Excursion probability of Gaussian random fields on sphere

0.2 MB·

by  Dan Cheng

Excursion Probability of Gaussian Random Fields on Sphere
Dan Cheng Yimin Xiao ∗
North Carolina State University Michigan State University
4
1 January 23, 2014
0
2

n
a
J Abstract

1
2 Let X = X(x) : x SN be a real-valued, centered Gaussian random field in-
dexedontheN{-dimension∈alunit}sphereSN. Approximationstotheexcursionprobability
]
R P supx∈SN X(x) u , as u , are obtained for two cases: (i) X is locally isotropic
≥ → ∞
P
and its sample path is non-smooth and; (ii) X is isotropic and its sample path is twice
(cid:8) (cid:9)
.
h differentiable. For case (i), it is shown that the asymptotics is similar to Pickands’ ap-
t
a proximationontheEuclideanspacewhichinvolvesPickands’constant;whileforcase(ii),
m
we use the expected Euler characteristicmethod to obtain a more precise approximation
[
such that the error is super-exponentially small.

1
v
Key Words: Excursion probability, Gaussian random fields on sphere, Pickands’ constant, Euler
8
9 characteristic.
4
5
.
1 1 Introduction
0
4
1 Eventhoughthecharacterizations ofisotropiccovariancefunctionsandvariogramsonspheres
:
v weregivenlongtimeagobySchoenberg(1942) andGangolli(1967), respectively, andrandom
i
X
fields on the sphere were studied by Obukhov (1947), Yaglom (1961) and Jones (1963), it
r
a is the applications in atmospherical sciences, geophysics, solar physics, medical imaging and
environmental sciences [see, e.g., Genovese et al. (2004), Oh and Li (2004), Stein (2007),
Cabella and Marinucci (2009), Tebaldi and Sans´o (2009), Hansen et al. (2012)] that have
stimulated therecent rapiddevelopments in statistics of randomfieldson thesphere. Various
new random field models have been constructed and new probabilistic and statistical meth-
ods have been developed. For example, Jun and Stein (2007, 2008), Huang et al. (2009),
Jun (2011), Hitczenko and Stein (2012), Ma (2012), Du et al. (2013) and Gneiting (2013)
have constructed severalclasses of realorvector-valued randomfieldsonspheres; Istas (2005,
∗Research partially supported byNSFgrants DMS-1309856 and DMS-1307470.
1
2006)hasconstructedsphericalfractionalBrownianmotion(SFBM),whichhasfractalsample
functions,andstudiedits Karhunen-Lo`eveexpansion andother properties. LangandSchwab
(2013) characterized sample H¨older continuity and sample differentiability of isotropic Gaus-
sian random fields on the two-dimensional sphere S2 in terms of their angular power spectra.
We refer to the recent book by Marinucci and Peccati (2011) for a systematic account on
theory and statistical inferences of random fields on the sphere SN, with a view towards
applications to cosmology.
In this paper we consider a real-valued, centered isotropic Gaussian random field X =
X(x) :x SN , indexed on the N-dimensional unit sphere SN, and investigate the asymp-
{ ∈ }
toticpropertiesoftheexcursionprobabilityP supx∈SN X(x) ≥ u asu → ∞. Suchexcursion
probabilities are important in probability theory, statistics and their applications. In par-
(cid:8) (cid:9)
ticular, we mention that the above excursion probability has appeared in Sun (1991), Park
and Sun (1998) to determine the P-value for studying exploratory projection pursuit and,
as illustrated by Sun (2001), is useful for constructing simultaneous confidence region for a
function f : SN R. In his studies of projection-based depth functions, Zuo (2003) has
→
shown that Gaussian random fields on sphere appear as scaling limit of sample projection
median [see Theorems 3.2 and 3.3 in Zuo (2003)] and the excursion probability of the limit-
ing Gaussian field is useful for constructing confidence regions for the true projection median
[see Remark 3.2 in Zuo (2003)]. For further information on extreme value theory of Gaussian
random fields on Euclidean spaces or manifolds and statistical applications, we refer to Adler
and Taylor (2007) and Adler et al. (2012).
