos�1⇣

1kaqk

qq

q�1��1�

qq+1

� ⌘<Aq�2(⇧(yq)).(145)

Asymptotic comparisons of the LECADs are extremely simple:

Proposition 6.1. For any sequence of nonzero vectors {yq 2 Mq�1 },

limq!1

Aq�2(⇧(yq)) = cos�1(0) = ⇡2 ,(146)

that is, the largest empty cap for ⇧(yq) approaches a hemisphere in terms ofits angular measure. Therefore no APD sequence exists.

Proof. From the lower bound in (144) we see that (146) holds for themaximal configurations {yq}. Because yq minimizes the largest empty cap,(146) holds for all nonzero sequences {yq}. ⇤

29

Next, the standardized limits of the univariate marginal distributions areas follows: for each fixed i � 1,

q12

q2�1(Uq�2yq )i

d! Uniform��p3,p3�;(147)

q12

q2�1(Uq�2yq )i = Op

�t2plog q

� p! 0;(148)q

12q2�1(U

q�2yq )i

d! N(0, 1);(149)q

12q2�1(U

q�2kyqk)i

d! N(0, 1).(150)

These asymptotic results for the LECDs, LECADs, and univariate marginaldistributions of the regular, maximal, and normal configurations are summa-rized in Table 1.

yq limq!1

Lq�2(⇧(yq)) limq!1

Aq�2(⇧(yq)) APF APU APE N(0,1)

yq regular (.0416, .0514) ⇡/2 no no yes noyq maximal 0 ⇡/2 yes no no noyq normal 1/2 ⇡/2 no no yes yeskyqk spherical “0” “⇡/2” “yes” “yes” “no” “yes”

Table 1: The first three rows refer to the discrete uniform distribution on thepermutations in ⇧(yq). The fourth row refers to the continuous uniform dis-tribution on the sphere Sq�2

kyqk, where the “entries” hold trivially. The secondand third columns show the limiting LECDs and LECADs, respectively. Thefinal column indicates whether or not the univariate marginal distributionsconverge to N(0, 1), a necessary condition for APU.

Thus, neither the regular nor normal sequences is APU, nor is the max-imal sequence APU even though it is APF. Therefore we conjecture, albeitsomewhat weakly, that the answer to the following question is no:

Question 4: Does any APU sequence {yq 2 Mq�1 } exist?

Some exact values of yq, yq, and yq, are shown in Table 2. For q = 3,y3 = y3 = y3, while for q � 4 the components of yq disperse more rapidlythan those of yq and yq as q increases. This is also seen from the followingasymptotic comparisons of the magnitudes of the ranges of the univariate

30

q yq yq yq

3 (0, 1) (0, 1) (0, 1)4 (.5, 1.5) (.242, 1.56) (.459, 1.51)5 (0, 1, 2) (0, .490, 2.18) (0, .909, 2.04)6 (.5, 1.5, 2.5) (.219, .756, 2.85) (.436, 1.37, 2.59)

Table 2: The regular, maximal, and normal configurations for q = 3, 4, 5, 6.The q components of each vector yq are symmetric about 0 so only thenonnegative components are shown.

marginal distributions: for each i = 1, . . . , q,��range[(Uq�2

yq )i]�� =

��[� q�12 , q�1

2 ]�� = q = O(q)

��range[(Uq�2yq )i]

�� =��⇥� kyqk

kaqk aqq,

kyqkkaqk a

qq

⇤�� = q�1kaqk = O

⇣q32p

log q

⌘

��range[(Uq�2yq )i]

�� =��⇥� kyqk

kaqk aqq,

kyqkkaqk a

qq

⇤�� ⇠q

q2�13 ��1( q

q+1) = O⇣qp

log q⌘

��range[(Uq�2kyqk)i]

�� =��[�kyqk, kyqk]

�� =q

q(q2�1)3 = O

�q

32�.

The four ranges satisfy

(151) regular ⌧ normal ⌧ maximal ⌧ spherical,

where “⌧” indicates o(·), whereas the limiting distributions of the univariatemarginals in (147)-(150) satisfy

(152) maximal ⌧p regular ⇡p normal ⇡p spherical,

where “⌧p” indicates op(·) and “⇡p” indicates Op(·). The ordering (152) issomewhat unexpected since the maximal configuration is the only one of thethree uniform permutation distributions that is APF.

7. The regular, maximal, and normal permutohedra.

The regular permutohedron6 Rq is defined to be the convex hull of ⇧(xq), the

set of all q! permutations of the regular configuration xq ⌘ (1, 2, . . . , q)0. It isa convex polyhedron in Mq�1

xq (cf. (4)) of a�ne dimension q�1. Equivalently

6 a.k.a. permutahedron.

31

we shall consider the congruent polyhedron Rq ⌘ ⌦q Rq, the translation ofRq into Mq�1, so Rq is the convex hull of ⇧(yq) (cf. (56)). Thus the uniformdistribution Uq�2

yq is the uniform distribution on the vertices of Rq.

