Normal Approximation for White Noise Functionals by Stein's Method and Hida Calculus

0.31 MB·

by  Louis H. Y. Chen

January3,2017 LectureNoteSeries,IMS,NUS—ReviewVol.9inx6in HidaStein7
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J Normal Approximation for White Noise Functionals

2 by Stein’s Method and Hida Calculus

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R
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Louis H.Y.Chen
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h Department of Mathematics
t National University of Singapore
a
m 10 Lower Kent Ridge Road, Singapore 119076
[email protected]
[

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v Yuh-JiaLee
0 Department of Applied Mathematics
6
National University of Kaohsiung
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Kaohsiung, TAIWAN 811
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[email protected]
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.
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0 Hsin-HungShih
7
Department of Applied Mathematics
1
National University of Kaohsiung
:
v Kaohsiung, TAIWAN 811
i [email protected]
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r
a
In this paper we establish a framework for normal approximation for
white noise functionals by Stein’s method and Hida calculus. Our work
isinspiredbythatofNourdinandPeccati(Probab.TheoryRelat.Fields
145,75-118,2009),whocombinedStein’smethodandMalliavincalculus
for normal approximation for functionals of Gaussian processes.
1. Introduction
Stein’s method,introducedbyC.Stein[39]inhis1972paper,isapowerful
way of determining the accuracy of normal approximation to the distri-
bution of a sum of dependent random variables. It has been extended to
approximations by a broad class of other probability distributions such as
thePoisson,compoundPoisson,andGammadistributions,andtoapproxi-
1
January3,2017 LectureNoteSeries,IMS,NUS—ReviewVol.9inx6in HidaStein7
2 L. H. Y. Chen, Y. J. Lee & H. H. Shih
mationsonfinite aswellasinfinitedimensionalspaces.Theresultsofthese
approximationshavebeenextensivelyappliedinawiderangeofotherfields
suchasthetheoryofrandomgraphs,computationalmolecularbiology,etc.
Forfurther details,see[1,2,3,6,35,36,40]andthe referencescitedtherein.
Analysis on infinite-dimensional Gaussian spaces has been formulated
in terms of Malliavin calculus and Hida calculus. The former, introduced
by Malliavin ([29]), studies the calculus of Brownian functionals and their
applications onthe classicalWiener space.The connectionbetween Stein’s
method and Malliavin calculushas been explored by Nourdin and Peccati
(see [31]). They developed a theory of normal approximation on infinite-
dimensional Gaussian spaces. In their connection, the Malliavin derivative
D plays an important role. See also [7,9].
Hida calculus, also known as white noise analysis, is the mathematical
theoryofwhitenoiseinitiatedbyT.Hidainhis1975CarletonMathematical
Lecture Notes [14]. Let B(t); t R be a standard Brownian motion and
{ ∈ }
let the white noise B˙(t) dB(t)/dt,t R, be represented by generalized
≡ ∈
functions.Byregardingthecollection B˙(t); t R asacoordinatesystem,
{ ∈ }
Hidadefinedandstudiedgeneralizedwhitenoisefunctionalsϕ(B˙(t), t R)
∈
throughtheirU-functionals.Werefertheinterestedreaderto[15,16,19,38].
The objective of this paper is to develop a connection between Stein’s
method and Hida calculus for normal approximation for white noise func-
tionals(seeSection5).Ourapproachisanalogoustothatfortheconnection
between Stein’s method and Malliavin calculus as established by Nourdin
andPeccati[31].TheconnectionbetweenStein’smethodandHidacalculus
will be built on the expression of the number operator (or the Ornstein-
Ulenbeck operator)intermsofthe Hida derivativesthroughintegrationby
partstechniques(seeSection4).Thedifficultythatwehaveencounteredso
faristhattheHidaderivative∂ ,thatistheB˙(t)-differentiation,cannotbe
t
defined on all square-integrable white noise functionals in (L2). Extending
the domainof∂ to a largersubclass of(L2) andstudying the regularityof
t
∂ will be a key contribution in our paper.
t
At the time of completing this paper we came to know about the PhD
thesisofChu[11],inwhichhedevelopednormalapproximation(inWasser-
stein distance) for L´evy functionals by applying Stein’s method and Hida
calculus. He achieved this by using the white noise approach of Lee and
Shih [28].
We list some notation which will be often used in this paper.
