A functional central limit theorem for
Lebesgue integrals of mixing random
fields
Joint work with E. Spodarev
Jürgen Kampf | Institute of Stochastics | October 2, 2015
WAGP 2015, Ulm
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
Random field = Set of random variables indexed by Rd
= Stochastic processes with index set Rd
Aim:
Examine asymptotic behavior of random variable
Z
f (X (t)) dt,
Wn
where (X (t))t∈Rd is a random field and
f : R → R is a deterministic function
as Wn approaches the whole space.
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2.10.2015
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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Theorem 1 (Bulinskii & Zhurbenko, 1976)
Let (X (t))t∈Rd be a stationary, measurable random field fulfilling
some α-mixing assumptions.
Let f : R → R be some measurable map such that (f (X (t)))t∈R
fulfills integrability assumptions.
Let (Wn )n∈N be a VH-growing sequence of compact sets of Rd ,
e.g. Wn = Bn (0) = {t ∈ Rd | ktk ≤ n}. Then
R
Wn f (X (t)) dt − λd (Wn ) · E[f (X (0))] d
p
→ N (0, σ 2 ),
λd (Wn )
as n → ∞, where
σ2 =
Z
Rd
Cov f (X (0)), f (X (t)) dt.
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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2.10.2015
Theorem 2 (Meschenmoser & Shashkin, 2011)
Let (X (t))t∈Rd be a stationary, measurable, associated random
field fulfilling some integrability and regularity assumptions.
Let (Wn )n∈N be a VH-growing sequence of compact sets of Rd .
Then the sequence of stochastic processes defined by
R
1[u,∞) (X (t)) dt − λd (Wn ) · E[1[u,∞) (X (0))]
p
Yn (u) := Wn
λd (Wn )
converges in distribution to a centered Gaussian process Y with
covariance function
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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2.10.2015
Cov(Y (u1 ), Y (u2 )) =
Z
P(X (0) > u1 , X (t) > u2 ) − P(X (0) > u1 ) · P(X (t) > u2 ) dt
Rd
as n → ∞ in the Skorokhod topology.
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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2.10.2015
Replace indicator functions by more general functions?
Consider the space V of Lipschitz continuous functions with norm
kf k := Lip f + |f (0)|.
Theorem 3
Let (X (t))t∈Rd be a stationary and measurable random field.
Assume there n ∈ N, δ > 4 and C , l > 0 with
n/d > max{l + δ/(δ − 2), δ/(δ − 4)} such that
αγ (r ) ≤ Cr −n γ l for all γ ≥ 2κd , r > 0,
and
EX (0)δ < ∞.
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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2.10.2015
Let (Wn )n∈N be a VH-growing sequence of compact sets of Rd .
Then the sequence of stochastic processes defined by
R
f (X (t)) dt − λd (Wn ) · E[f (X (0))]
p
, f ∈ V,
Φn (f ) := Wn
λd (Wn )
converges in distribution to a centered Gaussian process Φ with
covariance function
Z
Cov f (X (0)), g (X (t)) dt
Cov(Φ(f ), Φ(g )) =
Rd
as n → ∞ in the weak topology.
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
Sketch of the proof:
According to Oppel (1973) it suffices to show:
1. The finite-dimensional distributions converge appropriately.
2. The processes Φn have linear and continuous paths.
3. The process Φ exists and has a version with linear and
continuous paths.
For 1.: Employ Cramér-Wold-technique
2.: Trivial
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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2.10.2015
For 3. (Continuity of limiting process):
Employ theory of GB- and GC-sets
What is this theory about?
For a Hilbert space H with scalar product h·, ·i the isonormal
process is the centered Gaussian process Φ with
Cov(Φ(f ), Φ(g )) = hf , g i.
Sets, on which a version of the isonormal process has bounded /
continuous paths, are called GB / GC -sets.
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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2.10.2015
How to apply this theory to our problem?
Define scalar product such that Φ becomes the isonormal process:
hf , g i = Cov(Φ(f ), Φ(g ))
Z
=
Cov f (X (0)), g (X (t)) dt
Rd
This is a symmetric, non-negative definite, bilinear form.
Passing to equivalence classes and completion we obtain a Hilbert
space.
We have to show that (the projection of)
B := {f ∈ V | kf kLip ≤ 1}
is a GB-set.
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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Let N() be the minimal number of elements of an -net on B.
An -net on B are elements f1 , . . . , fn ∈ B such that
∀g ∈B : ∃i∈{1,...,n} : kfi − g kh·,·i < .
It is well known (see e.g. Dudley, 1999, Chapter 2) that it suffices
to show
Z
0
1
(log N())1/2 d < ∞.
(1)
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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2.10.2015
For m ∈ N and c > 0 consider the Lipschitz functions f with
I
f (0) = 0
I
On each interval [(k − 1)c, kc], k = −m + 1, . . . , m, the
function f is either
increasing with constant slope 1 or
decreasing with constant slope −1.
I
On (−∞, −mc] and [mc, ∞), the function f is
constant.
Under appropriate inequalities on , c and m, these functions form
an -net with (1).
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
Does the asymptotic variance Var(Φ(f )), f ∈ V , vanish?
For Gaussian random fields with non-negative
covariance function:
Var(Φ(f )) = 0 ⇐⇒ Var(X (0)) = 0
or f is constant
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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2.10.2015
Construction of a field X (t)t∈Rd that fulfills all assumptions of
Theorem 4, but Var(Φ(f )) = 0 for all f ∈ V :
Consider a Poisson-Voronoi-mosaic, i.e. a random partition of Rd
into convex polytopes.
X (t) := λd ({v ∈ C (t) | kv − ξt k ≤ kt − ξt k})/λd (C (t)),
where C (t) is the cell in which t lies and ξt is its nucleus.
t
ξt
t ∈ Rd ,
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A functional central limit theorem for Lebesgue integrals of mixing random fields
Z
Z
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Jürgen Kampf
1
f (x) dx · λd (C )
f (X (t)) dt =
C
0
for each (random) cell C .
Z
Z
f (X (t)) dt ≈
W
1
f (x) dx · λd (W )
0
for each (large, deterministic) compact set W ⊆ Rd ,
strengthened by mixing properties of Poisson-Voronoi-mosaic.
⇒ The variance vanishes asymptotically.
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
Further issue: Orthogonality in (V , h·, ·i)
For a certain class of random fields constructed with help of
Lévy-Meixner-systems we obtained explicit ONBs.
Open questions
Can Theorem 3 be derived
I
for a space larger than V ?
I
under association instead of mixing assumptions?
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2.10.2015
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A functional central limit theorem for Lebesgue integrals of mixing random fields
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Jürgen Kampf
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2.10.2015
Literature:
[1] A.V. Bulinskii, I.G. Zurbenko: A central limit theorem for additive
random functions, Theory of Probability and its Applications 21,
1976, 707–717.
[2] D. Meschenmoser, A. Shashkin: Functional central limit theorem for
the volume of excursion sets generated by associated random fields,
Statist. Probab. Lett. 81(6), 2011, 642-646.
[3] U. Oppel: Schwache Konvergenz kanonischer zufälliger Funktionale,
Manuscripta Math. 8, 1973, 323 – 334.
[4] R. Dudley: Uniform Central Limit Theorems, Cambridge University
Press, 1999.
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