arXiv:1607.07549v2 [] 11 Sep 2016

Concentration of Measure for Radial Distributions and
Consequences for Statistical Modeling
Ery Arias-Castro

Xiao Pu

Abstract
Motivated by problems in high-dimensional statistics such as mixture modeling for classification and clustering, we consider the behavior of radial densities as the dimension increases.
We establish a form of concentration of measure, and even a convergence in distribution, under
additional assumptions. This extends the well-known behavior of the normal distribution (its
concentration around the sphere of radius square-root of the dimension) to other radial densities. We draw some possible consequences for statistical modeling in high-dimensions, including
a possible universality property of Gaussian mixtures.

1

Introduction

Nonparametric density estimation can be quickly difficult in high-dimensions because of the curse of
dimensionality. Additional assumptions are often needed. The most popular one might well be the
Naive Bayes approach, popular in classification (Lewis, 1998), which presumes that the variables
are independent, or equivalently, that the density is the product of its marginals.
Another possibility is to assume that the density is elliptical, a classical assumption in multivariate analysis (Anderson, 2003), meaning that f is of the form f (x) = ∣A∣g(∥Ax∥), where A is a
positive definite matrix. Any centered and non-degenerate normal distribution is elliptical, with
base distribution the standard normal distribution and A = Σ−1/2 , where Σ is the covariance matrix.
Note that the same distribution is also the product of its marginals when Σ is diagonal — the assumption underlying Linear Discriminant Analysis, an important parametric special case of Naive
Bayes. High-dimensional density estimation is of course crucial in classification and clustering.
Our motivation comes from such problems, and in particular, mixture modeling in high-dimensions
using elliptical distributions. In the process of working on this problem we uncovered a difficulty
which we elaborate upon in the present paper: that of estimating the base function. Indeed, it is
quite tempting to extend the Gaussian mixture models to models of the form
K

∑ πk ∣Ak ∣gk (∥Ak x∥),

(1)

k=1

where we assume the mixture has K components, with the kth component having weight πk and density ∣Ak ∣gk (∥Ak x∥). For example, Bickel et al. (1998) and more recently Bhattacharyya and Bickel
(2015) consider models of this kind. Instead of smoothness assumptions, we are more interested here in shape assumptions, for example that gk is decreasing and/or log-concave on R+ .
Both authors are with the Department of Mathematics, University of California, San Diego, USA. Contact
Ery Arias-Castro or Xiao (Victor) Pu. We are grateful to Chris Sherlock and Daniel Elton for bringing their paper
to our attention, and to Gabor Lugosi for helpful discussions. This work was partially supported by a grant from the
US Office of Naval Research (N00014-13-1-0257).

1

Chang and Walther (2007) consider such mixture models but under the Naive Bayes assumption
instead of assuming the densities are elliptical. An EM approach to fitting such a model involves
being able to estimate gk based on a sample from gk (∥x∥). And this is what we found challenging
in our investigation.
Focusing on this task, suppose we have an IID sample from f , where f rotationally invariant (aka
radial), meaning that f (x) = g(∥x∥) for some function g, and consider the problem of estimating
g. In fact, we can work with the magnitudes (the norms of the observations), which are sufficient.
We explain the difficulty of estimating g by the fact that the magnitudes are highly concentrated
as the dimension becomes large.
The simplest case of this concentration of measure phenomenon arises when we assume that
that g ∝ ψ, where ψ is fixed, as is the case in the Gaussian setting. Specifically, we assume we are
in dimension d + 1 and we work with ψ satisfying the following assumptions (and some additional
assumptions specified later on)
ψ ∶ R+ → R+ is such that 1/cd ∶= ∫

∞
0

ud ψ(u)du < ∞ for all d ≥ 1.

