THEORY PROBAB. APPL.
Vol. 46, No. 3
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM
MEASURES AND MEASURE-VALUED PROCESSES∗
Z. G. SU†
Abstract. Let B be a separable Banach space. Suppose that (F, Fi , i 1) is a sequence of
independent identically distributed (i.i.d.) and symmetrical independently scattered
n (s.i.s.) B-valued
random measures. We first establish the central limit theorem for Yn = √1n
F by taking the
i=1 i
viewpoint of random linear functionals on Schwartz distribution spaces. Then, let (X, Xi , i 1) be
a sequence of i.i.d. symmetric B-valued random vectors and (B, Bi , i 1) a sequence of independent
standard Brownian motions on [0,1] independent
of (X, Xi , i 1). The central limit theorem for
n
measure-valued processes Zn (t) = √1n
Xδ
, t ∈ [0, 1], will be investigated in the same
i=1 i Bi (t)
frame. Our main results concerning Yn differ from D. H. Thang’s [Probab. Theory Related Fields,
88 (1991), pp. 1–16] in that we take into account F as a whole; while the results related to Zn are
extensions of I. Mitoma [Ann. Probab., 11 (1983), pp. 989–999] to random weighted mass.
Key words. central limit theorems, Gaussian processes, random vector measures, Schwartz
spaces
PII. S0040585X97979111
1. Introduction. The concept of measure-valued processes has its origin at the
evolution in time of the population. Consider a population, each of whose individuals
is represented by its state x ∈ R. Assume that the state of the population is completely described by the states of the individuals {xi , i ∈ I(t)}, where I(t) is the set
of living individuals at time t. A well-established representation for such a population
can be obtained by setting
(1.1)
X(t) =
ε δxi ,
i∈I(t)
where δx denotes the Dirac measure at point x, and ε is a normalizing factor which
may be one or may represent the mass of each particle if the individuals are particles.
In many situations one is led to consider sequences Xn (·) of this type of processes
and to study the limit of their laws. We refer to [5], [7], and [10]. Let {Bk (t), t ∈ [0, 1]},
k = 1, 2, . . . , be a sequence of independent one-dimensional Brownian motions with
Bk (0) = 0 for each k 1. Define a sequence of measure-valued processes Xn (t, ·) as
follows.
For a Borel subset A ∈ B(R),
Nn (t, A) =
n
δBk (t) (A)
k=1
∗ Received by the editors September 16, 1997. This work was supported by the Foundations of
National Natural Science of China and Zhejiang Province.
http://www.siam.org/journals/tvp/46-3/97911.html
† Department of Mathematics, Hangzhou University, 310028, China (zgsu@mail.hz.zj.cn).
448
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM MEASURES
449
and
(1.2)
1 Xn (t, A) = √ Nn (t, A) − ENn (t, A)
n
n
1
=√
δBk (t) (A) − P Bk (t) ∈ A
n
.
k=1
Since Xn (t, A) turns out to be extremely irregular as a measure in A when n becomes
large, one cannot expect the limit process X(t) to be a measure-valued process. However, letting S be the Schwartz space, we can consider Xn (t, ·) as a distribution-valued
stochastic process Xn (t) by setting
Xn (t)(ϕ) =
(1.3)
ϕ(x) Xn (t, dx)
n
1
=√
ϕ Bk (t) − E ϕ Bk (t)
n
R
,
ϕ ∈ S.
k=1
Both Itô [5] and Mitoma [7] show that there exists a distribution-valued stochastic
process X whose sample paths are elements in C([0, 1], S ) such that Xn converges
weakly to X.
One principal purpose of this paper is to extend the above result by assigning
a random mass to each individual. Let B be a separable Banach space, and let
(Xi , i 1) be a sequence of independent identically distributed (i.i.d.) symmetric
B-valued random variables and independent of all the Brownian motions. Define
(1.4)
n
1 Zn (t) = √
Xi δBi (t) ,
n i=1
t ∈ [0, 1].
Theorem 4.3 gives the limit law of the corresponding processes Zn , n 1.
For a fixed point t0 ∈ [0, 1], we have
n
1 Zn (t0 ) = √
Xi δBi (t0 ) ,
n i=1
n 1.