We will distinguish two cases: (i) the sample path of X is non-smooth and, (ii) X() C2
· ∈
a.s. For the non-smooth case, our argument is based on the extension of the asymptotic
results of Pickands (1969) and Piterbarg (1996) by Chan and Lai (2006). The main result
for Case (i) is Theorem 2.4, where one may observe that the asymptotics is very similar to
the classical Pickands’ approximation for locally isotropic Gaussian fields on the Euclidean
space [see Theorem 2.2]. The method for deriving Theorem 2.4 can also be applied to other
Gaussian fields on sphere with more complicated local covariance structures, see Section
2.2 for the example of standardized spherical fractional Brownian motion. For the smooth
case, we consider isotropic Gaussian fields on sphere. Thanks to the special representation
of covariance function [Theorem 3.1], we are able to apply the general theory of Adler and
Taylor (2007) to compute the Lipschitz-Killing curvatures induced by the field and hence
derive the approximation to the excursion probability, see Theorem 3.7 and Corollary 3.9
below. Such approximation is more precise than that in Theorem 2.4 for the non-smooth
case and the error is super-exponentially small.
We should mention that Mikhaleva and Piterbarg (1997) have established asymptotic
resultsfortheexcursionprobability ofGaussian randomfieldson afinitedimensionalsmooth
2
manifold in RN+1. Their theorems can be applied to obtain results similar to Theorem 2.4 in
this paper for a Gaussian random field X on the sphere SN, provided X is the restriction on
SN of a Gaussian randomfield definedon RN+1 and satisfies theconditions on the covariance
structures(i.e., thelocalstationarity condition)inMikhalevaandPiterbarg(1997). Sincenot
every Gaussian random field on SN can be obtained as the restriction of a Gaussian random
field on RN+1 [cf. e.g., Ma (2012, p. 775)] and the conditions on the covariance structure in
Mikhaleva and Piterbarg (1997) are not always easy to verify, we have decided in the present
paper to deal with Gaussian random fields on sphere directly.
For x = (x ,...,x ) SN, its corresponding spherical coordinate θ = (θ ,...,θ ) is
1 N+1 1 N
∈
defined as follows.
x = cosθ ,
1 1
x = sinθ cosθ ,
2 1 2
x = sinθ sinθ cosθ ,
3 1 2 3
(1.1)
.
.
.
x = sinθ sinθ sinθ cosθ ,
N 1 2 N−1 N
···
x = sinθ sinθ sinθ sinθ ,
N+1 1 2 N−1 N
···
where 0 θ π for 1 i N 1 and 0 θ < 2π.
i N
≤ ≤ ≤ ≤ − ≤
Throughout this paper, for two points x = (x ,...,x ) and y = (y ,...,y ) on SN,
1 N+1 1 N+1
we always denote by θ = (θ ,...,θ ) the spherical coordinate of x and by ϕ = (ϕ ,...,ϕ )
1 N 1 N
the spherical coordinate of y respectively.
Let , , be respectively the Euclidean norm and the inner product in RN+1 (or in
k·k h· ·i
RN, which will be clear from the context). Denote by d(, ) the spherical distance on SN,
· ·
i.e., d(x,y) = arccos x,y , x,y SN. For two functions f(t) and g(t), we say f(t) g(t)
h i ∀ ∈ ∼
as t t [ ,+ ] if lim f(t)/g(t) = 1.
→ 0 ∈ −∞ ∞ t→t0
2 Non-smooth Gaussian Fields on Sphere
2.1 Locally Isotropic Gaussian Fields on Sphere
Let X = X(x) : x SN be a centered Gaussian random field with covariance function C
{ ∈ }
satisfying
C(x,y) = 1 cdα(x,y)(1+o(1)) as d(x,y) 0, (2.1)
− →
for some constants c > 0 and α (0,2].
∈
Covariance functions satisfying (2.1) behave isotropically in a local sense, hence the cor-
responding random fields fall under the general category of locally isotropic random fields.
3
Similarly to Gaussian random fields defined on the Euclidean space [cf. Adler (2010)], one
can show that, when α (0,2), the sample function of X is not differentiable and the fractal
∈
dimensions of its trajectories are determined by α. See Andreev and Lang (2013), Hansan et
al. (2012) and Lang and Schwab (2013) for related results.