(2)

We let Xd denote a random variable with density cd ψ(∥x∥) on Rd+1 and let Ud denote its magnitude,
Ud = ∥Xd ∥, which has density cd ud ψ(u) on R+ .
In this context we show a form of concentration of measure, and convergence in distribution, as
the dimension d increases. Concentration is a well-known phenomenon in high-dimensions, in particular for product distributions (Naive Bayes), with far-reaching consequences (Boucheron et al.,
2013; Ledoux, 2005). For radial distributions, it is not as well-known, except for when the density is Gaussian or uniform on a ball. (The latter is often used to explain some forms of curse
of dimensionality.) This case was recently studied in detail by Sherlock and Elton (2012), who
cite older work by Diaconis and Freedman (1984, 1987) in rather special cases. We comment on
(Sherlock and Elton, 2012) in more detail in Section 4, after we present our results.
In the remaining of the paper we study what happens when d → ∞. In Section 2 we study the
case where ψ has compact support, which is the simplest situation. In Section 3 we consider the
case where ψ is not compactly supported. In Section 4 we discuss our results and some possible
implications for statistical modeling.

2

The case of compact support

In this whole section we assume that ψ has compact support. Define the supremum of the support
as follows
u
(3)
u∗ = sup {u ∶ ∫ ψ(u)du > 0 for all ε > 0}.
u−ε

Note that u∗ < ∞ by assumption and that the support of ψ is included in [0, u∗ ]. If ψ is continuous
(which the reader can assume without much loss of generality), then the following is an equivalent
definition u∗ = sup{u ∶ ψ(u) > 0}. The emblematic example is that of the uniform distribution
on the unit ball, in which case ψ(u) = I{u ≤ 1} and u∗ = 1. This distribution is well-known to
concentrate near the boundary of its support (the unit sphere). Our results below extend this to
other distributions with compact support.

2.1

Convergence in probability

We start by establishing a convergence in probability.
Theorem 1. In the setting considered here, Ud → u∗ in probability as d → ∞.
2

Proof. Assume u∗ = 1 without loss of generality. Then
P(Ud < 1 − ε) = cd ∫

1−ε

0

while

ud ψ(u)du ≤ cd (1 − ε)d ∫

P(Ud ≥ 1 − ε) ≥ P(Ud ≥ 1 − ε/2) = cd ∫

1
1−ε/2

1

(4)

ψ(u)du,
0

ud ψ(u)du ≥ cd (1 − ε/2)d ∫

1

1−ε/2

ψ(u)du.

(5)

Note that the last integral is strictly positive for all ε > 0 by definition of u∗ in (3) (recall that we
assumed that u∗ = 1). Hence
(1 − ε)d ∫0 ψ(u)du
P(Ud < 1 − ε)
→ 0,
≤
1
P(Ud ≥ 1 − ε) (1 − ε/2)d ∫1−ε/2
ψ(u)du
1

d → ∞,

(6)

when ε ∈ (0, 1) is fixed. Since P(Ud < 1 − ε) + P(Ud ≥ 1 − ε) = 1, we proved that P(Ud < 1 − ε) → 0
for all ε > 0. This, coupled with the fact that P(Ud ≤ 1) = 1, proves that Ud → 1 in probability as
d → ∞.

2.2

Convergence in distribution

Beyond a convergence in probability, we can establish a convergence in distribution. The limiting
distribution happens to depend on the behavior of ψ in the neighborhood of u∗ . We only cover the
case where ψ behaves as a power function near u∗ .
Theorem 2. In the setting considered here, assume in addition that ψ is bounded and that ψ(u) ∼
a(u∗ − u)b as u ↗ u∗ for some a > 0 and b > −1. Then d(u∗ − Ud ) converges weakly to the Gamma
distribution with shape parameter b + 1 and rate 1/u∗ .

Proof. Assume without loss of generality that u∗ = 1. We first control the behavior of cd as d → ∞.
Fix ε ∈ (0, 1). On the one hand, by the assumptions on ψ and Dominated Convergence, we have
d
d
b
∫1−ε u ψ(u)du ∼ ∫1−ε u a(1 − u) du ∼ aB(d + 1, b + 1),
1

1

d → ∞,

(7)

where B is the Beta function. On the other hand, by Theorem 1,
cd ∫

1

1−ε

Together, this proves that

ud ψ(u)du ∼ 1,

d → ∞.