This is a sequence of normalized sums of vector random measures and has a more
general form. Let (Fi , i 1) be a sequence of i.i.d. and symmetric independently
scattered (s.i.s.) B-valued random measures defined on (R, R). Let
(1.5)
n
1 Xn = √
Fi .
n i=1
In section 3 we will study the limit law for Xn . Our basic ideas differ from
Thang’s [8] in that we regard Xn as random linear functionals defined on the Schwartz
space S.
Section 2 contains some notation about vector random measures and Schwartz
distribution spaces. Lemma 2.1, Proposition 2.1, and Proposition 4.1 are of fundamental importance for weak convergence of distribution-valued random variables,
while Proposition 2.2 shows how to realize random linear functionals.
Throughout the paper, unless specifically stated otherwise, c will denote a positive
constant, which may be different from line to line.
450
Z. G. SU
2. Basic concepts and preliminary statements. Let B be a separable Banach space and (R, R, µ) be a Lebesgue measure space. A set function F : R →
L0 (Ω, F, P; B) is called a B-valued s.i.s. random measure on R if
(i) for every sequence (En ) of disjoint sets from R the random variables F (E1 ),
F (E2 ), . . . are symmetric and independent;
(ii) for every sequence (En ) of disjoint sets from R we have
∞
∞
(2.1)
En =
F (En )
F
n=1
n=1
a.s. in the norm topology of B.
The Lebesgue measure µ on R is called a control measure for F if F (E) = 0 a.s.
whenever µ(E) = 0.
We can define random integrals of real-valued functions with respect to vectorvalued s.i.s. random measures as usual. Suppose that F is a B-valued s.i.s. random
measure withthe control measure µ. If f is a real-valued simple function defined
n
on R, f =
i are disjoint sets, then the integral with respect
i=1 ai IAi , where
A
n
to F is defined by R f dF = i=1 ai F (Ai ). Moreover, a function f on R is said to
be F -integrable if there exists a sequence of simple functions fn such that
(i) fn (t) → f (t), µ-a.e.;
(ii) the sequence R fn dF converges in probability.
If f is F -integrable, then we put R f dF = P-limn→∞ R fn dF .
By saying that {Fn , n 1} are i.i.d. random measures, we mean that {Fn (E),
E ∈ R}, n 1, are i.i.d. as a sequence of random processes on (Ω, F, P) indexed by
σ-field R. We refer to Theorem 4.3 in [8] for comparison. The fact that {Fn , n 1}
are independent copies of F only means that, for each E, {Fn (E), n 1} are independent and have the same distributions as F (E). Suppose that {F, Fn , n 1} is
a sequence of B-valued s.i.s. random measures on R. Note for any finite sequence
(E1 , E2 , . . . , Ek ) in R there exists a finite family Ap = (A1 , A2 , . . . , Ap ) of disjoint
sets such that each Ei is the union of some members from the family Ap . This implies
that each Fn (Ei ) can be expressed as a linear combination of {Fn (Ai ), 1 i p},
Fn (Ei ) =
p
bij Fn (Aj ),
j=1
where bij = 0 or 1 and do not depend on n. Consequently, for all n 1,




Fn (E1 )
Fn (A1 )
 .. 
 . 
(2.2)
 .  = B  ..  ,
Fn (Ek )
Fn (Ap )
where B = (bij ). From this we see that “for each E ∈ R, Fn (E) has the same
distribution as F (E)” is equivalent to saying that {Fn (E), E ∈ R} has the same
distribution as {F (E), E ∈ R}.
Now assume that we are given an s.i.s. random measure
nF and a sequence
{Fn , n 1} of independent copies of F . Define Xn = √1n i=1 Fi , n 1. As
usual we hope to regard F as a random vector in some topological vector space. Denoting by F(R; B) the set of all countably additive set functions from R into B, we
see at a glance that such a topological vector space should be F(R; B) equipped with
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM MEASURES
451
the weak topology (i.e., Fn converges weakly to F if for each bounded continuous
function f , limn→∞ R f dFn = R dF in the norm of B; see [1] for the weak topology). Unfortunately, the preceding definition of random measure does not guarantee
F (ω) ∈ F(R; B) for all or almost all ω in Ω. In addition, it is very hard for us to
discuss the weak topology in F(R; B) itself. To avoid difficulties we shall turn to
the study of a sequence of random linear mappings induced by Xn , as Itô [5] and
Mitoma [7] did.
Next we recall some known facts on the Schwartz distributions. D and D denote
the C ∞ -functions of compact support and the Schwartz distributions, respectively.
