DETECTING INDEPENDENCE OF RANDOM VECTORS I.
GENERALIZED DISTANCE COVARIANCE AND GAUSSIAN
COVARIANCE

arXiv:1711.07778v1 [math.PR] 21 Nov 2017

BJÖRN BÖTTCHER, MARTIN KELLER-RESSEL, AND RENÉ L. SCHILLING
Abstract. Distance covariance is a quantity to measure the dependence of
two random vectors. We show that the original concept introduced and developed by Székely, Rizzo and Bakirov can be embedded into a more general
framework based on symmetric Lévy measures and the corresponding realvalued continuous negative definite functions. The Lévy measures replace the
weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework
and some proofs are streamlined.
Form a practical point of view this allows less restrictive moment conditions
on the underlying random variables and one can use other distance functions
than the Euclidean distance, e.g. the Minkowski distance. Most importantly,
it serves as the basic building block for distance multivariance, a quantity
to measure and estimate dependence of multiple random vectors, which is
introduced in the companion paper [BKRS17] to the present article.

Contents
1. Introduction
2. Auxiliary Results
2.1. Negative definite functions
2.2. Measuring independence with metrics
2.3. An elementary estimate for log-moments
3. Generalized distance covariance
3.1. Generalized distance covariance with finite Lévy measures
3.2. Generalized distance covariance with arbitrary Lévy measures
4. Estimating generalized distance covariance
5. Remarks on the uniqueness of the Cauchy distance covariance
6. Generalized distance correlation
7. Gaussian covariance
References

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3
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1. Introduction
The concept of distance covariance was introduced by Székely, Rizzo and Bakirov
[SRB07] as a measure of dependence between two random vectors of arbitrary dimensions. Their starting point is to consider a weighted L2 -integral of the difference
2010 Mathematics Subject Classification. Primary: 62H20; Secondary: 60E10; 62G20; 60G15.
Key words and phrases. Dependence measure; stochastic independence; negative definite function; characteristic function; distance variance; Gaussian random field.
We are grateful to our colleague, Ulrich Brehm, for insightful discussions on (elementary) symmetric polynomials. MKR acknowledges funding from the German Research Foundation (DFG)
under grant ZUK 64 and KE 1736/1-1.
1

2

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

of the (joint) characteristic functions fX , fY and f(X,Y ) of the (Rm - and Rn -valued)
random variables X, Y and (X, Y ),
ZZ
|f(X,Y ) (s, t) − fX (s)fY (t)|2 w(s, t) ds dt.
(1.1)
V 2 (X, Y ; w) =
Rm+n

The weight function w is given by w(s, t) := cα,m |s|−m−α cα,n |t|−n−α , α ∈ (0, 2).
We are going to embed this into a more general framework. In order to illustrate
the new features of our approach we need to recall some results on distance covariance. Among several other interesting properties, [SRB07] show that distance
covariance characterizes independence, in the sense that V 2 (X, Y ; w) = 0 if and
only if X and Y are independent. Moreover, they show that in the case α = 1
the distance covariance NV 2 (X, Y ; w) of the empirical distributions of two samples
(x1 , x2 , . . . , xN ) and (y1 , y2 , . . . , yN ) takes a surprisingly simple form. It can be
represented as
N
1 X
N
2
V (X, Y ; w) = 2
Akl Bkl
(1.2)
N
k,l=1

where A and B are double centrings (cf. Lemma 4.2) of the Euclidian distance
matrices akl = |xk − xl | and bkl = |yk − yl | of the sample points. If α 6= 1 then
Euclidian distance has to be replaced by its power with exponent α. The connection
between the weight function w in (1.1) and the (centred) Euclidian distance matrices
in (1.2) is given by Schoenberg’s representation of negative definite functions, i.e.
Z
ds
α
|x| = cp
(1 − cos s · x) m+α , x ∈ Rm .
|s|
m
R \{0}
where cp is a suitable constant, cf. Section 2.1. Finally, the representation (1.2) of
N
V 2 (X, Y ; w) and its asymptotic properties as N → ∞ are used by Székely, Rizzo
and Bakirov to develop a statistical test for independence in [SRB07].
Yet another interesting representation of distance covariance is given in the
follow-up paper [SR09a]: Let (Xcop , Ycop ) be an independent copy of (X, Y ) and
let W and W ′ be Brownian random fields on Rm and Rn , independent from each
other and from X, Y, Xcop, Ycop . [SR09a] define the Brownian covariance
i
h
′
W
W′
,
(1.3)
W 2 (X, Y ) = E X W Xcop
Y W Ycop

where X W := W (X) − E [W (X) | W ] for any random variable X and random
field W with matching dimensions. Surprisingly, as shown in [SR09a], Brownian
covariance coincides with distance covariance, i.e. W 2 (X, Y ) = V 2 (X, Y ; w) when
α = 1 is chosen for the kernel w.
The paper [SR09a] was accompanied by a series of discussion papers [New09,
BX09, Kos09, Cop09, Feu09, GFS09, Rém09, Gen09, SR09b] where various extensions, applications and open questions were suggested. Let us highlight the four
problems which we are going to address in the present paper and the companion
paper [BKRS17].
(1) Can the weight function w in (1.1) be replaced by other weight functions?
(Cf. [Feu09, GFS09])
(2) Can the Euclidian distance (or its α-power) in (1.2) be replaced by other
distances? (Cf. [Kos09, GFS09, Lyo13])
(3) Can the Brownian random fields W, W ′ in (1.3) be replaced by other random
fields? (Cf. [Kos09, Rém09])
(4) How can distance covariance be generalized from pairs (X, Y ) to d-tuples
X 1 , X 2 , . . . , X d of random variables? (Cf. [Feu09, BS11])

DETECTING INDEPENDENCE I. GENERALIZED DISTANCE COVARIANCE

3

While insights and partial results on these questions can be found in all of the
mentioned discussion paper, a definitive and unifying answer has so far been missing. In the present paper we propose a generalization of distance covariance which
resolves the closely related questions (1)–(3). In the companion paper [BKRS17]
we then turn to question (4) and introduce ‘distance multivariance’, which extends
distance covariance to multiple random variables.
More precisely, we introduce in Definition 3.1 the generalized distance covariance
Z Z
2
V (X, Y ) =
|f(X,Y ) (s, t) − fX (s)fY (t)|2 µ(ds) ν(dt)
Rn

Rm

where µ and ν are symmetric Lévy measures, as a natural extension of distance
covariance of Székely et al. The Lévy measures µ and ν are linked to negative
definite functions Φ and Ψ by the well-known Lévy–Khintchine representation, cf.
Section 2 where examples and important properties of negative definite functions
and other auxiliary results are discussed. In Section 3 we show that several different
representations (close to [Lyo13]) of V 2 (X, Y ) in terms of the functions Φ and Ψ
can be given. In Section 4 we turn to the finite-sample properties of generalized
distance covariance and show that the representation (1.2) of NV 2 (X, Y ) remains
valid, with the Euclidian distance matrices a and b replaced by
akl = Φ(xk − xl ),

2

bkl = Ψ(yk − yl ).

We also show asymptotic properties of V (X, Y ) as N tends to infinity, paralleling
those of [SR09a, SR12] for Euclidian distance covariance. After some remarks on
uniqueness and normalization, we show in Section 7 that also the representation
(1.3) remains valid, when the Brownian random fields W and W ′ are by centered
Gaussian random fields GΦ and GΨ with covariance kernel
N

E [GΦ (x)GΦ (x′ )] = Φ(x) + Φ(x′ ) − Φ(x − x′ )
and analogously for GΨ .
To use generalized distance covariance (and distance multivariance) in applications all necessary functions are provided in the R package multivariance [Böt17].
Notation. Most of our notation is standard or self-explanatory. Throughout
we use positive (and negative) in the non-strict sense, i.e. x ≥ 0 (resp. x ≤ 0) and
we write a ∨ b = max{a, b} and a ∧ b = min{a, b} for the maximum and minimum.
2. Auxiliary Results
In this section we collect some tools and concepts which will be needed in the
sequel.
2.1. Negative definite functions. A function Θ : Rd → C is called negative
definite (in the sense of Schoenberg)if for every m ∈ N and x1 , . . . , xm ∈ Rd the
∈ Cm×m is positive hermitian. It is not
matrix Θ(xi ) + Θ(xj ) − Θ(xi − xj )
i,j

hard to see, cf. Berg & Forst [BF75] or Jacob [Jac01], that this is equivalent to
saying that Θ(0) ≥ 0, Θ(−x) = Θ(x) and the matrix (−Θ(xi − xj ))i,j ∈ Cm×m is
conditionally positive definite, i.e.
m
X

i,j=1

[−Θ(xi − xj )]λk λ̄l ≥ 0 ∀λ1 , . . . , λm ∈ C such that

m
X

λk = 0.

k=1

Because of this equivalence, the function −Θ is also called conditionally positive
definite (and some authors call Θ conditionally negative definite). In general, Θ
being negative definite does not mean that −Θ is positive definite.