There are many examples of covariances of isotropic Gaussian fields on SN satisfying
(2.1). A well-known example is C(x,y) = e−cdα(x,y), where c > 0 and α (0,1] [cf. e.g.,
∈
Huang et al. (2011, p. 725)]. In their studies on germ-grain (or random balls) models on
the sphere SN, Estrade and Istas (2010, Remark 2.5 and Lemma 3.1) discovered an isotropic
Gaussian random field Wβ on SN with 0 < β < 1/2, whose covariance function satisfies
(2.1) for α = 2β (0,1]. [From here one can show that, even though Wβ and the spherical
∈
fractional Brownian motion B (x) introduced by Istas (2005) are different, they share some
β
local properties (e.g., they have the same H¨older continuity and fractal dimensions). In
Remark 2.6 below, we will compare the excursion probabilities of Wβ and the standardized
SFBM.] Moreover, as in Yadrenko (1983) and Ma (2012), one can apply the identity
d(x,y)
x y = 2sin , x, y SN
k − k 2 ∀ ∈
(cid:16) (cid:17)
to construct covariance functions that satisfy (2.1) from isotropic covariance functions K()
·
on RN which satisfy K(x) = 1 c x α(1+o(1)) as x 0. In particular, the following
1
− k k k k →
covariance function C given by Soubeyrand et al. (2008)
α
d(x,y)
C(x,y)= 1 sin 1l , (2.2)
− c1/α {d(x,y)≤πc1/α}
(cid:18) (cid:19)
wherec > 0andα (0,2)areconstants, satisfies(2.1). SeeHuangetal. (2011) andGneiting
∈
(2013) for further comments on (2.2) and more examples.
Recall the spherical coordinate representation in (1.1), we define the Gaussian random
field X = X(θ) : θ [0,π]N−1 [0,2π) by X(θ) := X(x) and denote by C the covariance
{ ∈ × }
function of X accordingly.
e e e e
The following lemma characterizes the local behavior of spherical distance of two points
e
on sphere. It provides a useful tool to establish the relation between the local behavior of
the covariance functions C and C.
Lemma 2.1 Let x,y SN and leet x be fixed. Then as d(y,x) 0,
∈ →
N−1
d2(y,x) (ϕ θ )2+(sin2θ )(ϕ θ )2+ + sin2θ (ϕ θ )2,
1 1 1 2 2 i N N
∼ − − ··· −
(cid:18) i=1 (cid:19)
Y
where θ = (θ ,...,θ ) and ϕ = (ϕ ,...,ϕ ) are the spherical coordinates of x and y respec-
1 N 1 N
tively.
4
Proof For x,y SN, we see that d(y,x) 0 implies y x 0 and
∈ → k − k →
1
cos y x 1 y x 2 = y,x .
k − k ∼ − 2k − k h i
Then as d(y,x) 0,
→
d2(y,x) = arccos2 y,x y x 2
h i ∼ k − k
= (cosϕ cosθ )2+(sinϕ cosϕ sinθ cosθ )2+
1 1 1 2 1 2
− − ···
+(sinϕ sinϕ sinϕ cosϕ sinθ sinθ sinθ cosθ )2
1 2 N−1 N 1 2 N−1 N
··· − ···
+(sinϕ sinϕ sinϕ sinϕ sinθ sinθ sinθ sinθ )2
1 2 N−1 N 1 2 N−1 N
··· − ···
= 2 2cos(ϕ θ )+2(sinϕ sinθ )[1 cos(ϕ θ )]
1 1 1 1 2 2
− − − −
N−1
+ +2 sinϕ sinθ [1 cos(ϕ θ )].
i i N N
··· − −
(cid:18) i=1 (cid:19)
Y
Itfollowsfromthesphericalcoordinatesthatd(y,x) 0isequivalentto ϕ θ 0. (There
→ k − k→
is an exception for θ with θ = 0, since for those ϕ such that d(y,x) 0 and ϕ tending to
N N
→
2π, ϕ θ does not tend to 0. In such case, we may treat θ as 2π instead of 0 and this
N
k − k
does not affect the result thanks to the periodicity.) Therefore, as d(y,x) 0, by Taylor’s
→
expansion,
N−1
d2(y,x) (ϕ θ )2+(sinϕ sinθ )(ϕ θ )2+ + sinϕ sinθ (ϕ θ )2
1 1 1 1 2 2 i i N N
∼ − − ··· −
(cid:18) i=1 (cid:19)
Y
N−1
(ϕ θ )2+(sin2θ )(ϕ θ )2+ + sin2θ (ϕ θ )2.
1 1 1 2 2 i N N
∼ − − ··· −
(cid:18) i=1 (cid:19)
Y
This completes the proof. (cid:3)
Next, werecallfromChanandLai(2006) someresultsonapproximationstotheexcursion
probability of Gaussian random fields over the Euclidean space.