Hn is the Hermite polynomial of degree n and hn (x) is the corresponding Hermite
2
functions, i.e., hn (x) = cn Hn (x) e−x /2 , n = 0, 1, 2, . . . , where cn = ( π2 2n n!)−1/2 . The
Hermite functions form an orthonormal basis (ONB) in L2 (R). The p-norm · p
in L2 (R) and the (−p)-norm · −p in D are defined as follows:
ϕ2p
=
(2.3)
f 2−p
=
∞
n=0
∞
(ϕ, hn )2 (2n + 1)p ,
h2n (2n + 1)−p ,
n=0
where p = 0, 1, 2, . . . and f (ϕ) denotes the value evaluated at ϕ ∈ D.
It is clear that
f p = sup f (ϕ) : ϕ ∈ D, ϕp 1 ,
which implies
f (ϕ) f −p ϕp ,
f ∈ D , ϕ ∈ D.
Define Sp , (ϕ, ψ)p , Sp , and (f, g)−p as follows:
Sp = ϕ ∈ L2 (R) : ϕp < ∞ ,
Sp = f ∈ D : f −p < ∞ ,
∞
(ϕ, ψ)p =
(ϕ, hn )(ψ, hn )(2n + 1)p ,
(2.4)
(f, g)−p =
n=0
∞
f (hn ) g(hn )(2n + 1)−p .
n=0
Then (Sp , (·, ·)p ) and (Sp , (·, ·)−p ) are Hilbert spaces with the norms · p and · −p ,
respectively. As p increases, · p increases and · −p decreases. So Sp decreases
and Sp increases. The intersection ∩p Sp coincides with the space S of rapidly decreasing functions and the union ∪p Sp with the space S of tempered distributions. It
follows from the definition that S0 = L2 (R), · 0 = · , and (·, ·)0 = (·, ·). By identifying ψ ∈ L2 (R) with [ψ] ∈ D , where [ψ](ϕ) =: (ψ, ϕ), we have L2 = S0 , · = · −0
and (·, ·) = (·, ·)−0 . Hence
(2.5)
D ⊂ S ⊂ · · · ⊂ S2 ⊂ S1 ⊂ S0 = L2 (R) = S0 ⊂ S1 ⊂ · · · ⊂ S ⊂ D .
In other words, S and S are nuclear Frechet spaces.
452
Z. G. SU
Let L(S, B) be the set of all continuous linear mappings from S into B. Let us
equip L(S, B) with the strong topology, i.e., the topology of uniform convergence on
all bounded subsets of S, and denote it by Lb (S, B). This is a completely regular
topological vector space. Analogously, Lb (Sp , B) is a Banach space.
The following lemma due to Le Cam (see [10, Thm. 6.7]) is an extension of
Prokhorov’s theorem.
Lemma 2.1. Let E be a completely regular topological space such that all compact
sets are metricable. If {Pn , n 1} is a sequence of probability measures on E which
is uniformly tight, then there exist a subsequence (nk ) and a probability measure Q
such that Pnk ⇒ Q.
To apply Lemma 2.1 to Lb (S, B) we need the following proposition.
Proposition 2.1. If K is compact in Lb (S, B), then there exists a natural
number p such that K is compact in Lb (Sp , B).
Proof. For each ϕ ∈ S define πϕ : Lb (S, B) → B as πϕ T = T ϕ, T ∈ Lb (S, B).
If Tn → T in the strong topology, then πϕ Tn − πϕ T = Tn ϕ − T ϕ → 0. So πϕ
is a continuous linear mapping. Thus {πϕ T, T ∈ K} is compact in B whenever K is
compact in Lb (S, B). This implies
sup sup T ϕ < ∞.
(2.6)
ϕ∈S T ∈K
By the Banach–Steinhaus theorem (see [9, Thm. 33.1]), K is equicontinuous
in Lb (S, B); i.e., for any neighborhood of zero V in B there is a neighborhood of
zero U in S such that for all mappings T ∈ K we have that ϕ ∈ U implies T ϕ ∈ V . In
particular, letting V = {x : x 1}, U contains a basis element, say {ϕ : ϕq < ε},
such that T ϕ ∈ V . Thus
(2.7)
sup T ϕ T ∈K
cϕq
ε
for all ϕ ∈ S.