4

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

Negative definite functions appear naturally in several contexts, for instance
in probability theory as characteristic exponents (i.e. logarithms of characteristic
functions) of infinitely divisible laws or Lévy processes, cf. Sato [Sat99] or [BSW13],
in harmonic analysis in connection with non-local operators, cf. Berg & Forst [BF75]
or Jacob [Jac01] and in geometry when it comes to characterize certain metrics in
Euclidean spaces, cf. Benyamini & Lindenstrauss [BL00].
The following theorem, compiled from Berg & Forst [BF75, Sec. 7, pp. 39–48,
Thm. 10.8, p. 75] and Jacob [Jac01, Sec. 3.6–7, pp. 120–155], summarizes some
basic equivalences and connections.
Theorem 2.1. For a function Θ : Rd → C with Θ(0) = 0 the following assertions
are equivalent
a)
b)
c)
d)

Θ is negative definite.
−Θ is conditionally positive definite.
e−tΘ is positive definite for every t > 0.
t−1 (1 − e−tΘ ) is negative definite for every t > 0.

If Θ is continuous, the assertions a)–d) are also equivalent to
e) Θ has the following integral representation
Z

1
1 − eix·r + ix · r1(0,1) (|r|) ρ(dr)
Θ(x) = il · x + x · Qx +
2
Rd \{0}

(2.1)

where l ∈ Rd , Q ∈ Rd×d is symmetric
and positive
semidefinite and ρ is a
R

measure on Rd \ {0} such that Rd \{0} 1 ∧ |s|2 ρ(ds) < ∞.

We will frequently use the abbreviation cndf instead of continuous negative definite function. The representation (2.1) is the Lévy–Khintchine formula and any
measure ρ satisfying
Z

d
(2.2)
1 ∧ |r|2 ρ(dr) < ∞
ρ is a measure on R \ {0} such that
Rd \{0}

is commonly
called RLévy measure. To keep notation simple,R we will from now on
R
write · · · ρ(dr) or Rd · · · ρ(dr) instead of the more precise Rd \{0} · · · ρ(dr).
The triplet (l, Q, ρ) uniquely determines Θ; moreover Θ is real (hence, positive)
if, and only if, l = 0 and ρ is symmetric, i.e. ρ(B) = ρ(−B) for any Borel set
B ⊂ Rd \ {0}. In this case (2.1) becomes
Z
1
Θ(x) = x · Qx +
(1 − cos x · r) ρ(dr).
(2.3)
2
Rd

Using the representation (2.1) it is straightforward to see that supx |Θ(x)| < ∞
if ρ is a finite measure, ρ(Rd \ {0}) < ∞, and Q = 0. The converse is also true, see
[ScS10, pp. 1390–1391, Lem. 6.2].
Table 1 contains some examples of continuous negative definite functions along
with the corresponding Lévy measures and infinitely divisible laws.
Lemma 2.2. Let p ∈ [1, 2]. The Minkowski distance function
1/p

ℓp (x) := (|x1 |p + · · · + |xd |p )

,

x = (x1 , . . . , xd ) ∈ Rd

is a continuous negative definite function on Rd . If p ∈ (1, 2), the Lévy measure
has full support.
It is interesting to note that the Minkowski distances for p > 2 and d ≥ 2 are
never negative definite functions. This is the consequence of Schoenberg’s problem,
cf. Zastavnyi [Zas00, p. 56, Eq. (3)].

DETECTING INDEPENDENCE I. GENERALIZED DISTANCE COVARIANCE

5

Proof of Lemma 2.2. Since each xi 7→ |xi |p , p ∈ [1, 2], is a one-dimensional continuous negative definite function, we can use the formula (2.3) to see that
Z
d
X

cp dxi

(1
−
cos
x
·
r)
⊗ δ0 (dx(i) ), if p ∈ [1, 2),
p
ℓp (x) =
|xi |1+p
Rd
i=1


x · x,
if p = 2,
where x(i) = (x1 , . . . , xi−1 , xi+1 , . . . xd ) ∈ Rd−1 and cp =

p2p−1 Γ( p+1
2 )
π 1/2 Γ(1− p
2)

is the con-

stant of the one-dimensional p-stable Lévy measure, cf. Table 1.
This means that ℓpp is itself a continuous negative definite function, but its Lévy
measure is concentrated on the coordinate axes. Writing ℓp (x) = fp (ℓpp (x)) with
Z ∞
1
1 1 
dt
1/p
,

∈ 2 , 1 , γ1/p =
fp (τ ) = τ
= γ1/p
(1 − e−τ t ) 1+1/p ,
p
t
0
pΓ 1 − p1

shows that ℓp can be represented as a combination of the Bernstein function fp
and the negative definite function ℓpp . In other words, ℓp is subordinate to ℓpp in the
sense of Bochner (cf. Sato [Sat99, Chap. 30] or [SSV12, Chap. 5, Chap. 13.1]) and
it is possible to find the corresponding Lévy–Khintchine representation, cf. [Sat99,

cndf Θ(x)
R
(1 − cos x · r) ρ(dr)
2

|x|
λ2 +|x|2 ,
2

|x|
λ2 +|x|2 ,
1

λ > 0, x ∈ R

λ > 0, x ∈ Rd

1 − e− 2 |x|

2

IDD

ρ finite measure

Compound Poisson (CP)

2λ−1 e−λ|r| dr

CP with exponential (1-d)

R∞
0

e

|r|2
λ2 u− 4u

λ2 du
(4πu)d/2
1

dr

2

(2π)−d/2 e− 2 |r| dr


1+d
Γ
2

|x|

π

1
2
2 |x|

1+d
2

dr
|r|d+1

no Lévy measure

|x|α , α ∈ (0, 2)
qP
d
p
p
k=1 |xk | , p ∈ [1, 2]
ln(1 +

Lévy measure ρ(dr)

x2
2 ),

x∈R

ln cosh(x), x ∈ R
|x|α + |x|β , α, β ∈ (0, 2)
1
β



α+d
α2α−1 Γ
2
α
π d/2 Γ(1− 2 )

dr
|r|d+α

see Lemma 2.2
1 −√ 1 |r|
2
e
dr
|r|
dr
2r sinh(πr/2)

CP, cf. [ScS10, Lem. 6.1]
CP with Normal
Cauchy
Normal
α-stable
see Lemma 2.2
Variance Gamma (1-d)
Meixner (1-d)
Mixture of stable

(1 + |x|α ) − 1,
Relativistic stable
α ∈ (0, 2), β ≥ α2
Table 1. Some real-valued continuous negative definite functions
(cndfs) on Rd and the corresponding Lévy measures and infinitely
divisible distributions (IDD).

6

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

Thm. 30.1]. We have
Z
d
X

cp dxi


⊗ δ0 (dx(i) ),
if p = 1,
(1
−
cos
x
·
r)


|xi |2
 Rd
Zi=1∞
Z
ℓp (x) =
dt

gt (x1 ) · gt (x2 ) · . . . · gt (xd ) 1+1/p , if p ∈ (1, 2),
(1 − cos x · r)


d

t
0

√R
x · x,
if p = 2,

where xi 7→ gt (xi ) is the probability density of the random variable tp X where X
is a one-dimensional, symmetric 1/p-stable random variable.
Although the 1/p-stable density is only for the cases 1/p ∈ {1, 2} explicitly
known, one can show – this follows, e.g. from [Sat99, Thm. 15.10] – that it is
strictly positive, i.e. the Lévy measure of ℓp , p ∈ (1, 2) has full support.

Using the Lévy–Khintchine representation (2.3) it is not hard to see, cf. [Jac01,
Lem. 3.6.21], that square-roots of real-valued cndfs are subadditive, i.e.
p
p
p
Θ(x + y) ≤ Θ(x) + Θ(y), x, y ∈ Rd
(2.4)
and, consequently,

Θ(x + y) ≤ 2 (Θ(x) + Θ(y)) ,

x, y ∈ Rd .

(2.5)

Using a standard argument, e.g. [BSW13, p. 44], we can derive from (2.4), (2.5)
that cndfs grow at most quadratically as x → ∞,
Θ(x) ≤ 2 sup Θ(y)(1 + |x|2 ).

(2.6)

|y|≤1

We will assume that Θ(0) = 0 is the only zero of the function Θ – incidentally,
this means that x 7→ e−Θ(x) is the characteristic function of a(n infinitely divisible) random
distribution. This and (2.4) show that
p variable which has no lattice
d
(x, y) 7→ Θ(x − y) is a metric on R and (x, y) 7→ Θ(x − y) is a quasi-metric,
i.e. a function which enjoys all properties of a metric, but the triangle inequality
holds with a multiplicative constant c > 1. Metric measure spaces of this type have
been investigated by Jacob et al. [JKLS12]. Historically, the notion of negative definiteness has been introduced by I.J. Schoenberg p
[Sch38] in a geometric context: he
observed that for a real-valued cndf dΘ (x, y) := Θ(x − y) is a metric on Rd and
that these are the only metrics such that (Rd , dΘ ) can be isometrically embedded
into Hilbert space. In other words: dΘ behaves like a standard Euclidean metric in
a possibly infinite-dimensional space.
2.2. Measuring independence with metrics. Let X, Y be a random variables
with values in Rm and Rn , respectively, and write L(X) and L(Y ) for the corresponding probability laws. For any metric d(·, ·) defined on the family of (m + n)dimensional probability distributions we have
d (L(X, Y ), L(X) ⊗ L(Y )) = 0 if, and only if, X, Y are independent.