Let 0 < α 2 and let W (s) : s [0, )N (t RN) be a family of Gaussian random
t
≤ { ∈ ∞ } ∈
fields such that
EW (s)= s αr (s/ s ),
t t
−k k k k
Cov W (s),W (v) = s αr (s/ s )+ v αr (v/ v ) (2.3)
t t t t
k k k k k k k k
(cid:0) (cid:1) s v αrt((s v)/ s v ),
−k − k − k − k
where r () :SN−1 R is a continuous function which satisfies
t +
· →
sup r (v) r (v) 0 as s t. (2.4)
t s
v∈SN−1| − | → →
5
Define
∞
Hr(t)= lim K−N euP sup W (s) u du. (2.5)
α K→∞ Z0 ns∈[0,K]N t ≥ o
Denote by H the usual Pickands’ constant, that is
α
∞
H = lim K−N euP sup Z(s) u du,
α
K→∞ Z0 ns∈[0,K]N ≥ o
where Z(s): s [0, )N is a Gaussian random field such that
{ ∈ ∞ }
EZ(s)= s α,
−k k
Cov(Z(s),Z(v)) = s α+ v α s v α.
k k k k −k − k
It is clear that Hr(t) becomes H when r 1.
α α t ≡
Let D RN be a bounded N-dimensional Jordan measurable set, that is, the boundary
⊂
of D has N-dimensional Lebesgue measure 0. Let Y = Y(t),t RN be a real-valued,
{ ∈ }
centered Gaussian field such that its covariance function C satisfies
Y
C (t,t+s) = 1 s αr (s/ s )(1+o(1)) as s 0, (2.6)
Y t
−k k k k k k →
for some constant α (0,2], uniformly over t D¯, the closure of D.
∈ ∈
We will make use of the following theorem of Chan and Lai (2006). See also Piterbarg
(1996), Mikhaleva and Piterbarg (1997) for similar results under somewhat more restrictive
conditions.
Theorem 2.2 [ChanandLai(2006, Theorem2.1)] Let D RN be a bounded N-dimensional
⊂
Jordan measurable set. Suppose the Gaussian random field Y(t) :t RN satisfies condition
{ ∈ }
(2.6), in which r () : SN−1 R is a continuous function such that the convergence (2.4)
t +
· →
is uniformly in D¯ and supt∈D¯,v∈SN−1rt(v) < ∞. Then as u → ∞,
P supY(t) u u2N/αΨ(u) Hr(t)dt.
≥ ∼ α
nt∈D o ZD
Here and in the sequel, Ψ(u) = (√2πu)−1e−u2/2.
The lemma below establishes the relation between Hr(t) and H for a special class of
α α
functions r.
Lemma 2.3 Let W (s) : s [0, )N (t RN) be a family of Gaussian random fields
t
{ ∈ ∞ } ∈
satisfying (2.3) with
r (v) = M v α, v SN−1,
t t
k k ∀ ∈
where, for every t RN, M is a non-degenerate N N matrix. Then for each t RN,
t
∈ × ∈
Hr(t)= detM H .
α | t| α
6
Proof Let t RN be fixed and consider the centered Gaussian random field W =
t
∈
W (s), s [0, )N defined by W (s) = W (M−1s). Then by (2.3), W satisfies
{ t ∈ ∞ } t t t t
EW (s)= s α,
t
−k k (2.7)
Cov(W (s),W (v)) = s α+ v α s v α.
t t
k k k k −k − k
Let B = [0,K]N and define M B = s RN : v B such that s = M v . Note that
K t K K t
{ ∈ ∃ ∈ }
Vol(M B ) = detM Vol(B ) and sup W (s) = sup W (s), it follows from (2.5)
t K | t| K s∈BK t s∈MtBK t
that
1 ∞
Hr(t) = lim euP sup W (s) u du
α K→∞Vol(BK) Z0 ns∈BK t ≥ o
Vol(M B ) 1 ∞
= lim t K euP sup W (s) u du
t
K→∞ Vol(BK) Vol(MtBK) Z0 ns∈MtBK ≥ o
1 ∞
= detM lim euP sup W (s) u du.
t t
| |K→∞Vol(MtBK)Z0 ns∈MtBK ≥ o
Because of (2.7), we can modify the proofs in Qualls and Watanabe (1973) to show that
1 ∞
H = lim euP sup W (s) u du.