For such a q there exists a natural number p > q such that the natural embedding
i : Sp → Sq is nuclear, and hence compact. The closure S of {ϕ : ϕp 1} is
compact in Sq and so is {T ϕ, T ∈ K, ϕ ∈ S}.
On the other hand, supT ∈K T ϕ − T ψ cϕ − ψq . If we consider K as a subset
of C(S, B), the set of continuous mappings from S into B, the required result is easily
obtained.
Let us turn back to the existence of the random integral R ϕ dF for any given
ϕ ∈ S and s.i.s. random measure F on (R, R). Suppose that B is of type 2 space
2
and
n E F (E) cµ(E) for all E ∈ R; then for any simple function ϕn (t) =
i=1 ai IEi (t),
E
R
n
2
2
n
2
ϕn (t) dF =
E
a
F
(E
)
c
a2i E F (Ei )
i
i
i=1
(2.8)
c
n
i=1
a2i µ(Ei ) = c
i=1
R
ϕ2n (t) dµ(t).
Thus ϕ must be integrable with respect to random measure F and
E
R
2
ϕ(t) dF c
R
ϕ2 (t) dµ(t) = cϕ20 .
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM MEASURES
453
Whenever the random integral R ϕ dF exists for each ϕ ∈ S can we define a random
linear mapping TF : TF (ϕ) = R ϕ dF from S into B.
Proposition 2.2. Let Y = {Y (ϕ) : ϕ ∈ S} be a family of B-valued random
variables such that Y (ϕ) is
(L) almost linear,
Y (C1 ϕ1 + C2 ϕ2 ) = C1 Y (ϕ1 ) + C2 Y (ϕ2 )
a.s.,
where the exceptional ω-set may depend on the choice of (C1 , ϕ1 ; C2 , ϕ2 );
(B) p-bounded,
2
E Y (ϕ) ϕ2p .
Then Y has an Lb (Sp+2 , B) regularization X satisfying
E X2−p−2 π2 c
.
8
Here and in what follows we denote by X−p−2 the operator norm of X in
Lb (Sp+2 , B).
Proof. Conditions (L) and (B) imply that the map Y : ϕ → Y (ϕ) is a bounded
linear operator from the pre-Hilbert space (S, · p ) into L2 (Ω, F, P; B). Since S
is dense in Sp , this map can be extended to a unique linear operator from Sp into
L
J 2 (Ω, F, P; B) still denoted by the same notation Y . Let hn be the Hermite functions;
then εn = (2n + 1)(p+2)/2 hn ∈ Sp , n = 0, 1, 2, . . . , form an ONB in Sp+2 . The ONB
in Sp+2
dual to εn is denoted by {en }. It is clear that
E
(2.9)
∞
∞
Y (εn )2 c
εn 2p < ∞.
n=1
n=1
2
∞ < ∞}. Define
This implies that P(Ω1 ) = 1, where Ω1 = {ω ∈ Ω :
n=1 Y (εn )(ω)
∞


Y (εn )(ω) en on Ω1 ,
X(ω) = n=1
(2.10)


0
otherwise.
Then X(ω) ∈ Lb (Sp+2 , B) for every ω, and X(ω) is measurable in ω. Hence X is an
Lb (Sp+2 , B)-valued variable and
∞
2
2
E X−p−2 = E sup X(ω)(ϕ)B = E sup sup f Y (ε) (ω)(en , ϕ)
f
ϕ p+2 1
ϕ p+2 1
(2.11)
∞
n=1
E Xn 2B n=1
2
π c
.
8
Since em (εn ) = δmn and P(Ω1 ) = 1, it follows that X(ϕ) = Y (ϕ) a.s. for ϕ = εn ,
n = 1, 2, . . . , and so on for every finite linear combination of {εn } by (L). For a general
ϕ ∈ Sp+2 ⊂ Sp we have
ϕ − ϕn p+2 → 0
(n → ∞),
ϕn =:
n
k=0
(ϕ, εk )p+2 εk .
454
Z. G. SU
Since X ∈ Lb (Sp+2 , B), it follows that
X(ϕ) − X(ϕn ) = X−p−2 ϕ − ϕn p+2 −→ 0
for every ω. Also, we can use (B) to check that
(2.12)
2
E Y (ϕ) − Y (ϕn ) cϕ − ϕn 2p cϕ − ϕn 2p+2 → 0.
Thus X(ϕ) = Y (ϕ) a.s. for every ϕ ∈ Sp+2 .