1/cd ∼ aB(d + 1, b + 1) ∼ aΓ(b + 1)d−(b+1) ,

(8)
d → ∞,

(9)

where Γ is the Gamma function.
We now consider the case where ε = εd → 0 as d → ∞. More precisely, we fix t > 0 and set
εd = t/d. By Dominated Convergence again, applied twice, and a change of variables, as d → ∞, we
have
P(Ud > 1 − t/d) = cd ∫

1

1−t/d
1

∼ cd ∫

1−t/d

ud ψ(u)du

(10)

ud a(1 − u)b du

(11)

= cd ad−(b+1) ∫ (1 − v/d)d v b dv

∼

t

0
t

1
−v b
∫ e v dv.
Γ(b + 1) 0
3

(12)
(13)

Recognizing the distribution function of the Gamma distribution with shape parameter b + 1 and
rate 1, the proof is complete.

3

The case of non-compact support

We now assume that ψ has non-compact support, which is equivalent to u∗ = ∞ in (3). We note
that here the emblematic example is that
which is known to
√ of the standard
√ normal distribution,
2
concentrate near the sphere of radius d, meaning Ud / d → 1. In
√ fact, Ud has the chi-squared
√
distribution with d degrees of freedom, and in particular, 2(Ud − d) is asymptotically standard
normal in the limit d → ∞. Our results below extend this phenomena to other radial distributions.
While we were able to handle the case of compact support, which we treated in Section 2,
with very natural assumptions, the case of non-compact support appears more challenging and our
working assumptions are more complicated. This is despite the fact that we favored simplicity over
generality. Nevertheless, our working assumptions include interesting (and natural) examples.

3.1

Convergence in probability

We start by establishing a convergence in probability.
We start by making the following assumptions. We assume there is u‡ such that, for u ≥ u‡ ,
Λ(u) ∶= − log ψ(u) is differentiable and L(u) ∶= uΛ′ (u) is increasing. In addition, we assume that
M (u) ∶= L(u)/ log(u) → ∞ as u → ∞ and
lim sup
u→∞

M ((1 − ε)u)
≤ 1,
M (u)

lim inf
u→∞

M ((1 + ε)u)
≥ 1,
M (u)

∀ε ∈ (0, 1).

(14)

Theorem 3. In the setting considered here, Ud /ud → 1 in probability as d → ∞, where ud ∶= L−1 (d).

Example 1. Consider the case where Λ(u) = c log(u + a)α (u + b)β , where a > 0, b ≥ 0, c > 0, α ∈ R
and β > 0. Surely, this defines a bonafide shape function ψ in the sense of (2). It can be shown
that ψ defined as such satisfies the conditions of Theorem 3, with
ud ∼ c−1/β β (α−1)/β (log d)−α/β d1/β ,

d → ∞.

(15)

Proof. The function u ↦ ud ψ(u) is increasing on [0, ud ) and decreasing on (ud , ∞). Indeed,
log(ud ψ(u)) = d log u − Λ(u) has derivative u1 (d − L(u)), which is positive for u < ud , zero at
u = ud , and negative at u > ud , by our assumptions and the definition of ud . Note that, necessarily,
ud → ∞ as d → ∞.
We have
v
(16)
P(Ud ≤ v) = cd ∫ ud ψ(u)du.
0

Fix ε ∈ (0, 1).
Left tail. Using the fact that ud ψ(u) ≤ ud0 ψ(u0 ) for any u ≤ u0 ≤ ud , we have
(1−ε)ud
1
P(Ud ≤ (1 − ε)ud ) = ∫
ud ψ(u)du
cd
0
≤ ((1 − ε)ud )d+1 ψ((1 − ε)ud ),

4

(17)
(18)

and we also have
∞
1
P(Ud ≥ (1 − ε)ud ) = ∫
ud ψ(u)du
cd
(1−ε)ud

≥∫

ud

(1−ε/2)ud

ud ψ(u)du

≥ (ε/2)ud ((1 − ε/2)ud )d ψ((1 − ε/2)ud ).

