Characterization theorems for the Gneiting class of space–time ...

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8 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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Bernoulli 17(1),2011,456–465
DOI:10.3150/10-BEJ278
Characterization theorems for the Gneiting
class of space–time covariances
VIKTOR P.ZASTAVNYI
1
and EMILIO PORCU
2
1
Department of Mathematics,Donetsk National University,Universitetskaya Str.24,Donetsk,34001,
Ukraine.E-mail:zastavn@rambler.ru
2
Institut für Mathematische Stochastik,University of Göttingen,Goldschmidtstrasse 7,37077 Göttingen,
Germany.E-mail:eporcu@uni-goettingen.de
We characterize the Gneiting class of space–time covariance functions and give more relaxed conditions
on the functions involved.We then show necessary conditions for the construction of compactly supported
functions of the Gneiting type.These conditions are very general since they do not depend on the Euclidean
norm.
Keywords:compact support;Gneiting’s class;positive definite;space–time
1.Introduction
The construction of space–time covariance functions is an important subject,the literature for
which can be traced back to at least the early 1990s [1,2],where it is emphasized how,under
the framework of geostatistical techniques for the study of,for instance,atmospheric and envi-
ronmental sciences,covariance functions are crucial for estimation and prediction since the best
linear predictor depends exclusively on the covariance matrix,which determines the weights of
any individual observation in the predictor itself [4].
There are several unsolved problems which are of interest to both the statistical and mathe-
matical communities and this paper provides solutions to two of them.
The first problemis related to the characterization of space–time covariance functions.To the
best of our knowledge,there is no literature related to this important problem.In particular,we
can find several permissibility criteria,that is,sufficient conditions to ensure that a candidate
function is positive definite (permissible) on the space–time domain,but no characterization
theorem,at least for given classes of covariance functions,is available.
A wide class of covariance functions can be obtained through Gaussian mixtures [4,7,8] for
which one can find a large number of contributions having as common origin the Gneiting class
of covariance functions [4]:for (x,t) ∈R
d+l
,the function
(x,t) →K(x,t):=h(t
2
)
−d/2
ϕ

x
2
h(t
2
)

(1.1)
is positive definite,where ϕ is completely monotone on the positive real line,h is a Bernstein
function and  ·  denotes the Euclidean norm.For l =1,the function above is a stationary and
non-separable space–time covariance.This function has been persistently used in the literature
1350-7265 © 2011 ISI/BS
Characterization theorems of space–time 457
and a Google Scholar search in September 2009 yielded over 90 papers where this covariance
has been used for applications to space–time data.
The first result in this paper states necessary and sufficient conditions for the permissibility of
the Gneiting class.Also,more general conditions for its permissibility are given.
The second problemconfronted in this paper relates to the construction of space–time covari-
ances that are compactly supported in the spatial component.Although such compactly supported
covariances are much in demand in the recent literature,there is no single contribution concern-
ing the construction of compactly supported correlations over space and time.This challenge is
considerable from a mathematical point of view.A natural perspective is to consider the Gneit-
ing class above and replace the completely monotone function ϕ(·) with a compactly supported
one,that is,a function which is identically zero outside a finite range.In particular,the tempting
choice t →ϕ(t):= (1 −t
α
)
λ
+
,for positive values of α and λ and where (x)
+
denotes the
positive part of x,creates an interesting connection to the celebrated Schoenberg [9] problem,
in which the positive definiteness of the function ϕ defined above is related to that of the func-
tion t →exp(−t
β
) for some positive β.The reader is referred to the survey in [14,15] and the
references therein for a thorough review.
In considering this problem,we work in a fairly general framework and let the function ϕ de-
pend on a general seminormand not on the Euclidean one,as the latter is a restrictive assumption
for spatial applications.
The paper is organized as follows.Section 2 completely characterizes the Gneiting class,for
which only sufficient conditions have been known.In Section 3,we present necessary conditions
for compactly supported covariances of the Gneiting type.
2.Characterization of the Gneiting class
In this section,we give a characterization of the Gneiting class.In doing so,we relax the permis-
sibility hypotheses stated in [4].Two technical lemmas are needed for a more elegant proof of
the main result,stated as Theorem2.1 below.
For a complex-valued function f:R
n
→C,we write f ∈ L(R
n
) when f is absolutely inte-
grable on R
n
.Similarly,we write f ∈C(R
n
) when f is continuous in R
n
.
For a real linear space E,we denote by FD(E) the set of all linear finite-dimensional subspaces
of E.
If dimE =n ∈N and e
1
,...,e
n
constitute a basis for E,then,by definition,we have
C(E) ={f:E →C| f(x
1
e
1
+· · · +x
n
e
n
) ∈C(R
n
)},
C
0
(E) ={f ∈C(E) | f has compact support} and
L(E) ={f:E →C| f(x
1
e
1
+· · · +x
n
e
n
) ∈L(R
n
)}.
Obviously,these classes do not depend on the choice of the basis in E.Thus,in this case,it is
possible to set E =R
n
.
If dimE = ∞,then,by definition,we have that C(E) = {f:E →C | f ∈ C(E
0
) for all
E
0
∈FD(E)}.
A complex-valued function f:E →C is said to be positive definite on E (denoted hereafter
f ∈ (E)) if,for any finite collection of points {ξ
i
}
n
i=1
∈ E,the matrix (f(ξ
i
− ξ
j
))
n
i,j=1
is
458 V.P.Zastavnyi and E.Porcu
positive definite,that is,
for all a
1
,a
2
,...,a
n
∈C
n

i,j=1
a
i
f(ξ
i
−ξ
j
)
a
j
≥0.
Let E = R
n
.By Bochner’s theorem,the function f is positive definite and continuous in R
n
if and only if f(x) =

R
n
e
−i(u,x)
dμ(u),where (u,x) =u
1
x
1
+u
2
x
2
+· · · +u
n
x
n
is a scalar
product in R
n
and μ is a non-negative finite Borel measure on R
n
.Additionally,if f ∈C(R
n
) ∩
L(R
n
),then f is positive definite on R
n
if and only if

f(u):=

R
n
e
i(u,x)
f(x) dx ≥0,u ∈R
n
.
Lemma 2.1.
(i) f ∈(E) ⇐⇒ f ∈(E
0
) ∀E
0
∈FD(E).
(ii) If dimE =n ∈N,then f ∈(E) ⇐⇒ fg ∈(E) for all g ∈(E) ∩C
0
(E).
Proof.For both parts,the necessity is obvious.For the sufficiency of part (i),for n ∈ N and
x
1
,...,x
n
in E,we have that x
1
,...,x
n
∈ E
0
,where E
0
is the linear span of these elements.
Obviously,dimE
0
≤n.
For the sufficiency of part (ii),let e
1
,...,e
n
be a basis in E.We then take g(x
1
e
1
+· · · +
x
n
e
n
) =(1 −ε|x
1
|)
+
· · · (1 −ε|x
n
|)
+
and ε ↓ 0.The proof is thus completed.￿
Lemma 2.2.Let the following conditions be satisfied:
(1) h,b ∈C(E) and h(t) >0 for all t ∈E;
(2) ϕ ∈C([0,+∞)) and for some m∈N,we have


0
|ϕ(u
2
)|u
m−1
du <∞;
(3) ρ ∈C(R
m
),ρ(tx) =|t|ρ(x) for all t ∈R,x ∈R
m
and ρ(x) >0,x =0.
Then K(x,t):= b(t)ϕ(
ρ
2
(x)
h(t)
) ∈ (R
m
×E) ⇐⇒ b(t)(h(t))
m/2
G
m
(

h(t)v) ∈ (E) for all
v ∈R
m
with G
m
(·) defined in equation (2.1) and
R
n
v →G
n
(v):=