Notation 1.1:
January3,2017 LectureNoteSeries,IMS,NUS—ReviewVol.9inx6in HidaStein7
Stein’sMethod and Hida Calculus 3
(1) ForareallocallyconvexspaceV,CV denotesitscomplexification.
If (V, ) is a real Hilbert space, then CV is a complex Hilbert
V
|·|
space with the |·|CV-normgivenby |φ|2CV =|φ1|2V +|φ2|2V for any
φ=φ +iφ ,φ ,φ V.Specially,forV = with -norm(see
1 2 1 2 p p
∈ S |·|
(2) STehcetisoynm2bfoolr(t,he)ddeenfiontietisotnh)e, we-w,ilClstil-lCuse, |·|p-to,doernCote |·-|CCSp.
′ ′ p p p p
· · S S S S S− S S− S
pairing.
(3) For a n-linear operator T on X X, Txn means T(x,...,x)
×···×
as well as Txn 1y =T(x,...,x,y), x,y X, where X is a real or
−
∈
complex locally convex space.
(4) The constant ω with r > 0 is given by the square root of
r
sup n/22nr; n N , where N =N 0 .
0 0
{ ∈ } ∪{ }
2. Stein’s method
Inthis sectionwe giveabrief expositionofthe basics ofStein’s method for
normal approximation.
2.1. From characterization to approximation
In his 1986 monograph [39], Stein proved the following characterization of
the normal distribution.
Proposition 2.1: (Stein’s lemma) The following are equivalent.
(i) W (0,1);
∼N
(ii) E[f (W) Wf(W)]=0 for all f 1.
′ − ∈CB
Proof: By integration by parts, (i) implies (ii). If (ii) holds, solve
f (w) wf(w)=h(w) Eh(Z)
′
− −
where h and Z has the standard normal distribution (denoted by
B
∈ C
Z N(0,1)). Its bounded unique solution f is given by
h
∼
fh(w) = e21w2 ∞e−21t2[h(t) Eh(Z)]dt
− −
Zw
w
= e21w2 e−12t2[h(t) Eh(Z)]dt. (2.1)
−
Z−∞
Using w∞e−21t2dt≤w−1e−21w2 for w >0, we can show that fh ∈CB1 with
f √2πe h and f 4 h . Substituting f for f in (ii)
k hk∞R≤ k k∞ k h′k∞ ≤ k k∞ h
leads to
Eh(W)=Eh(Z) for h .
B
∈C
January3,2017 LectureNoteSeries,IMS,NUS—ReviewVol.9inx6in HidaStein7
4 L. H. Y. Chen, Y. J. Lee & H. H. Shih
This proves (i).
The equation
EWf(W)=Ef (W) (2.2)
′
for all f 1, which characterizes the standard normal distribution, is
∈ CB
called the Stein identity for normal distribtution. In fact, if W N(0,1),
∼
(2.2) holds for absolutely continuous f such that Ef (W) < .
′
| | ∞
LetW bearandomvariablewithEW =0andVar(W)=1.Proposition
2.1 suggests that the distribution of W is “close” to N(0,1) if and only if
E[f′(W) Wf(W)] 0
− ≃
for f 1. How “close” the distribution of W is to the standard normal
∈ CB
distrbutionmaybequantifiedbydetermininghowcloseE[f (W) Wf(W)]
′
−
is to 0.To this endwe define adistance between the distribution ofW and
the standard normal distribution as follows.
dG(W,Z):= sup Eh(W) Eh(Z) (2.3)
h G| − |
∈
where G is a separating class and the distance dG is said to be induced by
G.ByaseparatingclassG,wemeanaclassofBorelmeasurablereal-valued
functions defined on R such that two random variables, X and Y, have the
same distribution if Eh(X) = Eh(Y) for h G. Such a separating class
∈
contains functions h for which both Eh(X) and Eh(Y) exist.
Let f be the solution, given by (2.1), of the Stein’s equation
h
f (w) wf(w)=h(w) Eh(Z) (2.4)
′
− −
where h G. Then we have
∈
dG(W,Z)=hsupG|E[fh′(W)−Wfh(W)]|. (2.5)
∈
So bounding the distance dG(W,Z) is equivalent to bounding
sup E[f (W) Wf (W)],forwhichweneedtostudytheboundedness
h∈G | h′ − h |
properties of f and the probabilistic structure of W.
h
The following three separating classes of Borel measurable real-valued
functions defined on R are of interest in normal approximation.