3. The CLT for vector random
n measures. After making the preceding
preparations, we can consider Xn = √1n i=1 Fi as an Lb (S, B)-valued random vector
n
by setting Xn (ϕ) = √1n i=1 R ϕ dFi , ϕ ∈ S. It is on this viewpoint that we base
our studies of weak convergence for random measures.
Now we shall state and prove our main results.
Theorem 3.1. Suppose that B is the space of type 2 and that {F , Fn , n 1} is
a sequence of i.i.d. and s.i.s. random measures on (R, R, µ). If E F (E)4 cµ2 (E),
and if {F (E)/µ1/2 (E), E ∈ R, µ(E) > 0} is uniformly tight in B, then there exists
an Lb (S, B)-valued Gaussian vector G such that {Xn (ϕ), ϕ ∈ S} converges weakly
to G.
Proof. By Lemma 2.1 it suffices to verify the following two statements.
(i) Xn is uniformly tight in Lb (S, B); i.e., for any ε > 0 there is a compact set
K ⊂ Lb (S, B) such that
(3.1)
/ K < ε;
sup P ω : Xn (ω) ∈
n
(ii) for each f ∈ Lb (S, B), f (TF ) satisfies the classical CLT.
Let us first prove (i). If K is compact in Lb (Sq , B) for some q, then K is also
compact in Lb (S, B) since the injection of Lb (Sq , B) into Lb (S, B) is continuous.
In the scalar case (i.e., B = R), for every p ∈ N there is a q > p, q ∈ N, such
that the natural embedding i : Sq → Sp is nuclear and so compact; thus its adjoint
i∗ : Sp → Sq is also compact (see [2, Thm. VI.5.2]). Equivalently, any bounded subset
of Sp is relatively compact in Sq .
In the general case (i.e., where B is a separable Banach space), however, it is
possible that a bounded subset of Lb (Sp , B) is not necessarily relatively compact
in Lb (Sq , B). So we need to make some necessary changes as follows.
For each p ∈ N, k > 0, and compact subset KB in B, define
(3.2)
K = T : T −p k
T : T ϕ, ϕq 1 ⊂ KB ,
where q is such that i : Sq → Sp is nuclear.
We claim that K is also relatively compact in Lb (Sq , B).
Indeed, let S be the closure of {ϕ : ϕq 1} in Sp ; then S is compact in Sp .
Since T ϕ − T ψ kϕ − ψq , the set K is equicontinuous in Lb (Sq , B). In addition,
{T ϕ : T ∈ K; ϕ ∈ S} ⊂ KB . Thus considering K as a subset of C(S, B), the set of
continuous mappings from S into B, we easily deduce that K is relatively compact
in C(S, B) and thus in Lb (Sq , B).
Thus in order to verify (i) it is enough to show that
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM MEASURES
455
(i ) for each ε > 0, there exist p, q ∈ N (q > p), k > 0, and a compact subset
KB ⊂ B such that
sup P ω : Xn −p > k < ε,
(3.3)
n
(3.4)
/ KB for some ϕ, ϕq 1 < ε.
sup P ω : Xn (ϕ) ∈
n
Equivalently, according to the well-known fact (see [6, Lem. 2.2]), a subset KB
of B is relatively compact if and only if it is bounded, and for each ε > 0 there is a
finite-dimensional closed subspace F of B such that supx∈KB qF (x) < ε, where qF is
the quotient norm, we need only check the following statement.
For each ε > 0 there exist p, q ∈ N (q > p), k > 0, and a finite-dimensional closed
subspace F of B such that
sup P ω : Xn −p > k < ε,
(3.5)
n
(3.6)
sup P ω : sup qF Xn (ϕ) > ε < ε.
n
ϕ
q 1
This will in turn be done in two steps.
Step 1. For each ε > 0 there exist p ∈ N and k > 0 such that
(3.7)
sup P ω : sup Xn (ϕ) > k < ε.
n
ϕ
p 1
The idea behind the proof of (3.7) is similar to that of Theorem 6.12 in [10].
(1) For any ε > 0, there exist m and δ > 0 such that
ϕm < δ implies sup E Xn (ϕ) ∧ 1 < ε.