R
n
ϕ(ρ
2
(y))e
i(y,v)
dy.(2.1)
Proof.Observe that ϕ(ρ
2
(x)) ∈L(R
m
).We have that
K(x,t) ∈(R
m
×E) ⇐⇒ K(x,t) ∈(R
m
×E
0
) for all E
0
∈FD(E)
⇐⇒ K(x,t)g(t) ∈(R
m
×E
0
) for all E
0
∈FD(E) and all g ∈(E
0
) ∩C
0
(E
0
)
⇐⇒

R
m

E
0
K(x,t)g(t)e
i(x,v)
e
i(t,u)
dx dt ≥0
for all E
0
∈FD(E),g ∈(E
0
) ∩C
0
(E
0
) and v ∈R
m
,u ∈E
0
.
As for the last integral,a change of variables of the type x =

h(t)y yields that the last inequality
is equivalent to

E
0
g(t)b(t)(h(t))
m/2
G
m


h(t)v


e
i(t,u)
dt ≥0 for all v ∈R
m
,u ∈E
0
,
Characterization theorems of space–time 459
which holds if and only if,for all g ∈(E
0
) ∩C
0
(E
0
) and v ∈R
m
,we have
g(t)b(t)(h(t))
m/2
G
m


h(t)v


∈(E
0
) for all E
0
∈FD(E)
⇐⇒ b(t)(h(t))
m/2
G
m


h(t)v


∈(E) for all v ∈R
m
.
The proof is thus complete.￿
A function f:(0,∞) →R is called completely monotone if it is arbitrarily often differen-
tiable and (−1)
n
f
(n)
(x) ≥ 0 for x >0,n =0,1,....By the Bernstein–Widder theorem [10],
the set M
(0,∞)
of completely monotone functions coincides with that of Laplace transforms L
of positive measures μ on [0,∞),that is,f(x) =Lμ(x) =

[0,∞)
e
−xt
dμ(t),x >0,where we
require e
−xt
to be μ-integrable for any x > 0.By Schoenberg’s theorem,the radial function
f(x) =ϕ(x
2
),ϕ ∈C([0,+∞)) belongs to (R
n
) for all n ∈N if and only if ϕ ∈M
(0,∞)
.
Theorem 2.1 gives our characterization of the Gneiting class.This has the feature,additional
to our introduction of the class in Section 1,that only negative definiteness of the function h is
required [8],while Gneiting’s assumptions are much more restrictive as it is required that h

is
completely monotone on the positive real line.Furthermore,the proof of this result is deferred to
the final section for reasons that will become apparent.
Theorem 2.1.Let h ∈ C(E),h(t) >0 for all t ∈ E.Let d ∈ N.The following statements are
equivalent:
(1) K(x,t):=(h(t))
−d/2
ϕ(
x
2
h(t)
) ∈(R
d
×E) for all ϕ ∈C([0,+∞)) ∩M
(0,∞)
;
(2) e
−λh(t)
∈(E) for all positive λ.
Let us consider examples of functions h for which statement (2) in Theorem2.1 holds.
Example 2.1.Let h(t) = t
α
p
+ c,c > 0,0 < p ≤ ∞,α ≥ 0,t = (t
1
,...,t
n
) ∈ R
n
,where
t
p
= (

n
k=1
|t
k
|
p
)
1/p
,0 < p < ∞,and t

= sup
1≤k≤n
|t
k
|.Then e
−λh(t)
∈ (E) for all
positive λ if and only if 0 ≤α ≤α
n,p
,where
α
n,p
=





2 if n =1,0 <p ≤∞,
p if n ≥2,0 <p ≤2,
1 if n =2,2 <p ≤∞,
0 if n ≥3,2 <p ≤∞.
(2.2)
The case 0 <p ≤2 corresponds to the result of Schoenberg [9].The other three cases have been
investigated by Koldobsky [5] and Zastavnyi [11–13] (2 <p ≤∞,n ≥2).Finally,Misiewiez
[6] gave the most recent result (p =∞,n ≥3).
Example 2.2.If ρ(t) is a norm on R
2
,then e
−ρ
α
(t)
∈ (R
2
) for all 0 ≤α ≤1 (see,e.g.,[14]).
Therefore,e
−λh(t)
∈(R
2
) for any λ >0,where h(t) =ρ
α
(t) +c,0 ≤α ≤1,c >0.
460 V.P.Zastavnyi and E.Porcu
Example 2.3.Let ψ(s) ∈ R,s >0.We then have e
−λψ
∈ M
(0,∞)
for all λ >0 if and only if
ψ

∈ M
(0,∞)
.Therefore,if ψ ∈ C([0,+∞)) and ψ(s) >0 for all s ≥0,and ψ

∈ M
(0,∞)
,then
e
−λh(t)
∈ (R
n
) for all λ > 0,n ∈ N,where h(t):= ψ(t
2
) and,hence (see Theorem 2.1),
K(x,t):=(ψ(t
2
))
−d/2
ϕ(
x
2
ψ(t
2
)
) ∈ (R
d
×R
n
) for all ϕ ∈ C([0,+∞)) ∩ M
(0,∞)
,d ∈ N.
This result was proven by Gneiting [4].
A complex-valued function h:E →C is called conditionally negative definite on E (denoted
h ∈ N(E) hereafter) if the inequality

n
k,j=1
c
k
¯c
j
h(x
k
− x
j
) ≤ 0 is satisfied for every posi-
tive integer n,every collection of points x
1
,...,x
n
in E and every set of complex numbers
c
1
,c
2
,...,c
n
,satisfying the condition

n
k=1
c
k
=0.
Example 2.4 (Schoenberg’s theorem[9]).e
−λh(t)
∈ (E) for any λ >0 if and only if h(−t) =
h(t) for all t ∈E and h(t) ∈N(E).
3.Necessary conditions for functions of the Gneiting type
Before presenting the main results contained in this section,some comments are in order.The
construction of compactly supported correlation functions is a non-trivial task that has conse-
quences for the estimation of space–time processes for the computational gains that follow.At
present,there is no contribution in the literature devoted to non-separable covariances that are
compactly supported.Until now,in order to obtain compactly supported correlations,the com-
monplace approach is to use tapering [3].We described a more natural approach in the Introduc-
tion and the results following subsequently highlight interesting solutions to this problem.
In order to be clear,we will henceforth write S
d−1
:={x ∈ R
d
:x =1} for the unit sphere
in R
d
.
Theorem3.1.Let the following conditions be satisfied:
(1) h ∈C(E),h(t) >0 for any t ∈E and h(t) ≡h(0) on E;
(2) ϕ ∈C([0,+∞)),ϕ(0) >0;
(3) for d ∈N,ρ ∈C(R
d
),ρ(tx) =|t|ρ(x) ∀t ∈R,x ∈R
d
and ρ(x) >0,x =0;
(4) K(x,t):=(h(t))
−d/2
ϕ(
ρ
2
(x)
h(t)
) ∈(R
d
×E).
Then:
1.(h(t))
−d/2
∈(E) and ϕ(ρ
2
(x)) ∈(R
d
);
2.if there exists an integer n ∈{1,...,d} such that


0
|ϕ(u
2
)|u
n−1
du <∞,then for all m=
1,...,n and v ∈ R
m
,the function s →f
m,v
(s):=s
m−d
G
m
(sv),with G
m
(·) as defined in
(2.1),is decreasing on (0,∞) and,furthermore,f
m,v
(∞) =0 for v =0.
3.if


0
|ϕ(u
2
)|u
d−1
du <∞,then G
d
(0) >0 and if,in addition,G
d
is real analytic,then for
any v ∈ R
d
,v =0,the function s →f
d,v
(s):=G
d
(sv) is strictly decreasing on [0,+∞)
and G
d
(v) >0 for all v ∈R
d
;
4.if