G := h; h(u) h(v) u v ,
W
{ | − |≤| − |}
G := h;h(w)=1 for w x and=0 for w>x,x R ,
K
{ ≤ ∈ }
G := h;h(w)=I(w A),A is a Borel subset of R .
TV
{ ∈ }
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Stein’sMethod and Hida Calculus 5
The distances induced by these three separating classes are respectively
called the Wasserstein distance, the Kolmogorov distance, and the total
variation distance. It is customary to denote dG , dG and dG respec-
W K TV
tively by d , d and d .
W K TV
Sinceforeachhsuchthat h(x) h(y) x y ,thereexistsasequence
| − |≤| − |
of h 1 with h 1 such that h h 0 as n , we have
n ∈C k ′nk∞ ≤ k n− k∞ → →∞
d (W,Z)= sup Eh(W) Eh(Z). (2.6)
W
h 1, h′ ∞ 1| − |
∈C k k ≤
By an application of Lusin’s theorem, we also have
d (W,Z)= sup Eh(W) Eh(Z). (2.7)
TV
| − |
h ,0 h 1
∈C ≤ ≤
It is also known that
1
d (W,Z)= sup Eh(W) Eh(Z)
TV
2 | − |
h ∞ 1
k k ≤
1
= sup Eh(W) Eh(Z).
2 | − |
h , h ∞ 1
∈Ck k ≤
It is generally much harder to obtain an optimal bound on the Kol-
mogorov distance than on the Wasserstein distance. There is a discussion
on this and examples of bounding the Wasserstein distance and the Kol-
mogorov distance are given in Chen [7].
We now state a proposition that concerns the boundedness properties
of the solution f , given by (2.1), of the Stein equation (2.4) for h either
h
boundedorabsolutelycontinuouswithboundedh.Theuseofthesebound-
′
edness properties is crucialfor bounding the Wasserstein,Kolmogorovand
total variation distances.
Proposition2.2: Letf betheuniquesolution,givenby(2.1),oftheStein
h
equation (2.4), where h is either bounded or absolutely continuous.
1. If h is bounded, then
kfhk∞ ≤ π/2kh−Eh(Z)k∞, kfh′k∞ ≤2kh−Eh(Z)k∞. (2.8)
2. If h is abpsolutely continuous with bounded h, then
′
kfhk∞ ≤2kh′k∞, kfh′k∞ ≤ 2/πkh′k∞, kfh′′k∞ ≤2kh′k∞. (2.9)
3. If h=I where x Rp, then, writing f as f ,
( ,x] h x
−∞ ∈
0<fx(w)≤√2π/4, |wfx(w)|≤1, |fx′(w)|≤1, (2.10)
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6 L. H. Y. Chen, Y. J. Lee & H. H. Shih
and for all w,u,v R,
∈
f (w) f (v) 1, (2.11)
| x′ − x′ |≤
(w+u)f (w+u) (w+v)f (w+v) (w +√2π/4)(u + v ). (2.12)
x x
| − |≤ | | | | | |
4. If h=h where x R, ǫ>0, and
x,ǫ
∈
1, w x,
≤
h (w)= 0, w x+ǫ,
x,ǫ  ≥
1+ǫ−1(x w), x<w <x+ǫ,
−
then, writing f as f , we have for all w,v,t R,
h x,ǫ
∈
0 f (w) 1, f (w) 1, f (w) f (v) 1 (2.13)
≤ x,ǫ ≤ | x′,ǫ |≤ | x′,ǫ − x′,ǫ |≤
and
f (w+t) f (w)
| x′,ǫ − x′,ǫ |
1 t∨0
(w +1)t + I(x w+u x+ǫ)du
≤ | | | | ǫ ≤ ≤
Zt∧0
(w +1)t +I(x 0 t w x 0 t+ǫ). (2.14)
≤ | | | | − ∨ ≤ ≤ − ∧
Except for (2.14), which can be found on page 2010 in Chen and Shao
[10],theboundsinProposition2.2andtheirproofscanbefoundinLemmas
2.3, 2.4 and 2.5 in Chen, Goldstein and Shao [8].
2.2. Stein identities and error terms
LetW be a randomvariablewithEW =0andVar(W)=1.Inadditionto
using the boundedness properties of the solution f of the Stein equation
h
(2.4), we also need to exploit the probabilistic structure of W, in order
to bound the error term in (2.5). This is done through the construction
of a Stein identity for W for normal approximation. This is perhaps best
understood by looking at a specific example.