α t
K→∞Vol(MtBK) Z0 ns∈MtBK ≥ o
This completes the proof. (cid:3)
For any T SN, we denote by D [0,π]N−1 [0,2π) the set corresponding to T under
⊂ ⊂ ×
the spherical coordinates (1.1). We say that T is an N-dimensional Jordan measurable set
on SN if D is an N-dimensional Jordan measurable set in RN. Now we can prove our main
result of this section.
Theorem 2.4 Let X(x) : x SN be a centered Gaussian random field satisfying condition
{ ∈ }
(2.1) and let T SN be an N-dimensional Jordan measurable set on SN. Then as u ,
⊂ → ∞
P supX(x) u cN/αArea(T)H u2N/αΨ(u),
α
≥ ∼
x∈T
n o
where Area(T) denotes the spherical area of T and c > 0 is the constant in (2.1).
Proof For any θ [0,π]N−1 [0,2π), let M be the N N diagonal matrix M =
θ θ
∈ × ×
c1/αdiag(1,sinθ ,..., N−1sinθ ). Here, we set M = c1/α if N = 1. By Lemma 2.1,
1 i=1 i θ
condition (2.1) implies
Q
C(θ,θ+ξ)= 1 ξ αr (ξ/ ξ )(1+o(1)) as ξ 0,
θ
−k k k k k k →
where rθ(τ) = Mθeτ α, τ SN−1. Then by Theorem 2.2, as u ,
k k ∀ ∈ → ∞
P supX(x) u = P supX(θ) u u2N/αΨ(u) Hr(θ)dθ. (2.8)
≥ ≥ ∼ α
nx∈T o nθ∈D o ZD
e
7
It follows from Lemma 2.3 that for any θ [0,π]N−1 [0,2π) such that M is non-degenerate
θ
∈ ×
(i.e., N−1sinθ = 0),
i=1 i 6
N−1
Q Hr(θ)= cN/α sinN−iθ H .
α i α
(cid:18) i=1 (cid:19)
Y
Note that ( N−1sinN−iθ )dθ is the spherical area element and M are non-degenerate for
i=1 i θ
θ D almost everywhere, we obtain
∈ Q
Hr(θ)dθ = cN/αArea(T)H .
α α
ZD
Plugging this into (2.8) gives the desired result. (cid:3)
Remark 2.5 Motivated byMikhalevaandPiterbarg(1997), wethinkitwouldbeinteresting
to study the excursion probability for Gaussian fields over Riemannian manifolds (beyond
sphere) whose covariance functions satisfy (2.6) with d(x,y) being the geodesic distance of x
and y. We conjecture that Pickands-type approximation similar to Theorem 2.4 still holds.
2.2 Standardized Spherical Fractional Brownian Motion
Theorem 2.4 provides a nice approximation to the excursion probability for locally isotropic
Gaussian random fields on SN whose covariance functions satisfy (2.1). When the local
behavior of the covariance function becomes more complicated, Theorem 2.4 may not be
applicable anymore. However, we can still apply Lemma 2.1 to find the corresponding local
behavior of covariance function under spherical coordinates and then apply Theorem 2.2
to obtain the asymptotics for the excursion probability. In the following, we use spherical
fractional Brownian motion on sphere as an illustrating example.
Let o be a fixed point on SN. The spherical fractional Brownian motion (SFBM) B =
β
B (x) : x SN is defined by Istas (2005) as a centered real-valued Gaussian random field
β
{ ∈ }
such that
B (o) = 0
β
E(B (x) B (y))2 = d2β(x,y), x,y SN,
β β
− ∀ ∈
where β (0,1/2]. It follows immediately that
∈
1
Cov(B (x),B (y)) = d2β(x,o)+d2β(y,o) d2β(x,y) .
β β
2 −
Without loss of generality, we take o = ((cid:0)1,0,...,0) RN+1, whose corre(cid:1)sponding spherical
∈
coordinate is (0,...,0) RN. We consider the standardized SFBM X = X(x) : x
∈ { ∈
SN o defined by
\{ }}
B (x)
X(x) = β , x SN o . (2.9)
dβ(x,o) ∀ ∈ \{ }
8
Then the covariance of X is
d2β(x,o)+d2β(y,o) d2β(x,y)
C(x,y) = Cov(X(x),X(y)) = − .