(2.7)

This equivalence can obviously be extended to finitely many random variables
Xi , i = 1, . . . , n, taking values in Rdi , respectively: Set d := d1 + · · · + dn ,
take any metric dNon the d-dimensional probability distributions and consider
n
d (L(X1 , . . . , Xn ), i=1 L(Xi )). Moreover, the random variables Xi , i = 1, . . . , n,
are independent if, and only if, (X1 , . . . , Xk−1 ) and Xk are independent for all
2 ≤ k ≤ n.1 In other words: X1 , . . . , Xn are independent if, and only if, for a
metric on the d1 + · · · + dk -dimensional probability distributions the distance of
1This is an immediate consequence of the characterization of independence using characteristic
functions: X, Y are independent if, and only if, Eeiξ·X+iη·Y = Eeiξ·X Eeiη·Y for all ξ, η.

DETECTING INDEPENDENCE I. GENERALIZED DISTANCE COVARIANCE

7

L(X1 , . . . , Xk ) and L(X1 , . . . , Xk−1 ) ⊗ L(Xk ) is zero. As in (2.7) only the concept of independence of pairs or random variables is needed. In the companion
paper [BKRS17, Sec. 3.1] we use a variant of this idea to characterize multivariate
independence.
Although (2.7) looks like a good starting point for the construction of (new)
estimators for independence, it turns out that the empirical evaluation for commonly used distances (e.g. χ2 , relative entropy, Hellinger distance, total variation,
Prohorov distance, Wasserstein distance) is computationally not feasible. For this
reason we use a different distance function.
Definition 2.3. Let U, V be d-dimensional random variables and denote by fU (x),
fV (y) their characteristic functions. For any symmetric measure ρ on Rd \ {0} with
full topological support we define the distance
Z
1/2
dρ (L(U ), L(V )) := kfU − fV kL2 (ρ) =
|fU (r) − fV (r)|2 ρ(dr)
.
(2.8)
The assumption that ρ has full topological support, i.e. ρ(G) > 0 for every
nonempty open set G ⊂ Rd \ {0}, ensures that dρ (L(U ), L(V )) = 0 if, and only
if, L(U ) = L(V ), hence dρ (L(U ), L(V )) is a metric. The symmetry assumption on
ρ is not essential since the integrand appearing in (2.8) is even; therefore, we can
always replace ρ(dr) by its symmetrization 21 (ρ(dr) + ρ(−dr)).
Note that dρ (L(U ), L(V )) is always well-defined in [0, ∞]. The following conditions ensure that dρ (L(U ), L(V )) is finite:
a) ρ is a finite measure;
b) ρ is a symmetric Lévy measure (cf. (2.2)) and E|U | + E|V | < ∞.
Indeed, dρ (L(U ), L(V )) < ∞ follows from the elementary estimates

|fU (r) − fV (r)| ≤ E|eir·U − eir·V | ≤ E|r · (U − V )| ≤ |r| · E|U − V |
and |fU (r) − fV (r)| ≤ 2, and the integrability properties (2.2) of the Lévy
measure ρ.
We obtain further sufficient conditions for V (X, Y ) < ∞ in terms of moments
of the real-valued cndf Θ, see (2.3), whose Lévy measure is ρ and with Q = 0.
d
Proposition 2.4. Let ρ be a symmetric
with full topoR Lévy measure on R \ {0}
logical support and denote by Θ(u) = (1 − cos u · r) ρ(dr), u ∈ Rd , the real-valued
cndf with Lévy triplet (l = 0, Q = 0, ρ). For all d-dimensional random variables
U, V the following assertions hold

a) Assume that (U ′ , V ′ ) is an iid copy of (U, V ). Then
EΘ(U − U ′ ) + EΘ(V − V ′ ) ≤ 2EΘ(U − V ′ ).

(2.9)

b) Let U ′ be an iid copy of U . Then
EΘ(U − V ) ≤ 2 (EΘ(U ) + EΘ(V )) ;

(2.10)

in particular, for V = U ′ one has EΘ(U − U ′ ) ≤ 4EΘ(U ).
c) In [0, ∞] we always have
d2ρ (L(U ), L(V )) ≤ 8 (EΘ(U ) + EΘ(V )) .

(2.11)

d) Let (U ′ , V ′ ) be an iid copy of (U, V ) and assume that EΘ(U ) + EΘ(V ) < ∞.
Then the following equality holds and all terms are finite
d2ρ (L(U ), L(V )) = 2EΘ(U − V ′ ) − EΘ(U − U ′ ) − EΘ(V − V ′ ).

(2.12)

8

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

Proof. Let us assume first that ρ is a finite Lévy measure, thus, Θ a bounded cndf.
We denote by U ′ an iid copy of U . Since L(U − U ′ ) is symmetric, we can use
Tonelli’s theorem to get
Z 
Z 


′
1 − fU (r)fU (r) ρ(dr) =
1 − E(eiU·r e−iU ·r ) ρ(dr)
Z
= E (1 − cos [(U − U ′ ) · r]) ρ(dr)
(2.13)
′
= EΘ(U − U )
Z
= Θ(u − u′ ) PU ⊗ PU (du, du′ ).

Now we consider an iid copy (U ′ , V ′ ) of (U, V ) and use the above equality in
(2.8). This yields

d2ρ (L(U ), L(V ))
Z 

fU (r)fU (r) − fU (r)fV (r) − fV (r)fU (r) + fV (r)fV (r) ρ(dr)
=
Z 
(2.14)



fU (r)fU (r) − 1 + 2 − 2 Re fU (r)fV (r) + fV (r)fV (r) − 1 ρ(dr)
=
= 2EΘ(U − V ′ ) − EΘ(U − U ′ ) − EΘ(V − V ′ ).

This proves (2.12) and, since dρ (L(U ), L(V )) ≥ 0, also (2.9). Combining Part b)
and (2.12) yields (2.11), while (2.10) immediately follows from the subadditivity of
a cndf (2.5).
If ρ is an arbitrary Lévy measure, the truncation ρǫ (dr) := 1(ǫ,∞) (|r|) ρ(dr) is a
finite measure and the corresponding cndf Θǫ is bounded. In particular, we have
a)–d) for ρǫ and Θǫ . Using monotone convergence we get
Z
Θ(u) = sup Θǫ (u) = sup
(1 − cos u · r) ρ(dr).
ǫ>0

ǫ>0

|r|>ǫ

Again by monotone convergence, we see that the assertions a)–c) remain valid
for general Lévy measures – if we allow them to attain values in [0, ∞]. Because of
(2.10), the moment condition assumed in Part d) ensures that the limits
lim EΘ(U − V ′ ) = sup EΘ(U − V ′ ) etc.

ǫ→0

ǫ>0

are finite, and (2.12) carries over to the general situation.
Remark 2.5.



a) Since U and V play symmetric roles in (2.14) it is clear that

d2ρ (L(U ), L(V ))

= EΘ(U − V ′ ) + EΘ(V − U ′ ) − EΘ(U − U ′ ) − EΘ(V − V ′ )
Z
= Θ(u − u′ ) (2PU ⊗ PV − PU ⊗ PU − PV ⊗ PV )(du, du′ ).

(2.15)

b) While dρ (L(U ), L(V )) ∈ [0, ∞] is always defined, the right-hand side of (2.12)
needs attention. If we do not assume the moment condition EΘ(U ) + EΘ(V ) < ∞,
we still have
d2ρ (L(U ), L(V )) = lim (2EΘǫ (U − V ′ ) − Θǫ (U − U ′ ) − Θǫ (V − V ′ ))
ǫ→0

(2.16)

but it is not clear whether the limits exist separately.
The moment condition EΘ(U ) + EΘ(V ) < ∞ is sharp in the sense that it
follows from EΘ(U − V ′ ) < ∞: Since U and V ′ are independent, Tonelli’s theorem

DETECTING INDEPENDENCE I. GENERALIZED DISTANCE COVARIANCE

9

entails that EΘ(u
− V ′ ) < ∞ for some u ∈ Rd . Using the symmetry and sub√
additivity of Θ, see (2.5), we get Θ(V ′ ) ≤ 2 (Θ(u − V ′ ) + Θ(u)), i.e. EΘ(V ′ ) < ∞;
EΘ(U ) < ∞ follows in a similar fashion.
c) Since a cndf Θ grows at most quadratically at infinity, see (2.6), it is clear
that


(2.17)
E |U |2 + E |V |2 < ∞ implies EΘ(U ) + EΘ(V ) < ∞.
One should compare this to the condition E|U |+E|V | < ∞ which ensures the finiteness of d2ρ (L(U ), L(V )), but not necessarily the finiteness of the terms appearing in
the representation (2.12).

It is worth mentioning that the metric dρ can be used to describe convergence
in distribution.
Lemma 2.6. Let ρ be a finite symmetric measure with full topological support, then
dρ given in (2.8) is a metric which characterizes convergence in distribution, i.e.
for random variables Xn , n ∈ N, and X one has
d

Xn −−−−→ X ⇐⇒ dρ (L(Xn ), L(X)) −−−−→ 0.
n→∞

n→∞

(2.18)

The proof below shows that the implication “⇐” does not need the finiteness of
the Lévy measure ρ.
Proof. Convergence in distribution implies pointwise convergence of the characteristic functions. Therefore, we see by dominated convergence and because of the
obvious estimate |fXn |, |fX | ≤ 1 that
Z
Z
2
lim |fXn (r) − fX (r)|2 ρ(dr) = 0.
|fXn (r) − fX (r)| ρ(dr) =
lim
n→∞

Rd n→∞

Rd

Conversely, assume that limn→∞ dρ (L(Xn ), L(X)) = 0. If we interpret this as
convergence in L2 (ρ), we see that there is a Lebesgue a.e. convergent subsequence
fXn(k) → fX ; since fXn(k) and fX are characteristic functions, this convergence is
already pointwise, hence locally uniform, see Sasvári [Sas94, Thm. 1.5.2]. By Lévy’s
continuity theorem, this entails the convergence in distribution of the corresponding
random variables. Since the limit does not depend on the subsequence, the whole
sequence must converge in distribution.