0
|ϕ(u
2
)|u
d+1
du <∞,then α
1
(v):=

R
d
ϕ(ρ
2
(y))(y,v)
2
dy ≥0 for all v ∈S
d−1
and
β
1
:=

R
d
ϕ(ρ
2
(y))y
2
dy ≥0 and,furthermore,α
1
(v) ≡0 on S
d−1
if and only if β
1
=0;
if,in addition,β
1
>0,then e
−λh(t)
∈(E) for any λ >0;
Characterization theorems of space–time 461
5.if


0
|ϕ(u
2
)|e
εu
du <∞for some ε >0,then for every λ >0 and every v ∈S
d−1
we have
e
−λh
p
(t)
∈(E),where
p =p(v):=min

k ∈N:α
k
(v) =

R
d
ϕ(ρ
2
(y))(y,v)
2k
dy =0

,v ∈S
d−1
;
the function p:S
d−1
→N is bounded on the unit sphere and
min
v∈S
d−1
p(v) =min

k ∈N:β
k
=

R
d
ϕ(ρ
2
(y))y
2k
dy =0

.
Proof.Part 1 is obvious.
As for part 2,by Lemma 2.2,we have
F
m,v
(t):=(h(t))
(m−d)/2
G
m


h(t)v


∈(E),m=1,...,n,v ∈R
m
.
Hence,F
m,v
(0) = (h(0))
(m−d)/2
G
m
(

h(0)v) ≥ 0 and |F
m,v
(t)| ≤ F
m,v
(0),t ∈ E.Therefore,
G
m
(v) ≥0,v ∈R
m
,and
(sh(t))
(m−d)/2
G
m


h(t)sv


≤(sh(0))
(m−d)/2
G
m


h(0)sv


for m=1,...,n,v ∈R
m
,s >0 and for all t ∈E.The latter inequality is equivalent to
f
m,v


h(t)
h(0)
· s

≤f
m,v
(s).
Since (h(t))
−d/2
∈ (E),we have h(t) ≥h(0),t ∈ E.Since h(t) ≡h(0) on E,there exists a
point t
0
∈ E such that q:=

h(t
0
)
h(0)
>1.By the mean value theorem,for any α ∈ [1,q],there
exists a ξ ∈ E such that

h(ξ)
h(0)
=α.Therefore,f
m,v
(αs) ≤f
m,v
(s) for all s >0 and α ∈ [1,q].
Hence,f
m,v

2
s) ≤f
m,v
(αs) ≤f
m,v
(s) for all s >0 and α ∈[1,q].Thus,f
m,v

p
s) ≤f
m,v
(s)
for all s >0,α ∈[1,q] and p ∈N.This implies that the function f
m,v
(s) decreases in s ∈(0,∞).
By the Riemann–Lebesgue theorem,it follows that G
m
(v) →0 as v →∞.Hence,f
m,v
(∞) =
0 for v =0.
3.i.From part 2,it follows that for all v ∈ R
d
,v =0,the function G
d
(sv) decreases in s ∈
(0,∞) and,hence,0 ≤G
d
(v) ≤G
d
(0).Therefore,G
d
(0) >0 (otherwise,G
d
(v) ≡0 on R
d

ϕ(ρ
2
(y)) ≡0 on R
d
,which contradicts the condition ϕ(0) >0).
ii.If,in addition,G
d
is real analytic,then for all v ∈R
d
,v =0,the function G
d
(sv) is strictly
decreasing on [0,∞).This can be proven by contradiction.Let us assume that,for some v
0
∈R
d
and v
0
= 0,the function G
d
(sv
0
) is constant on some interval (α,β) ⊂ (0,∞),α < β.This
would imply that G
d
is constant on [0,∞) and that G
d
(0) =lim
s→∞
G
d
(sv
0
) =0,which con-
tradicts part i above.Thus,for all v ∈R
d
,v =0,the function G
d
(sv) strictly decreases on [0,∞)
and,hence,G
d
(v) >lim
s→∞
G
d
(sv) =0.
462 V.P.Zastavnyi and E.Porcu
4.Let v ∈ S
d−1
and define f
d,v
(s):=G
d
(sv).Fromparts 2 and 3,it follows that the function
f
d,v
(s) decreases on [0,∞) and that f
d,v
(0) >0.Obviously,f
d,v
(s) ∈C
2
(R) and
f
d,v
(s) =f
d,v
(0) +
f

d,v
(0)
2
s
2
+o(s
2
),s →0,
where f

d,v
(0) =−α
1
(v).Note that f

d,v
(0) ≤0,otherwise the function f
d,v
(s) would be strongly
increasing on [0,c] for some c > 0,which would contradict part 2.Thus,α
1
(v) ≥ 0 for all
v ∈S
d−1
.For p >0,the following integral is constant on S
d−1
:

S
d−1
|(y,v)|
p
dσ(v) ≡c
d,p
>0,y ∈S
d−1
,
where dσ,if n ≥2,is the surface measure on S
d−1
and dσ(v) =δ(v −1) +δ(v +1),if d =1
(here,δ(v) is the Dirac measure with mass 1 concentrated in the point v =0).Therefore,

S
d−1
|(y,v)|
p
dσ(v) =c
d,p
y
p
,y ∈R
d
,p >0.(3.1)
Hence,

S
d−1
α
1
(v) dσ(v) =c
d,2
β
1
≥0
and α
1
(v) ≡0 on S
d−1
if and only if β
1
=0.
Let,in addition,β
1
>0.Then f

d,v
0
(0) =−α
1
(v
0
) <0 for some v
0
∈S
d−1
and
ψ
n
(t):=

G
d

n

h(t)v
0
)
G
d
(0)

n
=

1 +g
n
(t)


n
∈(E) ∀n ∈N,γ
n
>0.(3.2)
Now,let us take
γ
n
:=


2f
d,v
0
(0)
f

d,v
0
(0)
·
λ
n

1/2
>0,λ >0.
Obviously,γ
n
→+0 and
g
n
(t) =
f
d,v
0

n

h(t)) −f
d,v
0
(0)
f
d,v
0
(0)

f

d,v
0
(0)
2f
d,v
0
(0)
·

γ
n

h(t)


2
=−
λ
n
· h(t) as n →∞.
Therefore,ψ
n
(t) →e
−λh(t)
and,hence,e
−λh(t)
∈(E) for all λ >0.
5.In this case,G
d
is real analytic and
f
(2k)
d,v
(0) =(−1)
k
α
k
(v),
(3.3)
f
(2k−1)
d,v
(0) =0,

S
d−1
α
k
(v) dσ(v) =c
d,2k
β
k
,k ∈N.
Characterization theorems of space–time 463
Therefore,for all v ∈S
d−1
,there exists a natural number p ∈N such that
f
d,v
(s) =f
d,v
(0) +
f
(2p)
d,v
(0)
(2p)!
s
2p
+o(s
2p
) as s →0,
where f
(2p)
d,v
(0) =0;otherwise,the function f
d,v
(0) ≡f
d,v
(s) ≡f
d,v
(+∞) =0,which would
contradict the inequality G
d
(0) > 0 (see part 3).Hence,f
(2p)
d,v
(0) < 0,otherwise the function
f
d,v
(s) would be strongly increasing on [0,c] for some c >0,which would contradict part 2.
Let v ∈S
d−1
and p =p(v).Take the function in equation (3.2),where v
0
=v
γ
n
:=


