Intrinsic Volumes of Spherical Polar
Sets and Connections With Small
Ball Probabilities
Fuchang Gao
University of Idaho
Notations
1. E n denotes the n-dimensional Euclidean vector space.
2. B n and Sn−1 denote the unit ball and unit sphere in
E n.
3. λn and σn−1 denote the Lebesgue and the spherical
Lebesgue measure on Sn−1 .
4. Kn denotes the space of convex bodies (non-empty,
compact, convex sets) in E n .
5. Parallel body
Kε := {x ∈ E n : d(K, x) ≤ ε},
where d(K, ·) denotes the Euclidean distance from K.
Intrinsic Volumes in En and Hilbert spaces
Steiner’s formula:
λn (Kε ) =
n
X
εn−j κn−j Vj (K)
j=0
defines the jth intrinsic volume Vj .
Particular cases are V0 = 1; Vn , the volume of K; the
2Vn−1 , the surface area, and (2κn−1 /nκn )V1 , the mean
width.
(1)
Definition in Hilbert spaces
Either define directly by the following formula:
Z
−πd2 (x,tK)
e
n
∞
X
Vn (K)
t
√
dγ =
;
n!
2π
n=0
where γ is the Gaussian measure in the Hilbert space H;
or define it as the supremum of the intrinsic volumes of
inscribed finitely dimensional convex bodies.
Evaluation Formula
If P is a polytope and Fj (P ) denotes the set of its
j-dimensional faces, then
X
Vj (P ) =
γ(F, P )λj (F ),
(2)
F ∈Fj (P )
where γ(F, P ) is the normalized external angle of P at its
face F ; or in other words, the Gaussian measure of the
normal cone at face F .
For general sets, one can either approximate it with
polytopes, or use principal curvatures of ∂K. (We will use
the former.)
Intrinsic Volumes and Sample Boundedness
Let Xt , t ∈ T be a Gaussian process, and K the convex
hull of {Xt ∈ H : t ∈ T }. Sudakov (1971) proved that
√
E sup Xt = V1 (K)/ 2π
t∈T
where V1 (K) is the intrinsic volume of K.
The connections between supt∈T Xt and Vk (K), can be
found in Chevet (1976), Tsirel’son (1985,1986,1987) Vitale
(1996, 1999, 2000), Gao and Vitale (2001), and Gao
(2003). For example, the following can be found in Gao
(2003)
n
X
∞
λ2
Vn (K)
λ
2
√
E sup exp λXµ − E|Xµ | =
.
2
n!
µ
2π
n=0
A closer look
λ2
E sup exp λXµ − E|Xµ |2
2
µ
n
∞
X
Vn (K)
λ
√
=
.
n!
2π
n=0
where the sup Ris taken on all the probabilities measures on
T , and Xµ = T Xt µ(dt).
The left hand is called the Wills functional. By taking δ
measure at t ∈ T , one sees that the left hand side looks like
a moment generating function of supt∈T Xt .
Intrinsic volumes and deviations
The connection of Wills functional and large deviation has
been studied by Vitale (1996, 1999, 2000). Related ideas
are also found in Tsirel’son (1985, 1986, 1987). In
particular Vitale obtained a refined version of large
deviation inequality for Gaussian processes.
The investigation of connection between Wills functional
(intrinsic volumes, solid angles) and small deviation has
been tried, but there seems to be no immediate connection
to our knowledge.
Motivation
For Gaussian process X
small ball
t , t ∈ T , a typical
problem is to study P sup |Xt | < ε . If we write the
t∈T
Karhunen-Loève expansion
Xt =
∞ p
X
λn ψn (t)ξn ,
n=1
and let
p
K = absconv{( λn ψn (t)) ∈ l2 : t ∈ T },
then the above question can be reformulated as
γ({x ∈ H : hx, ai ≤ ε for all a ∈ K})
where γ is Gaussian measure in l2 .
Motivation (cont.)
If we let
K = conv{(λn ψn (t)) : t ∈ T },
then
2
γ({x ∈ l : hx, ai ≤ ε for all a ∈ K}) = P
sup Xt < ε
t∈T
which is a lower tail probability.
In particular, if the variance of Xt remains as a constant, for
example when Xt is stationary, one only needs to consider
{x ∈ l2 : kxk = 1, hx, ai ≤ ε for all a ∈ K}.