2.3. An elementary estimate for log-moments. Later on and in the companion paper [BKRS17] we need certain log-moments for the norm of a random vector.
The following lemma allows us to formulate the moment condition in terms of the
coordinate processes.

Lemma 2.7.√Let X, Y  be one-dimensional random variables and ǫ > 0. Then
E log1+ǫ 1 ∨ X 2 + Y 2 is finite if, and only if, the log-moments E log1+ǫ (1 + X 2)
and E log1+ǫ (1 + Y 2 ) are finite.
Proof. Assume that E log1+ǫ (1 + X 2 ) + E log1+ǫ (1 + Y 2 ) < ∞. Since
p
p
p
1 ∨ X 2 + Y 2 = (X 2 + Y 2 ) ∨ 1 ≤ (1 + X 2 )(1 + Y 2 )

we can use the elementary estimate (a + b)1+ǫ ≤ 2ǫ (a1+ǫ + b1+ǫ ), a, b ≥ 0, to get
"
1+ǫ #


p
1
1
1+ǫ
2
2
2
2
1∨ X +Y
E log
log(1 + X ) + log(1 + Y )
≤E
2
2
≤

1
1
E log1+ǫ (1 + X 2 ) + E log1+ǫ (1 + Y 2 ).
2
2

10

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

√

Conversely, assume that E log1+ǫ 1 ∨ X 2 + Y 2 < ∞. Then we have




E log1+ǫ (1 + X 2 ) = E 1{|X|<1} log1+ǫ (1 + X 2 ) + E 1{|X|≥1} log1+ǫ (1 + X 2 )


≤ log1+ǫ 2 + E log1+ǫ (2X 2 ) ∨ 2
h
1+ǫ i
≤ log1+ǫ 2 + E log 2 + log(1 ∨ X 2 )
≤ (1 + 2ǫ ) log1+ǫ 2 + E log1+ǫ (1 ∨ (X 2 + Y 2 ))


p
≤ (1 + 2ǫ ) log1+ǫ 2 + 21+ǫ E log1+ǫ 1 ∨ X 2 + Y 2 ,

and E log1+ǫ (1 + Y 2 ) < ∞ follows similarly.



3. Generalized distance covariance
Székely and co-authors [SRB07, SR09a] introduced distance covariance for two
random variables X and Y with values in Rm and Rn as
Z Z
2
V 2 (X, Y ; w) :=
f(X,Y ) (x, y) − fX (x)fY (y) w(x, y) dx dy
Rn

Rm

with a weight function w(x, y) = wα,m (x)wα,n (y) where wα,m (x) = c(p, α)|x|−m−α ,
m, n ∈ N, α ∈ (0, 2). It is well-known from the study of infinitely divisible distributions (see also Székely & Rizzo [SR05]) that wα,m (x) is the density of an
m-dimensional α-stable Lévy measure, and the corresponding cndf is just |x|α .
We are going to extend distance covariance to products of Lévy measures.
Definition 3.1. Let X be a random variable on Rm and Y on Rn and ρ := µ ⊗ ν
where µ and ν are symmetric Lévy measures on Rm \ {0} and Rn \ {0}, both having
full topological support. The generalized distance covariance V (X, Y ) is defined as
ZZ
V 2 (X, Y ) =
|f(X,Y ) (s, t) − fX (s)fY (t)|2 µ(ds) ν(dt).
(3.1)
Rm+n

By definition, V (X, Y ) = kf(X,Y ) − fX ⊗ fY kL2 (ρ) and in view of the discussion
in Section 2.2 we have
V (X, Y ) = dρ (L(U ), L(V )),
(3.2)
where U := (X1 , Y1 ), V := (X2 , Y3 ) and (X1 , Y1 ), (X2 , Y2 ), (X3 , Y3 ) are iid copies of
(X, Y ).2 It is clear that the product measure ρ inherits the properties “symmetry”
and “full topological support” from its marginals µ and ν.
From the discussion following Definition 2.3 we immediately get the following
lemma.

Lemma 3.2. Let V 2 (X, Y ) be generalized distance covariance of the m- resp. ndimensional random variables X and Y , cf. (3.1). The random variables X and Y
are independent if, and only if, V 2 (X, Y ) = 0.
3.1. Generalized distance covariance with finite Lévy measures. Let m, n ∈
N, set d := m + n, and assume that the measure ρ is of the form ρ = µ ⊗ ν where µ
and ν are finite symmetric Lévy measures on Rm \ {0} and Rn \ {0}, respectively.
If we integrate the elementary estimates
1 ∧ |s|2 ≤ 1 ∧ (|s|2 + |t|2 ) ≤ (1 ∧ |s|2 ) + (1 ∧ |t2 |)
1 ∧ |t|2 ≤ 1 ∧ (|s|2 + |t|2 ) ≤ (1 ∧ |s|2 ) + (1 ∧ |t2 |)

2This is a convenient way to say that L(U ) = L((X, Y )) while L(V ) = L(X) ⊗ L(Y ).

DETECTING INDEPENDENCE I. GENERALIZED DISTANCE COVARIANCE

11

with respect to ρ(ds, dt) = µ(ds) ν(dt), it follows that ρ is a Lévy measure if µ
and ν are finite Lévy measures.3 We also assume that µ and ν, hence ρ, have full
topological support.
Since V (X, Y ) is a metric in the sense of Section 2.2 we can use all results
from the previous section to derive various representations of generalized distance
covariance.
We write Φ, Ψ and Θ for the bounded cndfs induced by µ(ds), ν(dt), and ρ(dr) =
µ(ds) ν(dt),
Z
Z
Φ(x) =
(1 − cos x · s) µ(ds) and Ψ(y) =
(1 − cos y · t) ν(dt)
Rm
Rn
Z
(3.3)
and Θ(x, y) = Θ(u) =
(1 − cos u · r) ρ(dr),
Rm+n

with u = (x, y) ∈ Rm+n and r = (s, t) ∈ Rm+n . The symmetry in each variable
and the elementary identity

1
1− cos(x · s − y · t) + 1 − cos(x · s + y · t)
2
= (1 − cos y · t) + (1 − cos x · s) − (1 − cos x · s)(1 − cos y · t)
yield the representation
Θ(x, y) = Ψ(y)µ(Rm ) + Φ(x)ν(Rn ) − Φ(x)Ψ(y).

(3.4)

We can now easily apply the results from Section 2.2. To do so, we consider
six iid copies (Xi , Yi ), i = 1, . . . , 6, of the random vector (X, Y ), and set U :=
(X1 , Y1 ), V := (X2 , Y3 ), U ′ := (X4 , Y4 ), V ′ := (X5 , Y6 ). This is a convenient way to
say that
L(U ) = L(U ′ ) = L((X, Y ))

and L(V ) = L(V ′ ) = L(X) ⊗ L(Y )

and U, U ′ , V and V ′ are independent.4
The following formulae follow directly from Proposition 2.4.d) and Remark 2.5.a).
Proposition 3.3. Let X and Y be random variables with values in Rm and Rn
and assume that µ, ν are finite symmetric Lévy measures on Rm \ {0} and Rn \ {0}
with full support. Generalized distance covariance has the following representations
V 2 (X, Y ) = d2ρ (L(U ), L(V ))
′

(3.5)
′

′

= 2EΘ(U − V ) − EΘ(U − U ) − EΘ(V − V )

(3.6)

= 2EΘ(X1 − X5 , Y1 − Y6 ) − EΘ(X1 − X4 , Y1 − Y4 ) − EΘ(X2 − X5 , Y3 − Y6 )

(3.7)

= EΦ(X1 − X4 )Ψ(Y1 − Y4 ) + EΦ(X2 − X5 )EΨ(Y3 − Y6 ) − 2EΦ(X1 − X5 )Ψ(Y1 − Y6 ).

(3.8)

The last equality follows from (3.4) since the terms depending only on one of the
variables cancel as the random variables (Xi , Yi ) are iid. This gives rise to various
further representations of V (X, Y ).

3This argument also shows that ρ can only be a Lévy measure, if the marginals are finite

measures. In this case, ρ is itself a finite measure.
4In other words: in an expression of the form f (X , X , Y , Y ) all random variables are indei
j
k
l
pendent if, and only if, all indices are different. As soon as two indices coincide, we have (some
kind of) dependence.

12

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

Corollary 3.4. Let (X, Y ), (Xi , Yi ), Φ, Ψ and µ, ν be as in Proposition 3.3. Generalized distance covariance has the following representations
V 2 (X, Y )
= EΦ(X1 − X4 )Ψ(Y1 − Y4 ) − 2EΦ(X1 − X2 )Ψ(Y1 − Y3 ) + EΦ(X1 − X2 )EΨ(Y3 − Y4 )


= E Φ(X1 − X4 ) · {Ψ(Y1 − Y4 ) − 2Ψ(Y1 − Y3 ) + Ψ(Y2 − Y3 )}


= E {Φ(X1 − X4 ) − Φ(X4 − X2 )} · {Ψ(Y4 − Y1 ) − Ψ(Y1 − Y3 )}


= E {Φ(X1 − X4 ) − E(Φ(X4 − X1 ) | X4 )} · {Ψ(Y4 − Y1 ) − E(Ψ(Y1 − Y4 ) | Y1 )} .