(2p)!f
d,v
0
(0)
f
(2p)
d,v
0
(0)
·
λ
n

1/(2p)
>0,λ >0.
Then g
n
(t) ∼−
λ
n
· h
p
(t) as n →∞.Therefore,ψ
n
(t) →e
−λh
p
(t)
and,hence,e
−λh
p
(t)
∈ (E)
for all λ >0.
If α
k
(v
0
) =0 for some v
0
∈ S
d−2
,k ∈ N,then α
k
(v) =0 in some neighborhood of a point v
0
and,hence,p(v) ≤p(v
0
) in this neighborhood.Thus,the function p(v) is locally bounded on
S
d−1
and,hence,p(v) is bounded there.
Let m=min
v∈S
d−1
p(v) =p(v
0
) for some v
0
∈S
d−1
.Then α
m
(v
0
) =0 and,for all v ∈S
d−1
,
the equality
f
d,v
(s) =f
d,v
(0) +
f
(2m)
d,v
(0)
(2m)!
s
2m
+o(s
2m
) as s →0
holds.Obviously,(−1)
k
α
k
(v) =f
(2k)
d,v
(0) =0 for all 1 ≤k <m (if m≥2),and (−1)
m
α
m
(v) =
f
(2m)
d,v
(0) ≤ 0 (otherwise the function f
d,v
(s) is strongly increasing on [0,c] for some c > 0,
which would contradict 2).From (3.3),it follows that β
k
=0 for all 1 ≤k <m (if m≥2) and
(−1)
m
β
m
<0.Therefore,
m=min

k ∈N:β
k
=

R
d
ϕ(ρ
2
(y))y
2k
dy =0

and this completes the proof.￿
We are now able to give a simple proof of Theorem2.1.
Proof of Theorem 2.1.If h(t) ≡h(0) >0 on E,then the implication (1) ⇒(2) is obvious.If
h(t) ≡h(0) on E,then this implication follows from statement 4 of Theorem 3.1 for the choice
ϕ(s) =e
−s
∈C([0,+∞)) ∩M
(0,∞)
.
The reverse implication (2) ⇒(1) follows fromLemma 2.2 with the choice ϕ(s) =e
−s
,from
the equality

R
d
e
−1/(2σ)y
2
e
i(y,v)
dy =(2πσ)
d/2
e
−σ/2v
2
,v ∈R
d
,σ >0,
464 V.P.Zastavnyi and E.Porcu
and fromthe Bernstein–Widder theorem.￿
The next theoremis an addition to Theorem3.1 for the special case ρ(x) =x,that is,when
ρ is the Euclidean norm.If f(x) = ϕ(x
2
),ϕ ∈ C([0,+∞)),f ∈ L(R
n
),then the Fourier
transformabove simplifies to the Bessel integral

f(u) =(2π)
n/2
g
n
(u),where g
n
(s):=


0
ϕ(u
2
)u
n−1
j
n/2−1
(su) du (3.4)
and j
λ
(u):= u
−λ
J
λ
(u) with J
λ
a Bessel function of the first kind.In this case,the functions
G
n
(·) and g
n
(·) are related by the known equality G
n
(v) =(2π)
n/2
g
n
(v).
Theorem3.2.Let the following conditions be satisfied:
(1) h ∈C(E),h(t) >0 for all t ∈E and h(t) ≡h(0) on E;
(2) ϕ ∈C([0,+∞)),ϕ(0) >0;
(3) K(x,t):=(h(t))
−d/2
ϕ(
x
2
h(t)
) ∈(R
d
×E).
If


0
|ϕ(u
2
)|u
m−1
du <∞for some m∈ {1,...,d} and g
m
is real analytic,then the function
f
m
(s):=s
m−d
g
m
(s) is strictly decreasing on (0,∞) and g
m
(s) >0 for all s >0.
Proof.From Theorem 3.1,we have that f
m
decreases on (0,∞) and f
m
(s) ≥f
m
(∞) =0 for
s >0.Since f
m
is real-analytic on (0,∞),the function f
m
(s) is strictly decreasing on (0,∞).
Otherwise,the function f
m
is constant on some open interval (α,β) ⊂(0,∞),α <β,and,hence,
it is constant on (0,∞) and f
m
(s) =f
m
(∞) =0,s >0.Therefore,G
m
(v) =(2π)
m/2
g
m
(v) ≡
0 on R
m
.Hence,ϕ(x
2
) ≡0 on R
m
,which contradicts the condition ϕ(0) >0.Thus,the func-
tion f
m
is strictly decreasing on (0,∞) and,hence,f
m
(s) >f
m
(∞) =0 for all s >0.￿
Remark 3.1.The necessary conditions stated in Theorems 3.1 and 3.2 allow the following hy-
pothesis to be formulated.
Let the following conditions be satisfied:
(1) h ∈C(E) and h(t) >0 for all t ∈E;
(2) ϕ ∈C([0,+∞)),ϕ(0) >0 and ϕ has compact support;
(3) K(x,t):=(h(t))
−d/2
ϕ(
||x||
2
h(t)
) ∈(R
d
×E),d ∈N.
We then conjecture that h(t) ≡h(0) on E.
From Theorem 3.2,a weaker version of this hypothesis can be formulated:under the three
conditions stated above,and if:
(4) g
m
(s
0
) =0 for some m∈ {1,...,d} and for some s
0
>0,then we conjecture that h(t) ≡
h(0) on E.
Let us assume that h(t) ≡h(0) on E.Then (see Theorem 3.2) g
m
(s) >0 for all s >0,which
contradicts condition (4).
As an example,it is possible to take the function ϕ(u
2
):=(1 −|u|)
+
∈ (R),m=1.In this
case (see (3.4)),g
1
(s) =

2
π

1
0
(1 −u) cos(su) du =

2
π
1−cos s
s
2
.Condition (4) is fulfilled for
Characterization theorems of space–time 465
m=1 and s
0
=2π.Therefore,(h(t))
−1/2
ϕ(
x
2
h(t)
) ∈ (R×E),where h ∈ C(E) and h(t) >0
for all t ∈E ⇐⇒ h(t) ≡h(0) on E.
Acknowledgements
The authors are grateful to Daryl Daley for interesting discussions during the preparation of this
paper.They would also like to thank the Associate Editor and the two referees,whose remarks
allowed for a considerable improvement of an earlier version of the manuscript.Emilio Porcu
acknowledges the DFG-SNF Research Group FOR916,subproject A2.
References
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tist.Assoc.97 590–600.MR1941475
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(English translation in St.Petersburg Math.J.3 563–570).MR1150554
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325 901–903 (English translation in Russian Acad.Sci.Dokl.Math.46 (1993) 112–114).MR1198176
[13] Zastavnyi,V.P.(1993).Positive definite functions depending on the norm.Russian J.Math.Phys.1
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MR1766121
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Jaume I.
Received May 2009 and revised January 2010
Bernoulli 17(1),2011,441–455
DOI:10.3150/10-BEJ277
FromSchoenberg to Pick–Nevanlinna:
Toward a complete picture of the variogram
class
EMILIO PORCU
1
and RENÉ L.SCHILLING
2
1
Department of Mathematics,University Jaume I of Castellón,Campus Riu Sec,E-12071 Castellón,Spain.
E-mail:eporcu@uni-goettingen.de
2
Institute of Mathematical Stochastics,Technical University Dresden,D-01062 Dresden,Germany.
E-mail:rene.schilling@tu-dresden.de
We show that a large subclass of variograms is closed under products and that some desirable stability
properties,such as the product of special compositions,can be obtained within the proposed setting.We
introduce new classes of kernels of Schoenberg–Lévy type and demonstrate some important properties of
rotationally invariant variograms.
Keywords:complete Bernstein functions;isotropy;Schoenberg–Lévy kernels;variograms
1.Introduction
Positive and conditionally positive definite functions on groups or semigroups have a long his-
tory and appear in many applications in probability theory,operator theory,potential theory,
moment problems and various other areas.They constitute an important chapter in all treatments
of harmonic analysis and their origins can be traced back to papers by Carathéodory,Herglotz,
Bernstein and Matthias (see [3] and references therein),culminating in Bochner’s theoremfrom
1932;see the surveys by Berg [3] and Sasvári [28].Schoenberg’s theorem explains the possi-
bility of constructing rotationally invariant positive definite and (the negatives of) conditionally
positive definite functions on Euclidean spaces via completely monotone functions and Bernstein
functions.Positive and conditionally positive definite functions are a cornerstone of spatial statis-
tics where they are known,respectively,as covariances (or kernels) and variograms.The theory
of random fields,which began in the 1940s with the early works of Kolmogorov (see [10] and
references quoted therein) and was further developed by Gandin [13] and Matheron [24],among
others,is based on the specification of these classes.In particular,the kriging predictor,that is to
say,the best linear unbiased predictor,depends exclusively on the underlying covariance or vari-
ogramand we refer to the tour de force in Stein [33] for a rigorous assessment of this framework.
Let {Z(ξ),ξ ∈R
d
} be a stationary Gaussian randomfield.The associated covariance function
C:R
d
→Ris positive definite,that is,for any finite collection of points {ξ
i
}
n
i=1
∈R
d
,the matrix
(C(ξ
i
−ξ
j
))
n
i,j=1
is positive definite:
for all a
1
,a
2
,...,a
n
∈C
n
￿
i,j=1
a
i
C(ξ
i
−ξ
j
)
a
j
≥0.
1350-7265 © 2011 ISI/BS
442 E.Porcu and R.L.Schilling
Thus,a function C:R
d
→R is positive definite if and only if there exists a stationary Gaussian
random field having C(·) as covariance function.If C(·) is rotationally invariant,then the asso-
ciated Gaussian randomfield is called isotropic.
It is well known that the family of covariance functions is a convex cone which is closed under
products,pointwise convergence and scale mixtures;for these basic facts,the reader is referred
to standard textbooks on geostatistics such as Chilès and Delfiner [10].
Avariogramγ:R
d
→Ris the variance of the increments of an intrinsically stationary random
field,that is,for any two points ξ
1