Taking the ratio, we obtain

((1 − ε)ud )d+1 ψ((1 − ε)ud )
P(Ud ≤ (1 − ε)ud )
≤
P(Ud ≥ (1 − ε)ud ) (ε/2)ud ((1 − ε/2)ud )d ψ((1 − ε/2)ud )
1−ε 1−ε
ψ((1 − ε)ud )
(
)
.
ε/2 1 − ε/2 ψ((1 − ε/2)ud )

(19)
(20)
(21)

(22)

d

≤

(23)

Applying the logarithm, and ignoring the constant factor, we further get
1−ε
) − Λ((1 − ε)ud ) + Λ((1 − ε/2)ud )
1 − ε/2
(1−ε/2)ud
(1−ε/2)ud 1
du + ∫
Λ′ (u)du
= −d ∫
u
(1−ε)ud
(1−ε)ud
(1−ε/2)ud
1
= −∫
(d − L(u)) du
(1−ε)ud
u
1 − ε/2
),
≤ −(d − L((1 − ε/2)ud )) log (
1−ε

d log (

(24)
(25)
(26)
(27)

where we used the monotonicity of L in the last line. Therefore, to show that the fraction in (22)
converges to 0, it suffices to show that d − L((1 − ε/2)ud ) → ∞. The limit is as d → ∞ while ε
remains fixed. Using the fact that L(ud ) = M (ud ) log ud = d, we have
d − L((1 − ε/2)ud ) = d − M ((1 − ε/2)ud ) log((1 − ε/2)ud )

d
= d − M ((1 − ε/2)ud ) log(1 − ε/2) − M ((1 − ε/2)ud )
M (ud )
M ((1 − ε/2)ud )
= −M ((1 − ε/2)ud ) log(1 − ε/2) + d [1 −
].
M (ud )

(28)

(29)
(30)

In the last line, the first term tends to infinity because M (u) → ∞ as u → ∞ and ud → ∞, while
the second terms is nonnegative in the limit because of (14), so that the last expression tends to
infinity.
We conclude that, for the left tail,
P(Ud ≤ (1 − ε)ud ) → 0,

d → ∞.

(31)

Right tail. Using the fact that uℓ ψ(u) ≤ uℓ0 ψ(u0 ) for any u ≥ u0 ≥ uℓ , where uℓ = L−1 (ℓ) in
5

congruence with our definition above, and assuming for now that b ∶= L((1 + ε)ud ) > d + 1, we have
∞
1
ud ψ(u)du
P(Ud ≥ (1 + ε)ud ) = ∫
(1+ε)ud
cd

(32)

≤ ((1 + ε)ud )b ψ((1 + ε)ud ) ∫
= ((1 + ε)ud )b ψ((1 + ε)ud )
=

and we also have

∞

ud−b du

(33)

((1 + ε)ud )d+1−b
b−d−1

(34)

(1+ε)ud

((1 + ε)ud )d+1
ψ((1 + ε)ud ),
b−d−1

(1+ε)ud
1
P(Ud ≤ (1 + ε)ud ) = ∫
ud ψ(u)du
cd
0

≥∫

(1+ε/2)ud

ud

(36)

ud ψ(u)du

(37)

≥ (ε/2)ud ((1 + ε/2)ud )d ψ((1 + ε/2)ud ).