(3.9)
(3.10)
(3.11)
(3.12)

Corollary 3.4 shows, in particular, that V (X, Y ) can be written as a function of
(Xi , Yi ), i = 1, . . . , 4,
V 2 (X, Y ) = E [g ((X1 , Y1 ), . . . , (X4 , Y4 ))]

(3.13)

for an appropriate function g; for instance, the formula (3.9) follows with
g((x1 , y1 ), . . . , (x4 , y4 ))
= Φ(x1 − x4 )Ψ(y1 − y4 ) − 2Φ(x1 − x2 )Ψ(y1 − y3 ) + Φ(x1 − x2 )Ψ(y3 − y4 ).
For completeness, we include the following result which is shown in the companion paper [BKRS17, Cor. 3.10] in greater generality.
Corollary 3.5. Let (X, Y ), (Xi , Yi ), Φ, Ψ and µ, ν be as in Proposition 3.3 and
write
Φ = Φ(X1 − X4 ) − E(Φ(X4 − X1 ) | X4 ) − E(Φ(X4 − X1 ) | X1 ) + EΦ(X1 − X4 )
Ψ = Ψ(Y1 − Y4 ) − E(Ψ(Y4 − Y1 ) | Y4 ) − E(Ψ(Y4 − Y1 ) | Y1 ) + EΨ(Y1 − Y4 ).
for the “doubly centered” versions of Φ(X1 − X4 ) and Ψ(Y1 − Y4 ). Then the generalized distance covariance has the following representation
h
i
V 2 (X, Y ) = E Φ · Ψ .
(3.14)
In Section 4 we will encounter further representations of the generalized distance
covariance if X and Y have discrete distributions with finitely many values, as it is
the case for empirical distributions.

3.2. Generalized distance covariance with arbitrary Lévy measures. So
far, we have been considering finite Lévy measures µ, ν and bounded cndfs Φ and
Ψ (3.3). We will now extend our considerations to unbounded measures. The
product measure ρ := µ ⊗ ν satisfies the integrability condition
Z
Z
ZZ
(1 ∧ |x|2 )(1 ∧ |y|2 ) ρ(dx, dy) = (1 ∧ |x|2 ) µ(dx) (1 ∧ |y|2 ) ν(dy) < ∞. (3.15)
Rm+n

Rm

Rn

Other than in the case of finite marginals, ρ is no longer a Lévy measure, see the
footnote on page 10. Thus, the function Θ defined in (3.3) need not be a cndf
and we cannot directly apply Proposition 2.4; instead we need the following result
ensuring the finiteness of V (X, Y ).
Lemma 3.6. Let X, X ′ be iid random variables on Rm and Y, Y ′ be iid random
variables on Rn ; µ and ν are symmetric Lévy measures on Rm and Rn with full
support and with corresponding cndfs Φ and Ψ as in (3.3). Then
V 2 (X, Y ) ≤ EΦ(X − X ′ ) · EΨ(Y − Y ′ ) ≤ 16 EΦ(X) · EΨ(Y ).

(3.16)

DETECTING INDEPENDENCE I. GENERALIZED DISTANCE COVARIANCE

13

Proof. Following Székely et al. [SRB07, p. 2772] we get

and

2
|fX,Y (s, t) − fX (s)fY (t)|2 = E (eiX·s − fX (s))(eiY ·t − fY (t))

2
≤ E |eiX·s − fX (s)||eiY ·t − fY (t)|




≤ E |eiX·s − fX (s)|2 · E |eiY ·t − fY (t)|2




E |eiX·s − fX (s)|2 = E (eisx − fX (s))(e−isx − fX (s))
= 1 − |fX (s)|2 = 1 − fX (s)fX (s).

(3.17)

(3.18)

Using (2.13) for ρ = µ, Θ = Φ, U = X and (2.10) for Θ = Φ, U = X, V = X ′
shows
Z


E |eiX·s − fX (s)|2 µ(ds) = EΦ(X − X ′ ) ≤ 4EΦ(X),
Rm

and an analogous argument for ν and Y yields the bound (3.16).



Looking at the various representations (3.8)–(3.12) of V (X, Y ) it is clear that
these make sense as soon as all expectations in these expressions are finite, i.e. some
moment condition in terms of Φ and Ψ should be enough to ensure the finiteness
of V (X, Y ) and all terms appearing in the respective representations.
In order to use the results of the previous section we fix ǫ > 0 and consider the
finite symmetric Lévy measures
µǫ (ds) :=

ǫ2

|s|2
µ(ds)
+ |s|2

and νǫ (dt) :=

ǫ2

|t|2
µ(dt),
+ |t|2

(3.19)

and the corresponding cndfs Φǫ and Ψǫ given by (3.3); the product measure ρǫ :=
µǫ ⊗ νǫ is a finite Lévy measure and the corresponding cndf Θǫ is also bounded (it
can be expressed by Φǫ and Ψǫ through the formula (3.4)).
This allows us to derive the representations (3.8)–(3.12) for each ǫ > 0 and with
Φǫ and Ψǫ . Since we have Φ = supǫ Φǫ and Ψ = supǫ Ψǫ , we can use monotone
convergence to get the representations for the cndfs Φ and Ψ with Lévy measures
µ and ν, respectively. Of course, this requires the existence of certain (mixed) Φ-Ψ
moments of the random variables (X, Y ).
Theorem 3.7. Let µ and ν be symmetric Lévy measures on Rm \ {0} and Rn \ {0}
with full topological support and corresponding cndfs Φ and Ψ given by (3.3). For
any random vector (X, Y ) with values in Rm+n satisfying the following moment
condition
EΦ(X) + EΨ(Y ) < ∞
(3.20)
the generalized distance correlation V (X, Y ) is finite. If additionally

E Φ(X)Ψ(Y ) < ∞,
(3.21)
then also the representations (3.8)–(3.12) hold with all terms finite.5

Proof. We only have to check the finiteness. Using Lemma 3.6, we see that (3.20)
guarantees that V (X, Y ) < ∞. The finiteness of (all the terms appearing in) the
representations (3.8)–(3.12) follows from the monotone convergence argument, since
the moment condition (3.21) ensures the finiteness of the limiting expectations. 
Remark 3.8. a) Using the Hölder inequality the following condition implies
(3.20) and (3.21):
1 1
(3.22)
EΦp (X) + EΨq (Y ) < ∞ for some p, q > 1 with + = 1.
p q
5As before, we denote in these formulae by (X , Y ), i = 1, . . . , 6, iid copies of (X, Y ).
i
i

14

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

If Ψ, resp. Φ, is bounded, (3.22) can be replaced by EΦ(X) < ∞, resp. EΨ(Y ) < ∞.
Since cndfs grow at most quadratically, we see that (3.20) and (3.21) also follow
if
1 1
E|X|2p + E|Y |2q < ∞ for some p, q > 1 with + = 1.
(3.23)
p q
b) A slightly different set-up was employed by Lyons [Lyo13]: If the cndfs Φ
and Ψ are subadditive, then the expectation in (3.11) is finite. This is a serious
restriction on the class of cndfs one can consider since subadditivity means that Φ
and Ψ can grow at most linearly at infinity, whereas general cndfs grow at most
quadratically. Note, however, that square roots of real cndfs are always subadditive,
cf. (2.4).
4. Estimating generalized distance covariance
b (N ) , Yb (N ) ) the random
Let (xi , yi )i=1,...,N be a sample of (X, Y ) and denote by (X
variable which has the corresponding empirical distribution, i.e. the uniform distribution on (xi , yi )i=1,...,N ; to be precise, repeated points will have the corresponding
b (N ) , Yb (N ) ) is a bounded random variable and for
multiple weight. By definition, (X
(N
)
(N
)
b (N ) , Yb (N ) ) for i = 1, . . . , 4, we have using
b , Yb
)) = L(X
iid copies with L((X
i
i
(3.13)
h
i
b (N ) , Yb (N ) ), . . . , (X
b (N ) , Yb (N ) )
E g (X
1
1
4
4
Z
Z
(4.1)
2
=
f(Xb (N ) ,Yb (N ) ) (s, t) − fXb (N ) (s)fYb (N ) (t) µ(ds) ν(dt).
Rn \{0}

Rm \{0}

b (N ) , Yb (N ) ),
The formulae (3.6)–(3.12) hold for the empirical random variables (X
and so we get without further calculations

h
i
b (N) , Yb (N) )
b (N) , Yb (N) ), . . . , (X
E g (X
4
4
1
1
=

=

=

1
N4
1
N4

N
X

g (xi , yi ), (xj , yj ), (xk , yk ), (xl , yl )

N
X



i,j,k,l=1

i,j,k,l=1



(4.2)

Φ(xi − xk )Ψ(yi − yk ) − 2Φ(xi − xj )Ψ(yi − yl ) + Φ(xi − xj )Ψ(yk − yl )

N
N
2 X
1 X
Φ(x
−
x
)Ψ(y
−
y
)
−
Φ(xi − xj )Ψ(yi − yl )
i
i
k
k
N2
N3
i,k=1

+



(4.3)

(4.4)

i,j,l=1

N
N
1 X
1 X
Ψ(yk − yl )
Φ(x
−
x
)
i
j
N 2 i,j=1
N2
k,l=1

=




 X
 X
N
N
1
4
2
2
2
Φ(x
−
x
)Ψ(y
−
y
)
+
Φ(xi − xj )Ψ(yi − yl )
−
+
−
i
i
k
k
2
3
4
4
3
N
N
N
N
N
i,k=1
distinct

i,j,l=1
distinct

(4.5)
+

1
N4

N
X

Φ(xi − xj )Ψ(yk − yl ).

i,j,k,l=1
distinct

The sum in (4.2) is – for functions g which are symmetric under permutations of
their variables functions g – a V-statistic.
Definition 4.1. The estimator NV 2 := NV 2 ((x1 , y1 ), . . . , (xN , yN )) of V 2 (X, Y ) is
defined by (4.3).