Proposition 3.2 shows that ⇧(yq) occupies a vanishingly small portion ofthe sphere Sq�2

kyqk as q ! 1. Similarly, it will now be shown that Rq occupies

a vanishingly small portion of the corresponding ball Bq := Bq�2kyqk in which

Rq is inscribed.

Proposition 7.1. As q ! 1, Vol(Rq)

Vol(Bq)! 0 at a geometric rate.

Proof. From Proposition 2.11 of Baek and Adams [BA] with d = q � 1, thevolume of Rq is qq�

32 , while the volume of Bq is

⇡q�12 kyqkq�1

�( q+12 )

= ⇡q�12

�( q+12 )

⇥ q(q2�1)12

⇤ q�12 .

Therefore, using Stirling’s formula, the ratio of the volumes is given by

Vol(Rq)

Vol(Bq)=�12⇡

� q�12 qq�

32 �( q+1

2 )

[q(q2�1)]q�12

⇠�⇡e

� 12�

6⇡e

� q�12

⇡ 1.0750�0.7026

� q�12(153)

as q ! 1, which converges to zero at a geometric rate.

Remark 7.2. By comparison, the cube Cq inscribed in Bq has vertices

�± kyqkp

q , . . . ,±kyqkp

q

�=⇣±q

q2�112 , . . . ,±

qq2�112

⌘,

so

Vol(Cq)

Vol(Rq)=�q2�13

� q�12 q

32�q

⇠ q12

3q�12

⇡ q12 (0.3333)

q�12(154)

as q ! 1, which also converges to zero at a geometric rate. Therefore

(155) Vol(Cq) ⌧ Vol(Rq) ⌧ Vol(Bq)

32

for large q. ⇤

Next, define the maximal permutohedron Mq (normal permutohedron Nq)to be the convex hull of ⇧(yq) (⇧(yq)), the set of all q! permutations ofthe maximal configuration yq (normal configuration yq). Like the regularpermutohedron Rq defined in §7, Mq and Nq are convex polyhedrons inMq�1 (cf. (4)) of a�ne dimension q�1. Thus the uniform distribution Uq�2

yq

(Uq�2yq ) is the uniform distribution on the vertices of Mq (Nq). the following

question is suggested:

Question 5: What are the volumes of Mqand Nq

? As in Proposition 7.1 and

Remark 7.2, compare Vol(Rq), Vol(Mq), Vol(Nq), and Vol(Bq).

We conjecture, again somewhat weakly, that as q ! 1,

(156) Vol(Rq) ⌧ Vol(Mq) ⌧ Vol(Bq),

more precisely, that Vol(Rq)

Vol(Mq)! 0 at a geometric rate and Vol(Mq)

Vol(Bq)! 0 at a

slower rate. Similar results are expected if Mq is replaced by Nq.

8. Concluding remarks.

We conclude with a final question and remark.

Question 6: If the permutation group is replaced by some other finite subgroup

G of orthogonal transformations on Rq, how close to spherical uniformity is

the G-orbit ⇧G(yq) ⌘ {gyq | g 2 G} for nonzero yq 2 Rq?

Finite reflection groups (Coxeter groups) acting on Rq for all q � 2 are ofparticular interest, cf. [EP], [GB]. These include, and in fact are limited to,the permutation (= symmetric) group, the alternating group, and the groupgenerated by all permutations and sign changes of coordinates.

Remark 8.1. In coding theory, a finite set of N points on a d-sphere is calleda spherical code, cf. Leopardi [L1], [L2]. A question of major interest is theconstruction of spherical codes having small spherical discrepancy for largeN with d held fixed (recall Definition 2.2). Thus our sets ⇧(x) and ⇧(y),consisting of all q! permutations of x and y, can be viewed as spherical codesof a special type; we suggest that these be called permutation codes. We areinterested in a similar question: which if any permutation codes are APU,that is, have small spherical discrepancy as q ! 1? Here, however, N = q!and d = q � 2, so both N ! 1 and d ! 1 in our case. ⇤

33

Appendix. Subindependence of coordinate halfspaces.

The following inequality was used in the proof of Proposition 3.2:

Proposition A.1. Let Un ⌘ (U1, . . . , Un)0 be uniformly distributed on theunit (n� 1)-sphere Sn�1 in Rn. For any positive real numbers t1, . . . , tn,

(157) Pr[\ni=1{Ui ti}]

Yn

i=1Pr[Ui ti].

Proof. The proof is modelled on that of Proposition 2.10 in Barthe et al.

[BGLR]. We shall show more generally that for 1 r < n,

(158) Pr[\ni=1{Ui ti}] Pr[\r

i=1{Ui ti}] Pr[\ni=r+1{Ui ti}].