Taking the ratio, we obtain

(35)

d
ψ((1 + ε)ud )
P(Ud ≥ (1 + ε)ud )
b−d−1
≤
P(Ud ≤ (1 + ε)ud ) (ε/2)ud ((1 + ε/2)ud )d ψ((1 + ε/2)ud )

((1+ε)u )d+1

ψ((1 + ε)ud )
1+ε
1+ε
1
(
)
.
ε/2 b − d − 1 1 + ε/2 ψ((1 + ε/2)ud )

(38)

(39)

d

≤

(40)

We pause to show that b − d → ∞ eventually. This is because, using the fact that L(ud ) =
M (ud ) log ud = d,
b − d = L((1 + ε)ud ) − d = M ((1 + ε)ud ) log((1 + ε)ud ) − d
= M ((1 + ε)ud ) log(1 + ε) + [

M ((1 + ε)ud )
− 1] d.
M (ud )

(41)

(42)

In the last line, the first term tends to infinity because M (u) → ∞ as u → ∞ and ud → ∞, while
the second terms is nonnegative in the limit because of (14), so that the last expression tends to
infinity.
Returning to (39), applying the logarithm, and ignoring the first two factors whose product is
bounded by 1 eventually, we further get
1+ε
) − Λ((1 + ε)ud ) + Λ((1 + ε/2)ud )
1 + ε/2
(1+ε)ud 1
(1+ε)ud
= d∫
du − ∫
Λ′ (u)du
(1+ε/2)ud u
(1+ε/2)ud
(1+ε)ud
1
= −∫
(L(u) − d) du
u
(1+ε/2)ud
1+ε
≤ −(L((1 + ε/2)ud ) − d) log (
),
1 + ε/2

d log (

6

(43)
(44)
(45)
(46)

where we used the monotonicity of L in the last line. Therefore, to show that the fraction in (39)
converges to 0, it suffices to show that L((1 + ε/2)ud ) − d → ∞, and we already did this in (41).
We conclude that, for the right tail,
P(Ud ≥ (1 + ε)ud ) → 0,

d → ∞.

(47)

We can therefore conclude that Ud /ud → 1 in probability as d → ∞.

3.2

Convergence in distribution

We now turn to establishing a convergence in distribution. Although we speculate that other cases
may arise, we give (additional) sufficient conditions for a Gaussian limit.
We make the following additional assumptions. We assume that L is differentiable with νd ∶=
ud L′ (ud ) → ∞ and that there is ωd → ∞ such that
∣L(ud ) − L((1 − ε)ud ) − εud L′ (ud )∣ ≤ ∣ε∣νd /ωd ,

whenever ε2 ≤ ωd /νd .

(48)

Note that (48) is a form of first-order Taylor expansion around ud .
The following refines Theorem 3.
Proposition 1. In the setting considered here,
P(1 − εd ≤ U /ud ≤ 1 + εd ) → 1,

√
whenever εd ≫ 1/ νd .

(49)

Proof. By mononicity in εd > 0, it is enough to show that when ε2d ≤ ωd /νd . Then to prove (49), as
before, it suffices to show that
(left tail)

P(Ud ≤ (1 − εd )ud )
→0
P(Ud ≥ (1 − εd )ud )

and (right tail)

P(Ud ≥ (1 + εd )ud )
→ 0.
P(Ud ≤ (1 + εd )ud )

(50)

Left tail. As before, we can show that b ∶= L((1− εd )ud ) satisfies b− d → −∞, so that we may assume
that b < d − 1. Then using the fact that uℓ ψ(u) ≤ uℓ0 ψ(u0 ) for any u ≤ u0 ≤ uℓ , where uℓ = L−1 (ℓ),
we have
(1−εd )ud
1
P(Ud ≤ (1 − εd )ud ) = ∫
ud ψ(u)du
cd
0

(51)

≤ ((1 − εd )ud )b ψ((1 − εd )) ∫

=

(1−εd )ud
0

ud−b du

((1 − εd )ud )d+1
ψ((1 − εd )ud ).
d−b+1

(52)
(53)

Taking the ratio of (53) to (21) (but replacing ε by εd ), we obtain

P(Ud ≤ (1 − εd )ud )
1
1 − εd 1 − εd
ψ((1 − εd )ud )
≤
(
)
.
P(Ud ≥ (1 − εd )ud ) d − b + 1 εd /2 1 − εd /2 ψ((1 − εd /2)ud )
d