DETECTING INDEPENDENCE I. GENERALIZED DISTANCE COVARIANCE

15

In abuse of notation, we also write NV 2 ((x1 , . . . , xN ), (y1 , . . . , yN )) and NV 2 (x, y)
with x = (x1 , . . . , xN ), y = (y1 , . . . , yN ) instead of NV 2 ((x1 , y1 ), . . . , (xN , yN )).
Since all random variables are bounded, we could use any of the representations
(4.2)–(4.5) to define an estimator. A computationally feasible representation is
given in the following lemma.
Lemma 4.2. The estimator NV 2 of V 2 (X, Y ) has the following representation using
matrix notation:
V 2 ((x1 , y1 ), . . . , (xN , yN )) =

N

N
1 X
1
⊤
trace(B
A)
=
Akl Bkl
N2
N2

(4.6)

k,l=1

where
A = C ⊤ aC,
B = C ⊤ bC,
and C = I −

1
N1

a = (Φ(xk − xl ))k,l=1,..,N ,

(4.7)

b = (Ψ(yk − yl ))k,l=1,..,N ,

with 1 = (1)k,l=1,...,N and I = (δjk )j,k=1,...,N .

Remark 4.3. If Φ(x) = |x| and Ψ(y) = |y| the matrices a and b in (4.7) are
Euclidean distance matrices. For general cndfs Φ and Ψ, the matrices −a and −b
are conditionally positive definite (see Theorem 2.1), and A, B are positive definite.
This gives a simple explanation as to why the right-hand side of (4.2) is positive.
Proof of Lemma 4.2. By definition,
trace(b⊤ a) =

N
X

aij bij =

i,j=1

i,j=1

trace(b⊤ 1a) =

N
X

N
X

aij bik =

i,j,k=1

trace(1b⊤ 1a) =

N
X

Φ(xi − xj )Ψ(yi − yj ),

N
X

i,j,k=1

aij bkl =

N
X

i,j=1

i,j,k,l=1

(4.8)

Φ(xi − xj )Ψ(yi − yk ),
Φ(xi − xj )

N
X

k,l=1

(4.9)

Ψ(yk − yl ),

(4.10)

and this allows us to rewrite (4.3) as
V2 =

N

1
2
1
trace(b⊤ a) − 3 trace(b⊤ 1a) + 4 trace(1b⊤ 1a).
N2
N
N

(4.11)

Observe that C = C ⊤ and CC ⊤ = C. Using this and the fact that the trace is
invariant under cyclic permutations, we get
trace(B ⊤ A) = trace(C ⊤ b⊤ CC ⊤ aC) = trace(CC ⊤ b⊤ CC ⊤ a) = trace(Cb⊤ Ca).
Plugging in the definition of C now gives


⊤
1
1
N 1b )(a − N 1a)
− N1 trace(1b⊤ a) −
− N2 trace(b⊤ 1a) +

trace(B ⊤ A) = trace (b⊤ −
= trace(b⊤ a)

= trace(b⊤ a)

⊤
1
1
N trace(b 1a) + N 2
⊤
1
N 2 trace(1b 1a).

trace(1b⊤ 1a)

For the last equality we use that a and b are symmetric matrices.



We will now show that NV 2 is a consistent estimator for V 2 (X, Y ).
Theorem 4.4 (Consistency of NV 2 ). Let (Xi , Yi )i=1,...,N be iid copies of (X, Y )
and write X := (X1 , . . . , Xn ), Y := (Y1 , . . . , YN ). If E [Φ(X) + Ψ(Y )] < ∞, then
V 2 (X, Y ) −−−−→ V 2 (X, Y )

N

N →∞

a.s.

(4.12)

16

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

Proof. The moment condition E [Φ(X) + Ψ(Y )] < ∞ ensures that the generalized
distance covariance V 2 (X, Y ) is finite, cf. Lemma 3.6. Define µǫ and νǫ as in
(3.19), and write Vǫ2 (X, Y ) for the corresponding generalized distance covariance
and NVǫ2 (X, Y ) for its estimator. By the triangle inequality we obtain
|V 2 (X, Y ) − NV 2 (X, Y )|

≤ |V 2 (X, Y ) − Vǫ2 (X, Y )| + |Vǫ2 (X, Y ) − NVǫ2 (X, Y )| + |NVǫ2 (X, Y ) − NV 2 (X, Y )|.

We consider the three terms on the right-hand side separately. The first term
vanishes as ǫ → 0, since
lim Vǫ2 (X, Y ) = V 2 (X, Y )
ǫ→0

by monotone convergence. For each ǫ > 0, the second term converges to zero as
N → ∞, since
lim NVǫ2 (X, Y ) = Vǫ2 (X, Y ) a.s.
N →∞

by the strong law of large numbers (SLLN) for V -statistics; note that this is applicable since the functions Φǫ and Ψǫ are bounded (because of the the finiteness of
the Lévy measures µǫ and νǫ ).
For the third term we set µǫ = µ − µǫ , ν ǫ = ν − νǫ and write Φǫ , Ψǫ for
the corresponding continuous negative definite functions. Lemma 3.6 yields the
inequality
|NVǫ2 ((x1 , y1 ), . . . , (xN , yN )) − NV 2 ((x1 , y1 ), . . . , (xN , yN ))|
ZZ
2
fXb (N ) ,Yb (N ) (s, t) − fXb (N ) (s)fYb (N ) (t) µǫ (ds) ν ǫ (dt)
=
b (N ) ) · EΨǫ (Yb (N ) ) = 16
≤ 16 EΦǫ(X

N
N
X
X
1 ǫ
1 ǫ
Φ (xi ) ·
Ψ (yi ).
N
N
i=1
i=1

From the representation (3.3) we know that Φǫ (x) ≤ Φ(x), hence EΦǫ (X) ≤ EΦ(X)
and this is finite by assumption. Therefore, we can use monotone convergence to
conclude that limǫ→0 EΦǫ (X) = 0. Thus, the classical SLLN applies and proves
lim lim sup |NVǫ2 (X, Y ) − NV 2 (X, Y )| ≤ lim EΦǫ (X) · EΨǫ (Y ) = 0 a.s.

ǫ→0 N →∞

ǫ→0



Next we study the behaviour of the estimator under the hypothesis of independence.
Theorem 4.5. If X and Y are independent and satisfy the moment conditions


E [Φ(X) + Ψ(Y )] < ∞ and E log1+ǫ (1 + |X|2 ) + log1+ǫ (1 + |Y |2 ) < ∞ (4.13)

for some ǫ > 0, then
N

2

d

N · V (X, Y ) −−−−→
N →∞

ZZ

|G(s, t)|2 µ(ds) ν(dt) = kGk2µ⊗ν

(4.14)

in distribution, where (G(s, t))(s,t)∈Rm+n is a complex-valued Gaussian random field
with E(G(s, t)) = 0 and



Cov (G(s, t), G(s′ , t′ )) = fX (s − s′ ) − fX (s)fX (s′ ) fY (t − t′ ) − fY (t)fY (t′ ) .
(4.15)
Proof. Let X, Y and G be as above and let X ′ , Xi and Y ′ , Yi (i ∈ N) be independent
random variables with law L(X) and L(Y ), respectively. Define for N ∈ N, s ∈ RN ,
t ∈ Rm
N
N
1 X is·Xk +it·Yl
1 X is·Xk +it·Yk
e
− 2
e
.
(4.16)
ZN (s, t) :=
N
N
k=1

k,l=1

DETECTING INDEPENDENCE I. GENERALIZED DISTANCE COVARIANCE

Then

17

√
N · NV 2 (X, Y ) = k N ZN k2µ⊗ν .