2dβ(x,o)dβ(y,o)
Note that under the spherical coordinates, d(x,o) = θ and d(y,o) = ϕ , together with
1 1
Lemma2.1,weobtainthatthecovariance functionofcorrespondingGaussianfieldX satisfies
1
C(θ,ϕ) = Cov(X(θ),X(ϕ)) = 1 (1+o(1)) (ϕ θ )2+(sin2θ )(ϕ eθ )2
− 2θ2β 1− 1 1 2 − 2
1 (cid:20)
e e e N−1 β
+ + sin2θ (ϕ θ )2
i N N
··· −
(cid:18) i=1 (cid:19) (cid:21)
Y
as d(x,y) 0. Let
→
N−1
1
M = diag 1,sinθ ,..., sinθ ,
θ 21/(2β)θ 1 i
1 (cid:18) i=1 (cid:19)
Y
r (τ) = M τ 2β, τ SN−1,
θ θ
k k ∀ ∈
and ξ = ϕ θ, then as ξ 0,
− k k →
C(θ,θ+ξ) = 1 ξ 2βr (ξ/ ξ )(1+o(1)).
θ
−k k k k
Let T SN be an N-dimeensional Jordan measurable set such that o / T¯, and denote its
⊂ ∈
corresponding domain under the spherical coordinates by D, which implies θ = 0 for any
1
6
θ D¯. By Theorem 2.2, as u ,
∈ → ∞
P supX(x) u =P supX(θ) u uN/βΨ(u) Hr (θ)dθ.
≥ ≥ ∼ 2β
nx∈T o nθ∈D o ZD
e
It follows from Lemma 2.3 that for any θ such that M is non-degenerate( i.e., N−1sinθ =
θ i=1 i 6
0),
Q
N−1
1
Hr (θ) = sinN−iθ H .
2β 2N/(2β)θN i 2β
1 (cid:18) i=1 (cid:19)
Y
Therefore, as u ,
→ ∞
N−1
P supX(x) u uN/βΨ(u)2−N/(2β)H θ−N sinN−iθ dθ. (2.10)
≥ ∼ 2β 1 i
nx∈T o ZD (cid:18) Yi=1 (cid:19)
Remark 2.6 Comparing the excursion probabilities in (2.10) for the standardized SFBM
X and in Theorem 2.4 for the isotropic Gaussian field Wβ, which is defined in Estrade and
Istas (2010), we see that the constant in (2.10) is more complicated.
9
3 Smooth Isotropic Gaussian Fields on Sphere
In this section we study the excursion probability of smooth isotropic Gaussian fields on
sphere. Related to the results in this section, we mention that Cheng and Schwartzman
(2014) have determined the height distribution and overshoot distribution of local maxima
of smooth isotropic Gaussian random fields on sphere.
3.1 Preliminaries
Given λ > 0 and an integer n 0, the ultraspherical polynomial (or Gegenbauer polynomial)
≥
of degree n, denoted by Pλ(t), is defined by the expansion
n
∞
(1 2rt+r2)−λ = rnPλ(t), t [ 1,1].
− n ∈ −
n=0
X
For λ = 0, we follow Schoenberg (1942) and define P0(t) = cos(narccost) = T (t), where T
n n n
(n 0) are Chebyshev polynomials of the first kind defined by the expansion
≥
∞
1 rt
− = rnT (t), t [ 1,1].
1 2rt+r2 n ∈ −
− n=0
X
For reference later on, we recall the following formulae on Pλ.
n
(i). For all n 0, P0(1) = 1, and if λ > 0 [cf. Szego¨ (1975, p. 80)],
≥ n
n+2λ 1
Pλ(1) = − . (3.1)
n n
(cid:18) (cid:19)
(ii). For all n 0,
≥
d
P0(t) = nP1 (t), (3.2)
dt n n−1
and if λ > 0 [cf. Szego¨ (1975, p. 81)],
d
Pλ(t) = 2λPλ+1(t). (3.3)
dt n n−1
The following theorem by Schoenberg (1942) characterizes the covariance function of an
isotropic Gaussian field on sphere [see also Gneiting (2013)].
Theorem 3.1 Let N 1, then a continuous function C(, ) : SN SN R is the covariance
≥ · · × →
of an isotropic Gaussian field on SN if and only if it has the form
∞
C(x,y)= a Pλ( x,y ), x,y SN,
n n h i ∈
n=0
X
where λ = (N 1)/2, a 0 and ∞ a Pλ(1) < .
− n ≥ n=0 n n ∞
P
10