Because Un is the unique orthogonally invariant distribution on Sn�1,

Un ⌘✓

Ur

U�r

◆d=

✓ 00 In�r

◆✓Ur

U�r

◆=

✓ Ur

U�r

◆

for every orthogonal r ⇥ r matrix , where

Ur = (U1, . . . , Ur)0,

U�r = (Ur+1, . . . , Un)0.

Therefore ✓Wr

U�r

◆d=

✓ Wr

U�r

◆,

where

Wr ⌘ (W1, . . . ,Wr)0 = Ur

kUrk 2 Sr�1,

W�r ⌘ (Wr+1, . . . ,Wn)0 = U�r

kU�rk 2 Sn�r+1.

Thus the conditional distribution ofWr|U�r is the same as that of Wr|U�r

so, by uniqueness, is the uniform distribution on Sr�1. Therefore Wr is inde-pendent of U�r, hence Wr is independent of (W�r, kU�rk). Similarly, W�r

is independent of (Wr, kUrk). However, kUrk and kU�rk are statisticallyequivalent because kUrk2+kU�rk2 = kUnk2 = 1, hence W�r is independentof kU�rk, so Wr, W�r, and kU�rk are mutually independent. Thus Wr,

34

W�r, and kUrk are mutually independent, so

Pr[\ni=1{Ui ti}]

= EnPr[\r

i=1{Wi tikUrk�1} | kUrk]

· Pr[\ni=r+1{Wi ti(1� kUrk2)�1/2} | kUrk]

o

E�Pr[\r

i=1{Wi tikUrk�1} | kUrk]

· E�Pr[\n

i=r+1{Wi ti(1� kUrk2)�1/2} | kUrk]

= Pr[\ri=1{Ui ti}] · Pr[\n

i=r+1{Ui ti}].

The inequality holds because

Pr[\ri=1{Wi tikUrk�1} | kUrk]

is decreasing in kUrk while

Pr[\ni=r+1{Ui ti(1� kUrk2)�1/2} | kUrk]

is increasing in kUrk. ⇤

Remark A.2. The inequality (157) is a one-sided version for coordinatehalfspaces of a two-sided inequality for symmetric coordinate slabs, where|Ui| appears in place of Ui; see [BGLR] pp. 329-330 and the references citedtherein. As in [BGLR], it is straightforward to extend Proposition A.1 todistributions on the unit sphere in `p for 1 p < 1. ⇤

Acknowledgement. This paper was prepared with invaluable assistance fromSteve Gillispie. Warm thanks are also due to Persi Diaconis and Art Owenfor very helpful discussions about random permutations, spherical geometry,and uniform distributions on groups.

References

[AZ] Alishahi, K. and Zamani, M. S. (2015). The spherical ensemble anduniform distribution of points on the sphere. Electronic J. Prob. 20 1-27.

[BA] Baek, J. and Adams, A. (2009). Some useful properties of the permu-tohedral lattice for Gaussian filtering. http://graphics.stanford.edu/papers/permutohedral/permutohedral techreport.pdf.

35

[BP] Ball, K. and Perissinaki, I. (1998). The subindependence of coordinateslabs in `np balls. Israel J. Math. 107 289-299.

[BGLR] Barthe, F., Gamboa, F., Lozada-Chang, L., Rouault, A. (2010).Generalized Dirichlet distributions on the ball and moments. Latin American

J. Prob. Math. Statist. 7 319-340.

[DEOPSS] Das Gupta, S., Eaton, M. L., Olkin, I., Perlman, M. D., Savage,L. J., Sobel, M. (1972). Inequalities for the probability content of convexregions for elliptically contoured distributions. Proc. Sixth Berkely Symp.

Math. Statist. Prob. 2 241-265.

[E] Eaton, M. L. (1989). Group Invariance Applications in Statistics. Re-gional Conference Series in Probability and Statistics Vol. 1, Institute ofMathematical Statistics.

[EP] Eaton, M. L. and Perlman, M. D. (1977). Reflection groups, generalizedSchur functions, and the geometry of majorization. Ann. Prob. 5 829-860.

[FS] Fung, T. and Seneta, E. (2018). Quantile function expansion usingregularly varying functions. Method. Comp. Appl. Probab. 20 1091-1103.

[GB] Grove, L. C. and Benson, C. T. (1985). Finite Reflection Groups, 2ndEd. Springer, New York.

[L1] Leopardi, P. (2007). Distributing points on the sphere: partitions, sep-aration, quadrature and energy. Ph.D. thesis, The University of New SouthWales (2007).

[L2] Leopardi, P. (2013). Discrepancy, separation and Riesz energy of finitepoint sets. Adv. Comp. Math. 39 27-43.

[MO] Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Ma-

jorization and Its Applications. Academic Press, New York.

[QG] Qi, F. and Guo, B.-N. (2011). Sharp bounds for harmonic numbers.Applied Math. and Computation 218 991–995.

[W] Wendel, J. G. (1948). Note on the gamma function. Amer. Math.

Monthly 55 563-564.

36