2
∈R
d
,Var(Z(ξ
1
) −Z(ξ
2
)):=γ(ξ
1
−ξ
2
).Note that γ(0) =
0,γ(ξ) =γ(−ξ) and that −γ is conditionally positive definite,that is,for any finite collection
of points {ξ
i
}
n
i=1
∈R
d
,we have
for all a
1
,...,a
n
∈C such that
n
￿
i=1
a
i
=0,−
n
￿
i,j=1
a
i
γ(ξ
i
−ξ
j
)
a
j
≥0.(1)
With a slight abuse of notation,we will also use the name variogram for a function γ:R
d
→R
with γ(0) ≥ 0 and such that γ(ξ) −γ(0) is the variance of the increments of an intrinsically
stationary randomfield.
There is a close relationship between variograms γ and stationary covariance functions C.
The elementary estimate |C(ξ)| ≤C(0) =:Var Z shows that stationary covariance functions are
necessarily bounded;in particular,γ(ξ):= C(0) − C(ξ) is a variogram.Indeed,variograms
may be unbounded,as in the case of fractional Brownian motion.If,however,the variogram is
bounded,then it is necessarily of the form C(0) −C(ξ),ξ ∈R
d
,for some stationary covariance
function C(·);see,for instance,[10] or [4],Proposition 7.13,and for a more general result due
to Harzallah,see [18].
The terminology concerning positive and conditional positive definiteness is not uniform
throughout the literature;it depends very much on the mathematical context or the scientific
application.Christakos [11] and many other applied scientists use the notion of permissibility for
both concepts.We will use both conventions alongside each other whenever no confusion can
arise.
In this paper,we are mainly interested in rotationally invariant covariances and variograms.
This means that the associated Gaussian random field is weakly or intrinsically stationary and
isotropic.Isotropy and stationarity are independent assumptions,but we will assume both to keep
things simple.An isotropic covariance function,rescaled by its value at the origin,is the char-
acteristic function of a rotationally symmetric random vector on the sphere of R
d
.This class of
covariances is well understood and we refer to Gneiting [14,15] and the references therein for an
extensive survey of this topic.Much less is known about variograms.For instance,it is common
knowledge that the class of variograms is a convex cone which is closed in the weak topology of
pointwise convergence,but the product of two variograms is not necessarily a variogram.This is
a point that deserves a thorough discussion,in the light of a recent beautiful result in [23],The-
orem 3(i),where a simple permissibility condition is given for the product of two exponential
variograms composed with a homogeneous function.
We shall give a general answer to this question,as well as a complete characterization of those
variograms whose product is again permissible.We shall then focus on other challenging prob-
Pick–Nevanlinna variograms 443
lems related to special compositions of variograms,as well as to quasi-arithmetic compositions
of them.
The use of kernels of Schoenberg–Lévy type has been persistently emphasized in both old and
recent literature.In this paper,we give new forms of kernels of this type that may be appealing
for modeling in spatial statistics.
Another crucial problem faced in this paper regards the potential trade-off between,on the
one hand,the computational advantages induced by the use of compactly supported kernels and,
on the other hand,the fact that compactly supported kernels can be positive definite only on
finite-dimensional spaces,by a striking and beautiful result due to Wendland [35].We consider
this problemfromthe point of viewof variograms;this makes sense since variograms,which are
possibly unbounded,represent a larger class than covariance functions.
The paper is organized as follows.Section 2 contains the basic material required for a self-
contained exposition and for understanding the technical proofs of our statements.Section 3
assesses new stability properties of the variogram class,while Section 4 is dedicated to kernels
of Schoenberg–Lévy type.
2.Complete Bernstein functions and complete monotonicity
This section is mainly expository and we collect here some basic material needed later.We will
frequently use the following characterization of variograms,for which a proof can be found in
[4],Proposition 7.5.
Theorem1.A function γ:R
d
→R is a variogram if and only if the following three conditions
are satisfied:
(i) γ(0) ≥0;
(ii) γ(ξ) =γ(−ξ);
(iii) −γ is conditionally positive definite,that is,equation (1) holds for all ξ
1
,...,ξ
n
∈R
d
.
Let us remark that in harmonic analysis,functions satisfying conditions (i)–(iii) of Theorem1
are often called negative definite functions.We will not use this notion in this paper.
Often,Pólya’s theorem (see [4],Theorem 5.4) is useful if one wants to construct concrete
examples of variograms.
Theorem2.A continuous function φ:R→[0,∞) which is even (i.e.,φ(x) =φ(−x)),decreas-
ing and convex on the interval (0,∞) is positive definite.
Clearly,φ(0) −φ(x) is increasing,concave and a variogram;see,for example,[4],Corol-
lary 7.7.
Recall that a function f:(0,∞) →R is called completely monotone if it is arbitrarily often
differentiable and
(−1)
n
f
(n)
(x) ≥0 for x >0,n =0,1,....
444 E.Porcu and R.L.Schilling
By Bernstein’s theorem,the set CMof completely monotone functions coincides with the set of
Laplace transforms of positive measures μ on [0,∞),that is,
f(x) =Lμ(x) =
￿
[0,∞)
e
−xt
dμ(t),
where we only require that e
−xt
is μ-integrable for any x >0.CMis a convex cone which is
closed under multiplication and pointwise convergence.
Definition 3.A function f:(0,∞) →R is called a Stieltjes function if it is of the form
f(x) =a +
￿
[0,∞)
dμ(t)
x +t
,(2)
where a ≥0 and μ is a positive measure on [0,∞) such that
￿
[0,∞)
(1 +t)
−1
dμ(t) <∞.
The following properties of the family S of Stieltjes functions can be found in [4],Section 14,
and [3].S is a convex cone such that S ⊂ CM.For every f ∈ S,the fractional power f
α

S ⊂CM,0 <α ≤1,is again a Stieltjes function.Thus,for f ∈S,we see that f
α
is completely
monotone for any α > 0,so f belongs to the set L of logarithmically completely monotone
functions discussed in,for example,[3],Section 2.6.The formula
1
x(1 +x
2
)
=
￿
[0,∞)
e
−xt
(1 −cos t) dt
shows that x
−1
(1 +x
2
)
−1
is completely monotone;however,it cannot be a Stieltjes function
since it has poles at ±i and (2) indicates that a Stieltjes function has a holomorphic extension
to the cut plane C\(−∞,0].From the integral representation of f,it is immediate that this
extension satisfies ImzImf(z) ≤0,that is,f maps the upper complex half-plane to the lower
and vice versa.
Definition 4.A function f:(0,∞) →[0,∞) is called a Bernstein function if it is infinitely often
differentiable and f