Spherical Polar Sets
This motivates us to study the following so-called spherical
parallel set
Aε := {x ∈ Sn : ds (A, x) ≤ ε},
(3)
where A is a given set in Sn , and ds (A, x) denotes the
spherical distance of x from A; and to study spherical polar
sets
A∗ := {x ∈ Sn : hx, ai ≤ 0 for all a ∈ A}.
Spherical Intrinsic Volumes
There are at least three different sequences of functionals
that can be considered as spherical counterparts of the
Euclidean intrinsic volumes. (See Gao, Hug and Schneider,
2003) The basic one, and also the one that best fits the
study of Gaussian processes seems to be by the following
spherical Steiner’s formula in Sn
σn (Kε ) = βn Vn (K) +
n−1
X
fj (ε)βj βn−j−1 Vj (K),
j=0
where
Z
fj (ε) :=
0
ε
cosj t sinn−j−1 t dt.
(4)
Duality Relation
The spherical intrinsic volumes behave well under duality
(Note: there is no such nice property in Rn ).
Vj (K) = Vn−j−1 (K ∗ )
(5)
for j = 0, . . . , n − 1 and K ∈ Ksn . (See e.g.
Gao-Hug-Schneider 2003)
The means, the intrinsic volumes of a spherical polar set A∗
can be calculated using those of A.
A problem from Li’s and Shao’s talks
In Li’s and Shao’s talks, the following problem is raised: For
a stationary Gaussian process Xt with a given covariance
structure find
lim
T →∞
log P (sup0≤t≤T Xt < 0)
.
T
There are many more problems of this nature in Li-Shao
(2003).
We reformulate the problem into the following intrinsic
volume problem
How to use this method
Step 1: Discretization
It is enough to study P (max1≤i≤n Xti < 0) for large n.
Pn
Step 2: Write Xti = j=1 cij ξj , where ξj ’s are
independent standard normal random variables. Let
aij = cij /Var(Xti ). Then ai = (ci1 , ci2 , ..., cin ) ∈ Sn . Let
A be the spherical convex hull of {ai }ni=1 . Then
Vn (A∗ ) = σn (A∗ ) = P max Xti < 0 .
1≤i≤n
Thus, it becomes a problem of finding spherical intrinsic
volumes.
An Evaluation Formula
Let P ∈ Ksn be a spherical polytope, thus P = Sn ∩ CP ,
where CP is a polyhedral cone in E n+1 . For faces F, G of
CP with F ⊂ G, we denote by β(F, G) the internal and by
γ(F, G) the external (normalized) angle of G at F . Then
X
Vj (P ) =
β(0, F )γ(F, CP ),
(6)
F ∈Fj+1 (CP )
where Fj+1 (CP ) is the set of (j + 1)-faces of the
polyhedral cone CP .
Another Formula
Also, Vj (A∗ ) can be evaluated using Euler characteristic.
For example,
Z
1
Vn (A∗ ) =
χ(A ∩ S)dνn−1 (S)
n
2 Sn−1
n
where Sn−1
is the space of (n − 1)-dimensional great
subspaces in Sn , and νn−1 is the rotation invariant
n
probability measure on Sn−1
.
An Example
If Xt is Brownian motion on [0, 1], then the solid angle can
be expressed by the matrix in the next slides. Similar is the
case for Brownian Bridge (Gao-Vitale 2002), (Gao 2003).
For Gaussian Markov processes, the only change is to
replace the first row in the matrix below by a vector close
to (1, 0, ..., 0).
































1
−1 √1
√
2
2
..
.
−1 √1
√
2
2
−1 √1
√
2
2
..
.
−1 √1
√
2
2
..
.
−1 √1
√
2
2
..
.
−1
√
2
Solid Angle Approximation
Solid angle evaluation is typically difficult. However, since
in the application of small ball and lower tail problems, we
are interested only in the rates, most case even at the
logarithmic lever. Thus, even some rough estimate would
help.
The following type of approximation has been proved
possible: If Γ is Gaussian measure of the cone generated by
n vectors v1 , v2 , ..., vn in Rn , and Γ1 the Gaussian measure
of the cone generated by vectors n vectors w1 , v2 , ..., vn in
Rn . Suppose kv1 − w1 k = o(Γ) as n → ∞, then
Γ1 = O(Γ).
Stay tuned .....