Let X , ,X be independent random variables with EX = 0,
1 n i
···
Var(X ) = σ2 and EX 3 < . Let W = n X and W(i) = W X
i i | i| ∞ i=1 i − i
for i = 1, ,n. Assume that Var(W) = 1, which implies n σ2 = 1.
··· P i=1 i
Let f 2. Using the independence among the X and the property that
∈ CB i P
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Stein’sMethod and Hida Calculus 7
EX =0, we have
i
n n
EWf(W)= EX f(W)= EX [f(W(i)+X ) f(W(i))]
i i i
−
i=1 i=1
X X
n Xi
= E X f (W(i)+t)dt
i ′
i=1 Z0
X
n
= E ∞ f (W(i)+t)K (t)dt
′ i
Xi=i Z−∞
n b
= E ∞ f (W(i)+t)K (t)dt (2.15)
′ i
Xi=i Z−∞
where
K (t) =X [I(X >t>0) I(X <t<0)],
i i i i
−
K (t) =EK (t).
i i
b
b
Itcanbe shownthatforeachi,σi−2Ki isaprobabilitydensityfunction.So
(2.15) can be rewritten as
n
EWf(W)= σi2Ef′(W(i)+Ti) (2.16)
i=1
X
where T , ,T ,X , ,X are independent and T has the density
1 n 1 n i
··· ···
σi−2Ki. Both the equations (2.15) and (2.16) are Stein identities for W
for normal approximation.From (2.16) we obtain
n
E[f′(W)−Wf(W)]= σi2E[f′(W)−f′(W(i))]
i=1
X
n
σ2E[f (W(i)+T ) f (W(i))] (2.17)
− i ′ i − ′
i=1
X
where the error terms on the right hand side provide an expression for the
deviationofE[f (W) Wf(W)]from0.Nowletf bef whereh 1 such
′ h
− ∈C
that h 1. Applying Taylor expansion to (2.17), and using (2.6) and
′
k k∞ ≤
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8 L. H. Y. Chen, Y. J. Lee & H. H. Shih
(2.9), we obtain
n
d (W,Z) sup σ2EX f
W ≤Xi=1h∈C1,kh′k∞≤1 i | i|k h′′k∞
n
+ sup σ2ET f
Xi=1h∈C1,kh′k∞≤1 i | i|k h′′k∞
n n
2 σ2EX +2 σ2ET .
≤ i | i| i | i|
i=1 i=1
X X
Since σ2EX (EX 3)2/3)(EX 3)1/3 = EX 3 and σ2ET =
i | i| ≤ | i| | i| | i| i | i|
σ2(1/2σ2)EX 3 =(1/2)EX 3, we have
i i | i| | i|
n
d (W,Z) 3 EX 3. (2.18)
W i
≤ | |
i=1
X
The bound in (2.18) is of optimal order. However, it is much harder to
obtain a bound of optimal order for the Kolmogorov distance. Proofs of
such a bound on d (W,Z) can be found in Chen [7] and Chen, Goldstein
K
and Shao [8]. Stein’s identities for locally dependent random variables and
otherdependent randomvariablescanbe found inChenandShao[10]and
Chen, Goldstein and Shao [8].
2.3. Integration by parts
Let W be a random variable with EW = 0 and Var(W) = 1. In many
situations of normal approximation, the Stein identity for W takes the
form
EWf(W)=ETf (W) (2.19)
′
where T is random variable defined on the same probability as W such
that ET < , and f an absolutely continuous function for which the
| | ∞
expectationsexist.TypicalexamplesofsuchsituationsarecaseswhenW is
afunctionalofGaussianrandomvariablesorofaGaussianprocess.Insuch
situations,the Steinidentity(2.19)isoftencontructedusingintegrationby
parts as in the case of the Stein identity for N(0,1).
By letting f(w)=w, we obtain ET =EW2 =1. Let F be a σ-algebra
with respect to which W is measurable. From (2.19),
E[f′(W) Wf(W)]=E[f′(W)(1 T)]
− −
=E[f′(W)E(1 T F)].