∈CM.
The set of Bernstein functions is denoted BF;it is a convex cone which is closed under
pointwise convergence.Since a Bernstein function is non-negative and increasing,it has a non-
negative limit f(0+).Integrating the Bernstein representation of the completely monotone func-
tion f

gives the following integral representation of f ∈BF:
f(x) =αx +β +
￿
(0,∞)
(1 −e
−xt
)ν(dt),(3)
where α,β ≥0 are constants and ν is the Lévy measure,that is,a positive measure on (0,∞)
satisfying
￿
(0,∞)
t
1 +t
ν(dt) <∞.
Pick–Nevanlinna variograms 445
The following composition result will be useful throughout the paper;see [3].
Theorem5.Let X be either of the sets BF,CM.Then
f ∈X,g ∈BF ⇒ f ◦ g ∈X.
If we assume that the representing measure ν(dt) in (3) is of the form ν(dt) =m(t) dt,where
m(t) is completely monotone,then we get the family of complete Bernstein functions.We denote
the collection of all complete Bernstein functions by CBF.It is not hard to see that CBF is,like
BF,a convex cone which is closed under pointwise limits.Complete Bernstein functions are
widely used in various fields and they are closely related to the following concepts:Bondesson
T
2
-class (see [9] for the original definition and [5] for further information),operator-monotone
functions (the classical source is [12]) and Pick functions (which are also known as Nevanlinna
functions,i.e.,holomorphic functions in the upper half-plane with non-negative imaginary part
there).A detailed survey can be found in [29],and short introductions in [3,20,30].Among the
most prominent examples of complete Bernstein functions are
x →
λx
λ +x
(λ >0),x →x
α
(0 <α <1),
x →log(1 +x),x →

x arctan
1

x
.
Further examples are given belowin Table 1.Many Bernstein functions given in closed formare
already in CBF.There are not many known examples of functions in BF\CBF and they are all
finite or infinite sums of the form
￿
i
p
i
(1−e
−λ
i
x
);see [3].Some interesting examples are given
in terms of the q-versions of the digamma function ψ
q
(x) and Euler’s constant γ
q
:the function
x →ψ
q
(x +1) +γ
q
is in BF\CBF;see [22].
1
The following statements are taken from[29].
Table 1.Examples of complete Bernstein functions (
(a;x):=
￿

x
t
a−1
e
−t
dt is the incomplete Gamma
function)
Function Parameter restrictions Function Parameters restriction
1 −
1
(1+x
α
)
β
0 <α,β ≤1 ex −x(1 +
1
x
)
x

x
x+1
(
x
ρ
1+x
ρ
)
γ
0 <γ,ρ <1
1
a

1
x
log(1 +
x
a
) a >0
x
α
−x(1+x)
α−1
(1+x)
α
−x
α
0 <α <1
￿
x
2
sinh
2

2x
sinh(2

2x)

x(1 −e
−2a

x
) a >0 x
1−ν
e
ax

(ν;ax) a >0,0 <ν <1
x(1−e
−2

x+a
)

x+a
a >0 x
ν
e
a/x

(ν;
a
x
) a >0,0 <ν <1
1
We are grateful to a referee supplying this reference.
446 E.Porcu and R.L.Schilling
Theorem 6.A function f:(0,∞) →[0,∞) such that f(0+) exists is a complete Bernstein
function if and only if it has an analytic extension to the cut complex plane C\(−∞,0] such that
Imz · Imf(z) ≥0,that is,f preserves upper and lower half-planes.In particular,all complete
Bernstein functions are of the form
f(z) =bz +a +
￿
(0,∞)
z
z +t
σ(dt) (4)
with a,b ≥0 and a measure σ satisfying
￿
(0,∞)
(1 +t)
−1
dt <∞.
Proofs of this classic result can also be found in [3,20,30].Theorem 6 can be used to show
that,for any f ≡0,
f ∈CBF ⇐⇒
￿
x →
f(x)
x
￿
∈S ⇐⇒
￿
x →
x
f(x)
￿
∈CBF ⇐⇒
1
f
∈S.
(5)
Let us briefly indicate the argument:if f ∈CBF,then we can use (4) and divide by z.Comparing
the resulting formula with (2) reveals that f(z)/z is (the extension to C\(−∞,0] of) a Stieltjes
function.Therefore (see the comment following Definition 3),we know that f(z)/z maps the
upper to the lower complex half-plane.Consequently,the inverse g(z):=z/f(z) preserves up-
per and lower half-planes and is,by Theorem 6,in CBF.Using the integral representation (4)
for g and dividing by z,we get that g(z)/z =1/f(z) is (the extension of) a Stieltjes function.
As before,we see that f = 1/(1/f) preserves upper and lower half-planes and is,therefore,
a complete Bernstein function.This proves all equivalences in (5).
Using the fact that (the extensions of) functions in CBF preserve,and those in S swap,com-
plex half-planes,we immediately get the following result.If we let X be either CBF or S,then
f,g ∈X ⇒ f ◦ g ∈CBF.
The following stability properties are less obvious.
Theorem7.Let f,g,h ∈CBF and f ≡0.Then:
(i) (f
α
(x) +g
α
(x))
1/α
∈CBF for all α ∈[−1,1]\{0};
(ii) (f(x
α
) +g(x
α
))
1/α
∈CBF for all α ∈[−1,1]\{0};
(iii) f(x
α
) · g(x
1−α
) ∈CBF for all α ∈[0,1];
(iv) h(f(x)) · g(
x
f(x)
) ∈CBF.
Assertion (iv) was discovered by Uchiyama [34],Lemma 2.1,and since fractional powers
f(x) =x
α
,0 ≤α ≤1,are in CBF,(iv) implies (iii).For positive α >0,assertions (i),(ii) are
in [26] – his proofs are easily adapted to α <0 since f ∈ CBF if and only if 1/f ∈ S;see (5).
A unified approach will appear in [29].
Letting α →0 in Theorem7 proves lim
α↓0
(
1
2
f
α
+
1
2
g
α
)
1/α
=

fg and since pointwise limits
of complete Bernstein functions are complete Bernstein,we see that

fg ∈ CBF whenever
Pick–Nevanlinna variograms 447
f,g ∈ CBF.From this,we can easily deduce a new proof of the so-called log-convexity of the
convex cone CBF:
f,g ∈CBF,α ∈[0,1] ⇒ f
α
· g
1−α
∈CBF.(6)
Alternative proofs can be found in [2] and [29].
Indeed,if α is a dyadic number of the formα =
￿
n
i=1
α
i
2
−i
with α
i
∈{0,1} and α
n
=1,then
α

=1 −α is of the same type with α

n
=1.This is because
α

=

￿
i=1
2
−i

n
￿
i=1
α
i
2
−i
=
n−1
￿
i=1
(1 −α
i
)2
−i
+

￿
i=n+1
2
−i
=
n−1
￿
i=1
α

i
2
−i
+2
−n
with α

i
=1 −α
i
,i =1,...,n −1.This means that
f
α
g
1−α
=
n
￿
i=1
2
i
￿
f
α
i
g
α

i
=
￿
￿
￿
￿
h
1
￿
h
2
· · ·
￿
h
n−2
￿
h
n−1
￿
f
n
g
n
,
where h
i
stands for either f
i
or g
i
if α
i
=1 or α
i
=0,respectively.Thus,repeated applications
of (6) with α =α