(3.8)
n
To see this, consider the function F (ϕ) = supn E (Xn (ϕ) ∧ 1), ϕ ∈ S. Then
(1.i) F (0) = 0,
(1.ii) F (ϕ) 0 and F (ϕ) = F (−ϕ),
(1.iii) F (aϕ) < F (bϕ) if |a| < |b|,
(1.iv) F is lower-semicontinuous on S.
Indeed, if ϕj → ϕ in S, then Xn (ϕj ) → Xn (ϕ) in L
J 0 (Ω, F, P; B), and hence
Xn (ϕj ) ∧ 1 → Xn (ϕ) ∧ 1 in probability. Then
F (ϕ) = sup E Xn (ϕ) ∧ 1 sup lim inf E Xn (ϕj ) ∧ 1
n
n
j
lim inf sup E Xn (ϕj ) ∧ 1 = lim inf F (ϕj ).
j
n
j
(1.v) limn F (ϕ/n) = 0.
For this, note that for any given ε > 0 and ϕ ∈ S there exists a k > 0 such that
E X (ϕ)2
E
n
sup P ω : Xn (ϕ) > k =
2
k
n
ϕ dF 2
cϕ20
ε
< .
2
2
k
k
2
R
Choose M large enough so that k/M < ε/2 < 1; then
! "
! "
ϕ
k
k
ϕ Xn (ϕ)
F
= sup E ∧
1
sup
>
+
< ε.
X
P
ω
:
n M M
M
M
M
n
n
456
Z. G. SU
Let V = {ϕ : F (ϕ) ε}, where V is closed symmetric absorbing. We claim that
it is a neighborhood of 0. Indeed S = ∪n nV , so by the Baire category theorem, one
and hence all, of the nV must have a nonempty interior. In particular, 12 V does.
Then V ⊂ 12 V − 12 V must contain a neighborhood of 0. This in turn must contain
an element of basis, say {ϕm < δ}.
(2) For all n 1,
"
!
ϕ2m
E sup Re 1 − ei f,Xn (ϕ) 2ε 1 +
(3.9)
, ϕ ∈ S.
δ2
f ∈B In fact, in view of (1) we have
#
$ 1 #
$2
Re 1 − ei f,Xn (ϕ) = 1 − cos f, Xn (ϕ) f, Xn (ϕ) ∧ 2 .
2
If ϕm < δ, then
E sup Re 1 − ei
f,Xn (ϕ)
f ∈B #
$2
1
E sup f, Xn (ϕ) ∧ 2
2 f ∈B 2E sup f, Xn (ϕ) ∧ 1 < 2ε.
f ∈B On the other hand, if ϕm δ, we replace ϕ by ψ = δϕ/ϕm and obtain
#
$2 ϕ2m
1
ϕ2m
E sup Re 1 − ei f,Xn (ϕ) E sup f, Xn (ϕ)
∧
2
2ε
.
2 f ∈B δ2
δ2
f ∈B The required equation (3.9) is proved.
(3) There are p > m and M such that for all n and k > 0,
"
!
M
−k−1 supϕp 1 Xn (ϕ) 2
(3.10)
2ε 1 + 2 .
E 1−e
kδ
Indeed, there exists a p > m such that the natural embedding i : Sp → Sm is a
Hilbert–Schmidt operator. Let (ej ) be a complete orthogonal normed system (CONS)
in S relative to · p . Let Y1 , Y2 , . . . be i.i.d. N (0, 2/k) random variables and put
N
ΦN = j=1 Yj ej . Then by (2),
!
"
i f,Xn (ΦN )
i f,Xn (ΦN )
= EY EX sup Re 1 − e
E sup Re 1 − e
Y
f ∈B !
f ∈B E 2ε 1 +
On the other hand,
E sup Re 1 − ei
"
ΦN 2m
.
δ2
N
i
= E sup Re 1 − e j=1
f,Xn (ΦN )
f ∈B f,Xn (ej ) Yj
f ∈B N
i
E sup E Re 1 − e j=1
f,Xn (ej ) Yj
f ∈B However, given X, the conditional distribution of
!
N
j=1 f, Xn (ej )Yj
"
N
2
2
N 0,
f, Xn (ej ) .
k j=1
is
X .
457
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM MEASURES
So the right-hand side of the last inequality is
N
−k−1
j=1
E sup 1 − e
f,Xn (ej )
2
f ∈B Since
sup Xn (ϕ) = sup
ϕ
p 1
ϕ
p 1
sup f, Xn (ϕ)
f ∈B %
= sup sup
f ∈B = sup
f ∈B .