−
(cid:12)
(cid:12)
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Stein’sMethod and Hida Calculus 9
Now let f =f where h G, a separating class of functions. Assume that
h
∈
f < . Then by (2.3),
|| h′||∞ ∞
dG(W,Z) =hsupG|E[fh′(W)−Wfh(W)]|
∈
sup f EE(1 T F)
≤(cid:18)h∈Gk h′k∞(cid:19) | − (cid:12) |
sup f Var(E(T(cid:12)F)),
≤(cid:18)h∈Gk h′k∞(cid:19)q (cid:12)
whereforthelastinequalityitisassumedthatE(T F)(cid:12) issquareintegrable.
By Proposition 2.2,
(cid:12)
(cid:12)
2/π, G =G ,
W
hsu∈pGkfh′k∞ ≤p21,, GG ==GGTKV,. (2.20)
This implies 
d(W,Z) θEE(1 T F) θ Var(E(T F)) (2.21)
≤ | − |≤
q
where (cid:12) (cid:12)
(cid:12) (cid:12)
2/π, d(W,Z)=d (W,Z),
W
θ = 1, d(W,Z)=d (W,Z), (2.22)
p K
2, d(W,Z)=dTV(W,Z).
For the rest of thissection, we will present two approaches to the con-
struction of the Stein identity (2.19).
Let Z N(0,1) and let ψ be an absolutely continuous function such
∼
that Eψ(Z) = 0, Var(ψ(Z)) = 1 and Eψ (Z)2 < . Define W = ψ(Z).
′
∞
FollowingChatterjee[5],weuseGaussianinterpolationtoconstructaStein
identity for ψ(Z). Let f be absolutely continuouswith bounded derivative,
whichimplies f(w) C(1+ w)forsomeC >0.LetZ beanindependent
′
| |≤ | |
copy of Z and let W =ψ(√tZ+√1 tZ ) for 0 t 1. Then we have
t ′
− ≤ ≤
1 ∂W
EWf(W) =E(W W )f(W)=E f(W) tdt
1 0
− ∂t
Z0
1 Z Z
=E f(W) ′ ψ (√tZ+√1 tZ )dt.(2.23)
′ ′
2√t − 2√1 t −
Z0 (cid:18) − (cid:19)
Let
Ut =√tZ+√1 tZ′,
−
Vt =√1 tZ √tZ′.
− −
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10 L. H. Y. Chen, Y. J. Lee & H. H. Shih
Then U N(0,1), V N(0,1), and U and V are independent. This
t t t t
∼ ∼
together with f(w) C(1 + w) implies the integrability of the right
| | ≤ | |
hand side of (2.23). Solving for Z, we obtain
Z =√tU +√1 tV .
t t
−
The equation (2.23) can be rewritten as
1 1 1
EWf(W) = Ef(ψ(√tU +√1 tV ))V ψ (U )
t t t ′ t
2 t(1 t) −
Z0 −
= 1 1 p1 Ef (W)ψ (Z)ψ (U )dt
′ ′ ′ t
2 √t
Z0
1 1 1
= E f′(W)E ψ′(Z)ψ′(√tZ+√1 tZ′)dt Z
2 √t −
(cid:20) (cid:18)Z0 (cid:19)(cid:21)
(cid:12)
where for the second equality we used the indepedence of U and(cid:12)V , and
t t
appliedthe characterizationequationforN(0,1)to V .Note thatthe char-
t
acterizatonequationisobtainedbyintegrationbyparts.Hencewehavefor
absolutely continuous f with bounded derivative,
EWf(W)=ET(Z)f′(W) (2.24)
where
1 1
T(x)= E ψ (x)ψ (√tx+√1 tZ ) dt. (2.25)
′ ′ ′
2√t −
Z0 h i
In[5],Chatterjeeobtainedamultivariateversionof(2.24)whereψ :Rd
−→
R for d 1.
≥
Here is a simple application of (2.21) and (2.24). Let X , ,X be
1 n
···
independent and identically distributed as N(0,1). Let
n (X2 1)
W = i=1 i − .
√2n
P
TherandomvariableW hasthestandardizedχ2distributionwithndegrees
X2 1 n
of freedom. Let ψ(X ) = i − . Then W = ψ(X ). Let W(i) = W
i i
√2n −
i=1
X
ψ(X ) and let X , ,X be an independent copy of X , ,X . By the
i i′ ··· n′ 1 ··· n
independence of X , ,X , and by (2.24) and (2.25),
1 n
···
n n
EWf(W) = Eψ(Xi)f′(W(i)+g(Xi))= ET(Xi)f′(W)
i=1 i=1
X X
=ETf′(W)