(54)

As in (23) and (27), we apply a logarithm, and obtain the upper bound
log

1 − εd
εd
2 (d − b + 1)

− [d − L((1 − εd /2)ud )] log
7

1 − εd /2
.
1 − εd

(55)

By (48), which is applicable by our assumption ε2d ≤ ωd /νd , we have
Similarly,

d − b = L(ud ) − L((1 − εd )ud ) = εd νd ± ∣εd ∣νd /ωd ∼ εd νd ,
d − L((1 − εd /2)ud ) ∼ 21 εd νd ,

d → ∞.

d → ∞.

(56)
(57)

Also, note that ε2d νd → ∞ by assumption. Hence, the first term in (55) is ∼ − log(ε2d νd ) while the
second term (including sign) is ∼ − 14 ε2d νd , so that the sum tends to −∞.
Right tail. The treatment of the right tail is analogous, starting with (40) instead of (23). Details
are omitted.
In the following we examine the behavior of the normalizing constant cd as d → ∞.
Proposition 2. In the setting considered here,
√
1
2π d+1
∼
u ψ(ud ),
cd
νd d

d → ∞.

(58)

Proof. Let εd be such that 1/νd ≪ ε2d ≪ ωd /νd . Applying Proposition 1 and then performing a
change of variables, we get
(1+εd )ud
∞
1
ud ψ(u)du ∼ ∫
ud ψ(u)du
=∫
0
(1−εd )ud
cd

∼∫

εd

−εd

[(1 + t)ud ]d ψ[(1 + t)ud ]ud dt

= udd+1 ψ(ud ) ∫

As before,
log {(1 + t)d

(59)

εd

−εd

(1 + t)d

ψ[(1 + t)ud ]
dt.
ψ(ud )

ψ[(1 + t)ud ]
} = d log(1 + t) − Λ[(1 + t)ud ] + Λ(ud )
ψ(ud )
t s
t
= d∫
ds − ∫ ud Λ′ [(1 + s)ud ]ds
0 1+s
0
t 1
{d − L[(1 + s)ud ]}ds.
=∫
0 1+s

Noting that ∣s∣ ≤ εd , and using (48), we get
Hence,

with

− sνd − sνd /ωd ≤ d − L[(1 + s)ud ] = L(ud ) − L[(1 + s)ud ] ≤ −sνd + sνd /ωd .

−νd (1 + 1/ωd ) ∫

t
0

t 1
t s
s
ds ≤ ∫
{d − L[(1 + s)ud ]}ds ≤ −νd (1 − 1/ωd ) ∫
ds,
1+s
0 1+s
0 1+s

∫

0

since ∣t∣ ≤ εd , so that

∫

0

t

t

s
ds = 12 t2 + O(t3 ) = 12 t2 + O(ε3d ),
1+s

1
{d − L[(1 + s)ud ]}ds = − 21 t2 νd + O(εd + 1/ωd )ε2d νd ,
1+s
8

(60)
(61)

(62)
(63)
(64)

(65)

(66)

(67)

(68)

where the big-O is uniform in t ∈ [−εd , εd ]. We already took εd such that (1/ωd )ε2d νd → 0, and it
is compatible to choose εd such that, in addition, ε3d νd → 0. When we do so, the remainder term
above is o(1), and in particular,
∫

0

t

1
{d − L[(1 + s)ud ]}ds = − 12 t2 νd + o(1),
1+s

(69)

where the o(1) term is uniform in t ∈ [−εd , εd ]. With such a choice of εd , we continue our derivations
above
d
∫−ε (1 + t)
εd
d

εd
ψ[(1 + t)ud ]
dt = ∫
exp { − 21 t2 νd + o(1)}dt
ψ(ud )
−εd
√
ε d νd
1
1 √
∼ √ ∫ √ exp { − 21 s2 }ds ∼ √
2π,
νd −εd νd
νd

(70)
(71)

√
since εd νd → ∞ by assumption.