(4.17)
√
d
→ G and then
The essential idea is to show that (on an appropriate space) N ZN −
to apply the continuous mapping theorem.
The independence of X and Y yields6 E(ZN (s, t)) = 0 and
′
′
′
′
E(ZN (s, t)ZN (s′ , t′ )) = NN−1
2 (fX (s − s ) − fX (s)fX (s ))(fY (t − t ) − fY (t)fY (t )),
√
E| N ZN (s, t)|2 = NN−1 (1 − |fX (s)|2 )(1 − |fY (t)|2 ).
(4.18)
√
d
The convergence N ZN −
→ G on CT := (C(KT ), k.kKT ) with KT = {x ∈ Rm+n :
|x| ≤ T }, i.e. in the space of continuous
functions
p
 on KT equipped with the supre|X|2 + |Y |2 ∨ 1 < ∞, see Csörgő [Csö85, Thm. on
mum norm, holds if E log1+ǫ
p. 294] or Ushakov [Ush99, Sec. 3.7]. This log-moment condition is equivalent to
the log-moment condition (4.13), see Lemma 2.7.
In fact, the result in [Csö85] is cast in a more general setting, proving the convergence for vectors (X1 , . . . , XN ), but only one-dimensional marginals are considered.
The proof for multidimensional marginals is very similar, so we will only give an
outline:
Let FZ denote the distribution function of the random variable Z and NFz the
empirical distribution of a sample z1 , ..., zN ; if the sample is replaced by N independent copies of Z, we write NFZ . Using this notation and the independence of
X and Y yields the representation
!
N
√
√
1 X is·Xk +it·Yk
e
− f(X,Y ) (s, t)
N ZN (s, t) = N
N
k=1
!
!
N
N
√
1 X it·Yl
1 X is·Xk
e
− fX (s)
e
− N
N
N
k=1
l=1
!
N
√
1 X it·Yk
− N
e
− fY (t) fX (s)
N
(4.19)
l=1
Z
√
= ei(s·x+t·y) d( N (NF(X,Y ) (x, y) − F(X,Y ) (x, y)))
!
Z
N
√ N
1 X it·Yl
is·x
− e
d( N ( FX (x) − FX (x))) ·
e
N
l=1
Z
√
− eit·y d( N (NFY (y) − FY (y))) · fX (s).

The independence also implies that the distribution functions factorize, i.e.

FN,(X,Y ) (x, y) = FN,X (x)FN,Y (y) and F(X,Y ) (x, y) = FX (x)FY (y).
√
Therefore, a straightforward calculation shows that N ZN (s, t) equals
 N


Z 
√
P it·Yl
ei(s·x+t·y) − eis·x N1
− fX (t)eit·y d( N (NF(X,Y ) (x, y)−F(X,Y ) (x, y))).
e
l=1

(4.20)

Following Csörgő [Csö81b] we obtain for N → ∞ the limit
Z

eis·x+it·y − eis·x fY (t) − fX (s)eit·y dB(x, y)

(4.21)

Rm+n

6Since essentially the same calculations are needed in the multivariate setting, we do not give
details here but refer to the proof of Theorem 4.6 in the companion paper [BKRS17].

18

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

where B is a Brownian bridge; as in [Csö81b, Eq. (3.2)] one can show that it is an
(m + n)-variate Gaussian process satisfying
E(B(x, y)) = 0


E B(x, y)B(x′ , y ′ ) = P X ≤ x ∧ x′ , Y ≤ y ∧ y ′


− P X ≤ x, Y ≤ y P X ≤ x′ , Y ≤ y ′
lim B(x, y) = lim B(x, y) =

x→−∞

y→−∞

lim

(x,y)→(∞,∞)

(4.22)
(4.23)

B(x, y) = 0.

(4.24)

The limit (4.21) is continuous if, and only if, a rather complicated tail condition
is satisfied [Csö81b, Thm. 3.1]; Csörgő [Csö85, p. 294] shows that this condition is
√
d
implied by the simpler moment condition (4.13), cf. Lemma 2.7. Thus, N ZN −
→G
in CT := (C(KT ), k.kKT ).
Pick ǫ > 0, set δ := 1/ǫ and define
µǫ,δ (A) := µ(A ∩ {ǫ ≤ |s| < δ}) and µǫ,δ := µ − µǫ,δ ;
the measures νǫ,δ and ν

ǫ,δ

(4.25)

are defined similarly. Note that

khkµǫ,δ ⊗νǫ,δ − kh′ kµǫ,δ ⊗νǫ,δ

2

≤ kh − h′ k2µǫ,δ ⊗νǫ,δ
Z
= |h − h′ |2 dµǫ,δ ⊗ νǫ,δ

(4.26)

≤ kh − h′ k2KT · µǫ,δ (Rn ) · νǫ,δ (Rm )

shows that h 7→ khk2µǫ,δ ⊗νǫ,δ is continuous on CT . Thus, the continuous mapping
theorem implies
√
d
(4.27)
k N ZN k2µǫ,δ ⊗νǫ,δ −−−−→ kGk2µǫ,δ ⊗νǫ,δ .
N →∞

By the triangle inequality we have
√
N · NV 2 (X, Y ) − kGk2µ⊗ν ≤ N · NV 2 (X, Y ) − k N ZN k2µǫ,δ ⊗νǫ,δ
√
+ k N ZN k2µǫ,δ ⊗νǫ,δ − kGk2µǫ,δ ⊗νǫ,δ

(4.28)

+ kGk2µǫ,δ ⊗νǫ,δ − kGk2µ⊗ν
Thus, it remains to show that the first and last terms vanish uniformly as ǫ → 0.
Note that
kGk2µ⊗ν − kGk2µǫ,δ ⊗νǫ,δ = kGk2µǫ,δ ⊗ν ǫ,δ + kGk2µǫ,δ ⊗ν ǫ,δ + kGk2µǫ,δ ⊗νǫ,δ −−−→ 0 a.s.
ǫ→0

(4.29)
This follows from the dominated convergence theorem, since

2
2
E kGk2µ⊗ν = 1 − |fX |2 µ · 1 − |fY |2 ν = EΦ(X −X ′)·EΨ(Y −Y ′ ) < ∞. (4.30)
Moreover,


√
E N · NV 2 (X, Y ) − k N ZN k2µǫ,δ ⊗νǫ,δ
√
√
√
= Ek N ZN k2µǫ,δ ⊗ν ǫ,δ + Ek N ZN k2µǫ,δ ⊗ν ǫ,δ + Ek N ZN k2µǫ,δ ⊗νǫ,δ

and for the first term we have
√
2
2
E(| N ZN |2 )
= NN−1 · 1 − |fX |2
µǫ,δ ⊗ν ǫ,δ

2
µǫ,δ

· 1 − |fY |2

2
−−−→
ν ǫ,δ ǫ→0

(4.31)

0 (4.32)

by dominated convergence, since we have k1 − |fX |2 k2µǫ,δ ≤ EΦ(X − X ′ ) < ∞ and
k1 − |fY |2 k2ν ǫ,δ ≤ EΨ(Y − Y ′ ) < ∞. The other summands are dealt with similarly.
The result follows since the convergence in (4.29) and (4.32) is uniform in N . 

DETECTING INDEPENDENCE I. GENERALIZED DISTANCE COVARIANCE

19

Remark 4.6. a) If we symmetrize the expression for N · NV 2 in a suitable
way, we can transform it into a degenerate U-statistic. For random variables with
bounded
second moments we can use classical results to show the convergence to
P∞
2
λ
X
k=1 k k , where λk are some coefficients and Xk are iid standard normal random
variables, see e.g. Serfling [Ser09, Sec. 5.5.2] or Witting & Müller-Funk [WMF95,
Satz 7.183]. In order to relax the bounds on the moments one would have to show
convergence of the corresponding λk .
b) The log-moment condition can be slightly relaxed, but this leads to a much
more involved expression, cf. Csörgő [Csö85], and for the case ǫ = 0 a counterexample is known, see Csörgő [Csö81a, p. 133]. Unfortunately, the convergence of the
characteristic functions is stated without any moment condition in Murata [Mur01,
Thm. 4] which is based on Feuerverger & Mureika [FM77, Thm. 3.1].
Corollary 4.7. Assume that EΦ(X) + EΨ(Y ) < ∞ and set
aN :=

N
1 X
Φ(Xi − Xk )
N2

and

bN :=

i,k=1

N
1 X
Ψ(Yj − Yl ).
N2
j,l=1

a) If X and Y are independent random variables satisfying the log-moment
conditions E log1+ǫ (1 + |X|2 ) < ∞ and E log1+ǫ (1 + |Y |2 ) < ∞ for some
ǫ > 0, then
∞
X
N · NV 2
d
αk Xi2 ,
(4.33)
−−−−→
aN bN N →∞
k=1

wherePXi are independent standard normal random variables and αk ≥ 0
with ∞
k=1 αk = 1.
b) If the random variables X and Y are not independent, then
N · NV 2
d
−−−−→ ∞.
aN bN N →∞

(4.34)

Proof. We can almost literally follow the proof of Corollary 2 and Theorem 6 in
Székely et al. [SRB07]. Just note that aN bN is an estimator for EΦ(X − X ′ ) ·
EΨ(Y − Y ′ ) and this is, by (4.18) and (4.30), the limit of the expectation of the
numerator.

5. Remarks on the uniqueness of the Cauchy distance covariance
It is a natural question whether it is possible to extend distance covariance
further by taking a measure ρ in the definition (3.1) which does not factorize. To
ensure the finiteness of V (X, Y ) one still has to assume
Z
(1 ∧ |s|2 )(1 ∧ |t|2 ) ρ(ds, dt) < ∞,
(5.1)

see also Székely & Rizzo [SR12, Eq. (2.4)] and (3.1) at the beginning of Section 3.2.
Furthermore, it is no restriction to assume that ρ is symmetric, since the integrand
in (3.1) is symmetric, hence we can always symmetrize any non-symmetric measure
ρ without changing the value of the integral. Thus, the function
Z Z
Θ(x, y) :=
(1 − cos(x · s))(1 − cos(x · t)) ρ(ds, dt)
(5.2)
Rn

Rm

is well-defined and symmetric in each variable. The corresponding generalized
distance covariance V 2 (X, Y ) can be expressed by (3.7), if the expectations on
the right-hand side are finite. Note, however, that the nice and computationally
feasible representations of V 2 (X, Y ) make essential use of the factorization of ρ
which means that they are no longer available in this setting.

