MULTIDIMENSIONAL PROBABILITY INEQUALITIES VIA SPHERICAL SYMMETRY IOSIF PINELIS

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MULTIDIMENSIONAL PROBABILITY INEQUALITIES VIA

SPHERICAL SYMMETRY

IOSIF PINELIS

Abstract. Spherical symmetry arguments are used to produce a general de-

vice to convert identities and inequalities for the pth absolute moments of

real-valued random variables into the corresponding identities and inequalities

for the pth moments of the norms of random vectors in Hilbert spaces. Partic-

ular results include the following: (i) an expression of the pth moment of the

norm of such a random vector Xin terms of the characteristic functional of X;

(ii) an extension of a previously obtained von Bahr–Esseen-type inequality for

real-valued random variables with the best possible constant factor to random

vectors in Hilbert spaces, still with the best possible constant factor; (iii) an

extension of a previously obtained inequality between measures of “contrast

between populations” and “spread within populations” to random vectors in

Hilbert spaces.

1. Introduction

In this note, we will use averaging with respect to spherically symmetric measures

over Rdto extend certain probability identities and inequalities for real-valued

random variables (r.v.’s) to random vectors in Hilbert spaces.

Let µbe a ﬁnite spherically symmetric measure over Rd, so that µ(T B) =

µ(B) for all orthogonal transformations T:Rd→Rdand Borel sets B⊆Rd. Let

g:R→Rbe a Borel-measurable function such that RRdµ(dt)|g(t·x)|<∞for

all x∈Rd, where t·xdenotes the dot product of vectors tand xin Rd. Let

e1:= (1,0,...,0) ∈Rd. Let |·|denote the Euclidean norm on Rd, for all natural d.

Then, for any x∈Rd\ {0}and the unit vector y:= x/|x|,

Hg,µ(x) := ZRd

µ(dt)g(t·x)

=ZRd

µ(dt)g(|x|t·y)

=ZRd

µ(dt)g(|x|t·e1) =: hg,µ(|x|),

and also Hg,µ(0) = g(0) RRdµ(dt) = hg,µ(0), so that the equality

(1.1) Hg,µ(x) = hg,µ(|x|)

holds for all x∈Rd. It follows that the function Hg,µ is spherically symmetric.

2010 Mathematics Subject Classiﬁcation. 60E15; 60B11; 60E10; 60E05.

Key words and phrases. Spherical symmetry; probability identities and inequalities; von Bahr–

Esseen-type inequalities; random vectors in Hilbert spaces; pth moments of the norm.

arXiv:2210.04391v2 [math.PR] 12 Oct 2022

2 IOSIF PINELIS

In particular, for any real p > 0 and the function gp:R→Rdeﬁned by the

formula

gp(u) := |u|p

for real u, identity (1.1) can be rewritten as

|x|p=1

cp,µ ZRd

µ(dt)|t·x|p

for x∈Rd, assuming that

cp,µ := ZRd

µ(dt)|t·e1|p∈(0,∞);

the latter condition will hold if e.g. µis the uniform distribution on the unit sphere

Sd−1in Rd.

It follows immediately that, for any random vector Xin Rd,

(1.2) E|X|p=1

cp,µ ZRd

µ(dt)E|t·X|p.

Thus, the pth moment of the norm |X|of the random vector Xis expressed as a

mixture of the pth absolute moments of the real valued r.v.’s t·X.

Another kind of integral representation of E|X|p, for any random vector Xin

Rdwith E|X|p<∞, in terms of the distributions of the real-valued r.v.’s t·Xfor

t∈Rdwill be established in Theorem 2.1 in Section 2.

2. Moments of the norm of a random vector in Rdvia the

characteristic functional

Let Xbe a random vector in Rd, with the characteristic functional (c.f.) fX, so

that

fX(t) = Eeit·X

for t∈Rd.

Take any real p > 0 which is not an even integer, and let

m:= bp/2c.

For nonnegative integers kand real z, let

(2.1) ck(z) := cos z−

j=0

(−1)jz2j

(2j)!, sk(z) := sin z−

j=0

(−1)jz2j+1

(2j+ 1)!.

Noting that c0

k=−sk−1and s0

k=ck−1if k>1, and ck(0) = 0 = sk(0), it is

easy to check by induction that (−1)kck(z)60 and (−1)kzsk(z)60, again for all

nonnegative integers kand real z.

So, for any nonzero vector x∈Rdand the unit vector y:= x/|x|, by the Tonelli

theorem we have

ZRd

|t|p+dcm(t·x) = ZRd

|t|p+dcm(|x|t·y)

=|x|pZRd

|s|p+dcm(s·y)

=|x|pΩdZSd−1

dω Z∞

dr rd−1

rp+dcm(rω ·y),

MULTIDIMENSIONAL PROBABILITY INEQUALITIES VIA SPHERICAL SYMMETRY 3

where

(2.2) Ωd:= 2π(d+1)/2

Γ((d+ 1)/2)

is the surface area of the unit sphere Sd−1in Rdand RSd−1dω ··· is the integral with

respect to the uniform distribution on Sd−1. Using now the spherical symmetry of

the uniform distribution on Sd−1and recalling the deﬁnition e1:= (1,0,...,0) ∈

Sd−1, we see that for any nonzero vector x∈Rd

ZRd

|t|p+dcm(t·x) = Cd,p|x|p,

where

Cd,p := ΩdZSd−1

dω Z∞

rp+1 cm(rω ·e1)

= ΩdZSd−1

dω Z∞

rp+1 cm(r|ω·e1|)(2.3)

= ΩdKpZSd−1

dω |ω·e1|p

and

(2.4) Kp:= Z∞

zp+1 cm(z) = −π

2Γ(p+ 1) sin(πp/2);

the equality in (2.3) holds because the function cmis even, and the latter equality

in (2.4) is the special case of formula (5) in [10] obtained by replacing xthere by 1.

To evaluate the integral RSd−1dω |ω·e1|p, note that the distribution of ω·e1

coincides with that of G1

(G2

1+··· +G2

d)1/2,

where G1, . . . , Gdare independent standard normal random variables. Hence,

(ω·e1)2has the beta distribution with parameters 1/2,(d−1)/2. So,

ZSd−1

dω |ω·e1|p=Γ(d/2)

Γ(1/2)Γ((d−1)/2) Z1

dz zp/2−1/2(1 −z)(d−1)/2−1

=Γ(d/2)Γ((p+ 1)/2)

√πΓ((p+d)/2) .

It follows that

(2.5) Cd,p =−πd/2

sin(πp/2)

Γ(d/2)Γ((p+ 1)/2)

Γ((d+ 1)/2)Γ(p+ 1)Γ((p+d)/2)

and

(2.6) |x|p=1

Cd,p ZRd

|t|p+dcm(t·x)

for all nonzero vectors x∈Rd. Identity (2.6) also trivially holds if xis the zero

vector, because cm(0) = 0.

Thus, using the Tonelli theorem again, we immediately obtain the following

expression of the pth moment of the norm |X|of a random vector Xin terms of

the c.f. fXof X:

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MULTIDIMENSIONALPROBABILITYINEQUALITIESVIASPHERICALSYMMETRYIOSIFPINELISAbstract.Sphericalsymmetryargumentsareusedtoproduceageneralde-vicetoconvertidentitiesandinequalitiesforthepthabsolutemomentsofreal-valuedrandomvariablesintothecorrespondingidentitiesandinequalitiesforthepthmomentsofthenormsofrand...

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