We are finally equipped to establish a convergence in distribution for Ud .
√
Theorem 4. In the setting considered here, νd (Ud /ud −1) converges weakly to the standard normal
distribution as d → ∞.
Example 1 (Continued). It can be checked that the same example of shape function ψ satisfies the
conditions assumed here, with uL′ (u) ∼ cβ 2 (log u)α uβ as u → ∞, so that
νd = ud L′ (ud ) ∼ βd,

d → ∞.

Proof. Fix r ∈ R and let εd be as before. As in (61), we get
√
√
P( νd (Ud /ud − 1) ≤ r) = P(Ud ≤ (1 + r/ νd )ud )

(72)

√
∼ P((1 − εd )ud ≤ Ud ≤ (1 + r/ νd )ud )
= cd udd+1 ψ(ud ) ∫

Again, as before,
∫

√
r/ νd

−εd

d ψ[(1 + t)ud ]

(1 + t)

ψ(ud )

dt ∼ ∫

√
r/ νd

−εd

√
r/ νd

−εd

(1 + t)d

exp { − 12 t2 νd }dt

ψ[(1 + t)ud ]
dt.
ψ(ud )

r
1
1 √
= √ ∫ √ exp { − 21 s2 }ds ∼ √
2πΦ(r),
νd −εd νd
νd

(73)
(74)
(75)

(76)
(77)

where Φ is the standard normal distribution function. We then combine this with Proposition 2.

4

Discussion

While there is relatively little related work, a detailed comparison with (Sherlock and Elton, 2012)
is in order.1 Sherlock and Elton focus on the non-compact case — corresponding to Section 3
here. They derive the same result as our Theorem 3 under different conditions. They require that
η(u) ∶= Λ(exp(u)) is twice differentiable with η ′′ (u) → ∞ as u → ∞, while our condition is a bit
1
Note that we only became aware of that work after we posted our paper on arxiv.org, which explains some of
the nontrivial overlap.

9

weaker than requiring that η is once differentiable with η ′ (u)/u → ∞ and increasing. Note that
their condition is equivalent to requiring that L is differentiable with uL′ (u) → ∞, a condition
that arises in Section 3.2. Sherlock and Elton do not establish weak convergence, however, but
they obtain other results. In particular, they establish concentration for a marginal of Xd and also
for the maximum of the marginals. In addition, they extend their results to the case of elliptical
distributions under conditions on the eigenvalues of the scaling matrix.
What are possible consequences for statistical modeling? Because of the weak convergence of the
sort established here, the behavior of Ud is asymptotically characterized solely by a few parameters
of the underlying distribution. For example, if the conditions of Theorem 2 are fulfilled, then
the distribution of Ud in the large-dimension limit (d → ∞) only depends on u∗ (irrelevant in
practice because scale is typically estimated) and the behavior of ψ near u∗ . In particular, whether
ψ(u) = I{u ≤ 1} or ψ(u) = (2 − u)I{u ≤ 1}, in both cases, d(1 − Ud ) converges weakly to the
exponential distribution with rate 1. This means that, in order to even distinguish two such
distributions with nontrivial accuracy, we require a sample of size that increases with d. (We
did not attempt to quantify this further, although this is possible by framing the problem as a
hypothesis testing problem.) A similar phenomenon arises with certain distributions with noncompact support, based on our Theorem 4. Thus, if the sample size is small relative to the
dimension, very different shape functions (meaning, different ψ’s) could yield indistinguishable
models.
The flip side of this is a form of universality of the Gaussian distribution, in particular, in context
such as Linear Discriminant Analysis (classification) or Gaussian Mixture Modeling (clustering).
Surely, both LDA and GMM have computational advantages over other methods (the latter using
the EM algorithm, for example). Beyond this important computational aspect, our results indicate
that when the sample is small relative to the dimension, fitting a Gaussian model may be, in fact,
indistinguishable from fitting another model base on a shape function having similar characteristics
as the standard normal distribution that dictate the asymptotic behavior of Ud .

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