=
1
2
lead to (6) for all dyadic α ∈(0,1).Since (0,1) α →f
α
is continuous,
we get (6) for all α ∈(0,1).
3.Variograms and their stability properties
As already emphasized in Section 1,the starting point for this work is a result in [23],Theo-
rem3(i),which is reported below with a short alternative proof.
Theorem8 ([23]).Let γ:R
d
→R be a homogeneous function.Then
￿
1 −e
−a
1
γ(ξ)
￿￿
1 −e
−a
2
γ(ξ)
￿
,(7)
a
i
>0,i =1,2,is a variogramif and only if γ(ξ) =|Aξ| for the Euclidean norm| · | and a d ×d
matrix A.
It is natural to ask whether Ma’s theoremworks only for the exponential class of variograms or
whether it can be generalized.The subsequent result gives an answer to this problem,supplying
a wide class of variograms closed under products.
Here and hereafter,we will use a famous result of Schoenberg and Bochner;see [31] (in the
context of covariance functions and complete monotonicity) and [8],page 99 (in the context of
variograms and Bernstein functions).We restate Bochner’s version in the setting of the current
paper.Alternative proofs can be found in the Appendix of Jacob and Schilling [21] and Steerne-
man and van-Perlo-ten Kleij [32].
448 E.Porcu and R.L.Schilling
Lemma 9.All variograms γ which are rotationally invariant and permissible in all (or at least
infinitely many) dimensions d =1,2,...are of the formγ(ξ) =f(|ξ|
2
) with a Bernstein function
f ∈BF.
The next result is not only a generalization of Ma’s result,but also the key to a simple proof
of Theorem8.
Theorem 10.Let g
1
,g
2
be Bernstein functions and 0 ≤ α
1

2
such that α
1
+ α
2
≤ 1.Then
g
1
(x
α
1
)g
2
(x
α
2
) is a Bernstein function.
Proof.Set h
α,β
(x):=g
1
(x
α
) · g
2
(x
β
),x >0.It is enough to show that h

α,β
∈CM.Clearly,
h

α,β
(x) =x
α+β−1
￿
αg

1
(x
α
)
g
2
(x
β
)
x
β
+βg

2
(x
β
)
g
1
(x
α
)
x
α
￿
.
Since g
i
∈ BF,we have that g

i
∈ CM and x
−1
g
i
(x) ∈ CM.This will also be the case for
the compositions g

1
(x
α
) and g

2
(x
β
),g
1
(x
α
)/x
α
and g
2
(x
β
)/x
β
,by a straightforward applica-
tion of Theorem 5.Moreover,for α +β ≤1,x →x
α+β−1
is completely monotone.The proof
is completed since completely monotone functions form a convex cone which is closed under
products.￿
Corollary 11.Let R
d
ξ →γ
i
(ξ) =g
i
(|ξ|
2
) be rotationally invariant variograms for all d ∈N,
i =1,2.Let α,β ∈[0,1] be such that α +β ≤1 and let A be a d ×d matrix.Then
f
α,β
(ξ):=g
1
(|Aξ|

)g
2
(|Aξ|

)
is still a variogram on R
d
for all d ∈N.
Remark 12.The result of Theorem 10 extends immediately to the product of n Bernstein func-
tions:for
￿
n
i=1
α
i
≤1,α
i
≥0 and g
i
∈ BF,the function h(x):=
￿
n
i=1
g
i
(x
α
i
) is again in BF.
This generalizes the case where α
i
=
1
n
,g
i
= g,i = 1,...,n,leading to h(x) = (g(x
1/n
))
n
,
which is due to [7].
The proof of the result above offers a considerably easier way to show Ma’s result.
Proof of Theorem 8.If γ(ξ) =|Aξ|,Corollary 11 with g
i
(x) =1 −exp(−a
i
x),i =1,2 and
α =β =
1
2
shows that (7) is a variogram.
Now,assume that (7) is a variogram.Then
ξ →
(1 −e
−a
1
γ(ξ)
)
a
1
·
(1 −e
−a
2
γ(ξ)
)
a
2
Pick–Nevanlinna variograms 449
is a variogramfor all a
1
,a
2
>0 and so is its pointwise limit γ
2
(ξ) as a
1
,a
2
→0;thus,γ
2
(ξ) is
a real-valued variogram.As such,it has a Lévy–Khinchine representation
γ
2
(ξ) =Qξ · ξ +
￿
x=0
￿
1 −cos(x · ξ)
￿
ν(dx),
where Q∈R
d×d
is positive semi-definite and ν is a measure with
￿
x=0
|x|
2
∧1ν(dx) <∞.Since
γ(ξ) is homogeneous,we get
γ
2
(ξ) =
γ
2
(nξ)
n
2
n→∞
−→Qξ · ξ =
￿
￿
￿

￿
￿
2
for the uniquely determined,positive semidefinite square root A=

Qof Q.￿
Several examples of Bernstein functions may be found in [3,4] or in [21];an extensive list
will be included in the monograph [29].Three celebrated classes of Bernstein functions are well
known in the spatial statistics literature:
(1) the Matérn class [25] f
α,ν
=1 −2
1−ν
/
(ν)(α

x)
ν
K
ν


x),x >0,for α,ν >0 and
where K
ν
is the modified Bessel function of the second kind of order ν;
(2) the Cauchy class [16] f
α,β
(x):=1 −(1 +x
α
)
−β
,x >0,where 0 <α ≤1 and 0 <β;
(3) the Dagum class [6] f
ρ,γ
(x):=(
x
ρ
1+x
ρ
)
γ
,x >0,where ρ,γ ∈(0,1).
Let us mention a few more stability properties that make some classes of functions appealing
for their use in spatial statistics.We again work within the framework of rotationally invariant
functions.
Proposition 13.Let γ:R
d
→R be rotationally invariant for all dimensions d =1,2,...such
that γ(ξ) =g(|ξ|
2
) for some g ∈CBF.Then:
(i) R
d
 ξ →
|ξ|
2
g(|ξ|
2
)
is a rotationally invariant variogram which is permissible for every d ∈
N;
(ii) R
d
 ξ →
1
g(1/|ξ|
2
)
and ξ →|ξ|
2
g(
1
|ξ|
2
) are rotationally invariant variograms which are
permissible for every d ∈N.
Proof.Part (i) is a simple application of the first equivalence in (5) which states that g ∈CBF if
and only if g(x)/x is a Stieltjes function.
Part (ii) follows immediately by noting that,for g ∈ CBF,the function x →1/g(1/x) is
a composition of the type σ ◦ g ◦ σ(x),where σ is the Stieltjes function x →
1
x
.Since the
composition σ ◦ g is a Stieltjes function and since the composition of two Stieltjes functions is
in CBF,we have the first assertion of part (ii).If we apply part (i) to this variogram,the second
assertion follows.￿
For further (stability) properties of the class CBF,the reader is referred to [29];some examples
of complete Bernstein functions are given below.
450 E.Porcu and R.L.Schilling
Another interesting problem arises when quasi-arithmetic operators,in the sense of Hardy,
Littlewood and Pólya [17],are applied to variograms.This means that we seek conditions which
preserve the permissibility of the underlying structure.This has been considered in [27] for quasi-
arithmetic composition of covariance functions.We believe that the same question in connection
with variograms is even more challenging from the mathematical point of view and is equally
important as far as statistics are concerned.
Recall that a power mean is a mapping of the form (u,v) →ψ
α
(u,v):=(u
α
+v
α
)
1/α
for
(u,v) ∈R
2
and α ∈R\{0}.
Proposition 14.Let γ
i
:R
d
→R,i =1,2,be rotationally invariant variograms for all dimen-
sions d ∈N.We write g
i
for the radial function such that γ
i
(ξ) =g
i
(|ξ|
2
):
(i) If g
1
,g
2
∈CBF,then ξ →(γ
α
1
(ξ) +γ
α
2
(ξ))
1/α
is a variogram for all α ∈[−1,1]\{0}.
(ii) If g
1
,g
2
∈CBF,then ξ →(g
1
(|ξ|