20

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

Let X and Y be random variables with values in Rm and Rn , respectively, such
that for some x ∈ Rm and y ∈ Rn
P(X = 0) = P(X = x) = P(Y = 0) = P(Y = y) =

1
.
2

(5.3)

A direct calculation of (3.7), using Θ(0, y) = Θ(x, 0) = 0, gives
V 2 (X, Y ) = γ · Θ(x, y)

(5.4)

with
γ = 2P(X1 6= X5 , Y1 6= Y6 )−P(X1 6= X4 , Y1 6= Y4 )−P(X2 6= X5 )P(Y3 6= Y6 ) (5.5)
where (Xi , Yi ), i = 1, . . . , 6, are iid copies of the random vector (X, Y ).
Now suppose that V 2 (X, Y ) is homogeneous and/or rotationally invariant, i.e.
for some α, β ∈ (0, 2), all scalars a, b > 0 and orthogonal matrices A ∈ Rn×n , B ∈
Rm×m
V 2 (aX, bY ) = aα bβ V 2 (X, Y )
V 2 (AX, BY ) = V 2 (X, Y )
hold. The homogeneity, (5.6), yields
Θ(x, y) = |x|α |y|β Θ



y
x
|x| , |y|

(5.6)
(5.7)



(5.8)

and the rotational invariance, (5.7), shows that Θ(x/|x|, y/|y|) is a constant. In
particular, homogeneity of degree r = s = 1 and rotational invariance yield that
Θ(x, y) = const · |x| · |y|. Since the Lévy–Khintchine formula furnishes a oneto-one correspondence between the cndf and its Lévy triplet, see (the comments
following) Theorem 2.1, this determines uniquely ρ: it factorizes into two Cauchy
Lévy measures. This means that – even in a larger class of weights – the assumptions
(5.6) and (5.7) imply a unique (up to a constant) choice of weights, and we have
recovered Székely-and-Rizzo’s uniqueness result from [SR12].
Lemma 5.1. Let V 2 (X, Y ) := kfX,Y − fX ⊗ fY k2L2 (ρ) be a generalized distance
covariance as in Definition 2.3 and assume that the symmetric measure ρ satisfies
the integrability condition (5.1). If V 2 (X, Y ) is homogeneous of order α ∈ (0, 2)
and β ∈ (0, 2) and rotationally invariant in each argument, then the measure ρ
defining V 2 (X, Y ) is – up to a multiplicative constant – of the form
ρ(ds, dt) = c(α, m)c(β, n)|s|−α−m |t|−β−n ds dt.

Moreover, V 2 (X, Y ) can be represented by (3.7) with Θ(x, y) = const · |x|α · |y|β .
For completeness, let us mention that the constants c(α, m) and c(β, n) are of
the form


c(α, m) = α2α−1 π −m/2 Γ α+m
Γ 1 − α2 ,
2
see e.g. [BSW13, p. 34, Example 2.4.d)] or [BF75, p. 184, Exercise 18.23].
6. Generalized distance correlation
We continue our discussion in the setting of Section 3.2. Let ρ = µ ⊗ ν and
assume that µ and ν are symmetric Lévy measures on Rm \ {0} and Rn \ {0}, each
with full topological support. For m- and n-dimensional random variables X and
Y the generalized distance covariance is
sZ Z
|f(X,Y ) (s, t) − fX (s)fY (t)|2 µ(ds) ν(dt).
(6.1)
V (X, Y ) =

DETECTING INDEPENDENCE I. GENERALIZED DISTANCE COVARIANCE

We set

21

sZ
V (X) :=
|f(X,X) (s, t) − fX (s)fX (t)|2 µ(ds) µ(dt)

(6.2)

sZ
|f(Y,Y ) (s, t) − fY (s)fY (t)|2 ν(ds) ν(dt)
V (Y ) :=

(6.3)

and define generalized distance correlation as


 p V (X, Y ) , if V (X) · V (Y ) > 0,
V (X)V (Y )
R(X, Y ) :=

0,
otherwise.

(6.4)

=kf(X,X) − fX ⊗ fX kL2 (µ⊗µ) ,

=kf(Y,Y ) − fY ⊗ fY kL2 (ν⊗ν) ,

Using the Cauchy-Schwarz inequality it follows from (3.17) that, whenever R(X, Y )
is well-defined, one has
0 ≤ R(X, Y ) ≤ 1

and R(X, Y ) = 0 iff X, Y are independent.

The sample distance correlation is given by


(6.5)

 21
A
B
kl
kl
k,l=1
N

q
R((x1 , y1 ), . . . , (xN , yN )) =  q
1 PN
1 PN
A
A
·
B
B
kl
kl
kl
kl
2
2
k,l=1
k,l=1
N
N
(6.6)
where we use the notation introduced in Lemma 4.2.
1
N2

PN

Example 6.1. For standard normal random variables X, Y with ρ = Cor(X, Y ),
Φ(x) = |x| and Ψ(y) = |y| the distance correlation becomes
!1/2
p
p
1 − ρ2 − 4 − ρ2 + ρ(arcsin ρ − arcsin ρ2 ) + 1
√
≤ |ρ|, (6.7)
R(X, Y ) =
1 − 3 + π/3
cf. Székely & Rizzo [SR09a, Thm. 6].
7. Gaussian covariance
Let us finally show that the results on Brownian covariance of Székely & Rizzo
[SR09a, Sec. 3] do have an analogue for the generalized distance covariance. As
we will encounter similar (and more general) results in the multivariate setting,
we only state the results and refer for proofs to the companion paper [BKRS17,
Sec. 3.3].
For a symmetric Lévy measure with corresponding continuous negative definite
function Φ : Rm → R let GΦ be the Gaussian field indexed by Rm with
EGΦ (x) = 0

and E(GΦ (x)GΦ (x′ )) = Φ(x) + Φ(x′ ) − Φ(x − x′ ).

(7.1)

Analogously we define the random field GΨ .
For a random variable Z with values in Rd and a Gaussian random field G
indexed by Rd we set
Z G := G(Z) − E(G(Z) | G).
(7.2)
Definition 7.1. Let GΦ , GΨ be mean-zero Gaussian random fields indexed by Rm
and Rn and with covariance structure given by the cndfs Φ and Ψ, respectively. For

22

B. BÖTTCHER, M. KELLER-RESSEL, AND R. L. SCHILLING

any two m- and n-dimensional random variables X and Y the Gaussian covariance
is defined as
G2 (X, Y ) := Cov2GΦ ,GΨ (X, Y ) := E(X1GΦ X2GΦ Y1GΨ Y2GΨ ),

(7.3)

where (X1 , Y1 ), (X2 , Y2 ) are iid copies of (X, Y ).
We can now identify Gaussian covariance and generalized distance covariance.
Theorem 7.2 (Gaussian covariance is generalized distance covariance). Assume
that X and Y have finite moments EΦ(X) + EΨ(Y ) < ∞. If GΦ and GΨ are
independent, then
G2 (X, Y ) = V 2 (X, Y ).
(7.4)
Proof. The proof is similar to Székely & Rizzo [SR09a, Thm. 8]. By conditioning
and the independence of GΦ and GΨ we see


E X1GΦ X2GΦ Y1GΨ Y2GΨ


 (7.5)
= E E X1GΦ X2GΦ | X1 , X2 , Y1 , Y2 · E Y1GΨ Y2GΨ | X1 , X2 , Y1 , Y2 .

Using E(GΦ (x)GΦ (x′ )) = Φ(x) + Φ(x′ ) − Φ(x − x′ ) =: φ(x, x′ ) yields

E X1GΦ X2GΦ | X1 , X2 , Y1 , Y2

= φ(X1 , X2 ) − E(φ(X1 , X2 ) | X1 ) − E(φ(X1 , X2 ) | X2 ) + Eφ(X1 , X2 )

= −Φ(X1 − X2 ) + E(Φ(X1 − X2 ) | X1 ) + E(Φ(X1 − X2 ) | X2 ) − EΦ(X1 − X2 ),

where the second equality is due to cancellations. An analogous calculation for
E(Y1GΨ Y2GΨ | X1 , X
2 , Y1 , Y2 ) turns (7.5) into (3.14). Finally, note that for (7.5) it

is required that E X1GΦ X2GΦ Y1GΨ Y2GΨ < ∞, which is satisfied under the given
assumptions, cf. [BKRS17, proof of Thm. 3.13].

References
[BF75]

Christian Berg and Gunnar Forst. Potential Theory on Locally Compact Abelian
Groups. Springer, Berlin, 1975.
[BKRS17] Björn Böttcher, Martin Keller-Ressel, and René L. Schilling. Detecting independence
of random vectors II. Distance multivariance and Gaussian multivariance. 2017.
[BL00]
Yoav Banyamini and Joram Lindenstrauss. Geometric Nonlinear Functional Analysis.
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TU Dresden, Fachrichtung Mathematik, Institut für Mathematische Stochastik, 01062
Dresden, Germany
E-mail address: bjoern.boettcher@tu-dresden.de
URL: https://www.math.tu-dresden.de/~boettch/
E-mail address: martin.keller-ressel@tu-dresden.de
URL: https://www.math.tu-dresden.de/~mkeller/
E-mail address: rene.schilling@tu-dresden.de
URL: http://www.math.tu-dresden.de/sto/schilling/