) +g
2
(|ξ|

))
1/α
is a variogramfor all α ∈[−1,1]\
{0}.
(iii) ξ →g
1
(|ξ|

)g
2
(|ξ|
2−2α
) is a variogram for all 0 <α <1.
Proof.Since,by Lemma 9,g
i
∈BF,assertion (iii) is a simple consequence of Corollary 11.We
should mention at this point that for g
1
,g
2
∈CBF,the resulting rotationally invariant variogram
would again be of the form h(|ξ|
2
) with h ∈ CBF;see Theorem 7(iv).Both (i) and (ii) follow
immediately from7(i) and (ii),respectively.￿
Finally,we combine two aspects treated separately until now.Given two or three isotropic var-
iograms,we seek permissibility conditions for the products of special compositions.The propo-
sition below results from a simple application of Theorem 7(iv) with h(s) =s,f =g
1
,g =g
2
,
respectively,h =g
3
,f =g
1
,g =g
2
.
Proposition 15.Let R
d
 ξ →γ
i
(ξ),i =1,2,3,be rotationally invariant and isotropic vari-
ograms for all d ∈N and assume that γ
i
(ξ) =g
i
(|ξ|
2
),where g
i
∈CBF.Then
ξ →γ
1
(ξ)γ
2
￿
ξ

γ
1
(ξ)
￿
and γ
3
￿
￿
γ
1
(ξ)
￿
γ
2
￿
ξ

γ
1
(ξ)
￿
are still permissible for all d ∈N and of the form h(|ξ|
2
) with some h ∈CBF.
We conclude this section by presenting another curious way to construct continuous vari-
ograms and,more generally,complex-valued conditionally positive definite functions,with the
help of Bernstein functions.The interesting fact in the example below is the product structure,
which is quite unusual for conditionally positive definite functions.
Proposition 16.Let f be a Bernstein function such that the representing measure ν in the Lévy–
Khinchine formula (3) has a monotone decreasing density m,that is,f(x) =α+βx+
￿
(0,∞)
(1−
e
−xt
)m(t) dt.
Then ξ →iξf(iξ) is conditionally positive definite and ξ →−Re(iξf(iξ)) is a continuous
variogram.
Pick–Nevanlinna variograms 451
Proof.By the monotonicity of m,we see that m(t) =ν[t,∞) for a (Lévy) measure ν,that is,
a measure ν on (0,∞) satisfying
￿
(0,∞)
t (1 +t)
−1
ν(dt).The integration properties of ν become
clear fromthe calculation belowsince we have only used Fubini’s theoremfor positive integrands
to swap integrals.For x ≥0,we get
xf(x) =αx +βx
2
+
￿

0
x(1 −e
−xt
)
￿

t
ν(ds) dt
=αx +βx
2
+
￿

0
￿

t
x(1 −e
−xt
)ν(ds) dt
=αx +βx
2
+
￿

0
￿
s
0
x(1 −e
−xt
) dtν(ds)
=αx +βx
2
+
￿

0
[e
−xs
−1 +sx]ν(ds),
which,as by-product,shows that
￿

0
s
2
∧sν(ds) <∞.We may,therefore,plug in z =iξ and
get
iξf(iξ) =−
￿
−iαξ +βξ
2
+
￿

0
[1 −e
−isξ
−isξ]ν(ds)
￿
.
Thus,−γ(ξ):=iξf(iξ) is conditionally positive definite and Reγ(ξ) is a variogram.￿
4.Kernels and variograms of the Schoenberg–Lévy type
This section explores some results that may be obtained when working with kernels of the
Schoenberg–Lévy type.These kernels are extensively used in the literature and we refer to Ma
[23] and the references therein.For ξ
1

2
∈ R
d
,these are non-stationary covariance functions
obtained froma non-negative function g:[0,∞) →[0,∞) such that g(0) =0 through the linear
combination
g(|ξ
1
|) +g(|ξ
2
|) −g(|ξ
1
−ξ
2
|).
A celebrated example is the fractional Brownian sheet [1] with g(ξ) =|ξ|
α
,α ∈ (0,2].Ma [23]
shows that for a fixed ξ
0
∈R
d
,the function
C
ξ
0
(ξ):=g(|ξ +ξ
0
|) +g(|ξ −ξ
0
|) −2g(|ξ|)
is a covariance function,provided that g(|ξ|) is a variogram.Indeed,we are going to show that
this is a simple consequence of the following,more general,result.
Lemma 17.Let γ:R
d
→R be a continuous variogram and let ξ,η ∈R
d
,d ∈N.Then
φ
η
(ξ):=γ(ξ +η) +γ(ξ −η) −2γ(ξ)
452 E.Porcu and R.L.Schilling
is a continuous covariance function as a function of ξ.Moreover,if
γ
η
(ξ):=2γ(η) +2γ(ξ) −γ(ξ +η) −γ(ξ −η),
then ξ →γ
η
(ξ) is a continuous variogram.
Note that in Lemma 17,we have γ
η
(ξ) =γ
ξ
(η),that is,η →γ
η
(ξ) is also a continuous vari-
ogram.
Proof of Lemma 17.Recall the following elementary formula for the cosine:cos(a + b) +
cos(a −b) =2cos a cos b.Since γ(ξ) has the Lévy–Khinchine representation
γ(ξ) =Qξ · ξ +
￿
x=0
(1 −cos x · ξ)ν(dx),
we find that
φ
η
(ξ) =Q(ξ +η) · (ξ +η) +Q(ξ −η) · (ξ −η) −2Qξ · ξ
+
￿
x=0
￿
2cos x · ξ −cos x · (ξ +η) −cos x · (ξ −η)
￿
ν(dx)
=2Qη · η +
￿
x=0
(2cos x · ξ −2cos x · ξ cos x · η)ν(dx)
=2Qη · η +2
￿
x=0
(1 −cos x · η) cos x · ξν(dx).
This shows that ξ →φ
η
(ξ) is symmetric and positive definite,hence a covariance function.Now,
consider
γ
η
(ξ) =2γ(η) −φ
η
(ξ)
=2a +2Qη · η +2
￿
x=0
(1 −cos x · η)ν(dx)
−2Qη · η −2
￿
x=0
(1 −cos x · η) cos x · ξν(dx)
=2a +2
￿
x=0
(1 −cos x · η)(1 −cos x · ξ)ν(dx).
Thus,γ
η
(ξ) is a variogramin ξ.The proof is thus complete.￿
Lemma 17 has an obvious extension to continuous complex-valued functions γ:R
d
→C sat-
isfying γ(0) ≥ 0,γ(ξ) =
γ(−ξ) and the permissibility condition (1) for all ξ
1
,...,ξ
n
∈ R
d
.
Since such functions also enjoy a (complex) Lévy–Khinchine representation (see [4]),exactly
Pick–Nevanlinna variograms 453
the same argument as in the proof of Lemma 17 shows that for every fixed ξ
0
∈R
d
,
γ
ξ
0
(ξ):=2γ(ξ) +2Reγ(ξ
0
) −γ(ξ −ξ
0
) −γ(ξ +ξ
0
)
is permissible and has the Lévy–Khinchine representation
γ
ξ
0
(ξ) =2
￿
y=0
(1 −e
iy·ξ
)
￿
1 −cos(y · ξ
0
)
￿
ν(dy),
where ν is the Lévy measure of γ.Lemma 17 is a very special case of [4],Proposition 18.2,
which goes back to Harzallah [19].
Acknowledgements
We are grateful to three anonymous referees for their valuable comments which helped to im-
prove the presentation of this paper.This work was initiated when the first-named author was
visiting the Technical University of Dresden.He is grateful to Professor Zoltán Sasvári for the
invitation and the hospitality he was shown.Emilio Porcu acknowledges the DFG-SNF Research
Group FOR916,subproject A2.
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Received December 2008 and revised March 2010