∞
ai f, Xn (ej ),
i,j=1
∞
∞
&
a2i 1
i=1
1/2
f, Xn (ej )2
,
j=1
∞
then setting M =: 2 j=1 ej 2m , we easily obtain (3.10) as follows:
"
!
"
!
−1
−1
2
2
= E 1 − e−k supϕp 1 Xn (ϕ)
E 1 − e−k supϕp 1 Xn (ϕ)
!
"
∞
−k−1 supf ∈B f,Xn (ej ) 2
j=1
=E 1−e
∞
−k−1
f,Xn (ej ) 2
j=1
= E sup 1 − e
f ∈B = E sup lim
f ∈B N →∞
(3.11)
−k−1
1−e
N
j=1
N
−k−1
j=1
lim E sup 1 − e
N →∞
f ∈B (4) Markov’s inequality gives
2
P ω : sup Xn (ϕ) > k ϕ
p 1
f,Xn (ej )
2
f,Xn (ej )
2
"
!
M
2ε 1 + 2 .
kδ
−1
e
E 1 − e−k supϕp 1
e−1
Xn (ϕ)
2
.
The proof of (3.7) is complete.
Step 2. For any ε > 0 there exists a finite-dimensional closed subspace F in B
such that
sup P ω : sup qF Xn (ϕ) > ε < ε,
(3.12)
n
ϕ
q 1
where q is determined by p of Step 1.
The proof for (3.12) involves the same idea as for (3.7) but requires stronger
assumptions. Given ε > 0, let p be chosen as in Step 1. There exists q > p such that
∞
if (ej ) is a CONS in Sq ⊂ Sp , then C 2 =: j=1 ej 2p < ∞. Letting Y1 , Y2 , . . . be a
sequence of i.i.d. normal random variables N (0, 2/ε), we have
2
& !
%
"2 ∞
∞
ε
4C
P ω: Yj ej >
E
Yj ej ε
4C
j=1
j=1
p
p
"2 !
∞
ε
ε
(3.13)
E
Yj2 ej 2p .
4C
8
j=1
458
Z. G. SU
We continue to prove that there exists a finite-dimensional closed subspace F in B
such that
ε
ε
sup
(3.14)
< .
sup P ω : qF Xn (ϕ) >
8
8
ϕ p 4C/ε n
Since B is the space of type 2, then for any closed subspace F in B, B/F is also
of type 2. So
!
"
ε
64c
64
2 E qF2 Xn (ϕ) 2 E qF2
P ω : qF Xn (ϕ) >
(3.15)
ϕ dF .
8
ε
ε
R
Now the key to our problem is to estimate E qF2 ( R ϕ dF ). If we are able to prove
that, for any η > 0, there is a finite-dimensional closed subspace F in B such that for
any E ∈ R,
E qF2 F (E) c ηµ(E),
(3.16)
then it is not hard to obtain
!
2
E qF
(3.17)
"
R
ϕ dF
cη
R
ϕ2 dµ = c ηϕ20 .
Thus (3.14) can be proved by choosing η small enough, i.e., taking a finite-dimensional
closed subspace F .
Let us return to the proof of (3.16). Since {F (E)/µ1/2 (E), E ∈ R} is uniformly
tight, then for any η > 0 there exists a finite-dimensional closed subspace F such that
sup P ω : qF2 F (E) > ηµ(E) < η 2 .
E
Moreover, we have
E qF2 F (E) ηµ(E) +
{ω :
2 (F (E))>ηµ(E)}
qF
qF2 F (E) dP
4 1/2 P ω : qF2 F (E) > ηµ(E)
ηµ(E) + E F (E)
1/2
ηµ(E) + cµ(E)η (1 + c) ηµ(E).
After presenting the previous argument, we can now prove (3.12). In fact we
have, in a similar way to the proof of (3.10),
%
&
−1
2
e
2
P ω : sup qF Xn (ϕ) > ε E 1 − e−ε supϕq 1 qF (Xn (ϕN ))
e−1
ϕ q 1
e
(3.18)
lim E sup Re 1 − ei f,Xn (ΦN ) ,
e − 1 N →∞ f ∈F ⊥
where
ΦN =
N
j=1
Yj ej ,
{Yj , j 1}
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM MEASURES
459
is a sequence of i.i.d. N (0, 2/ε), {ej } is a CONS in S relative to · q , and F ⊥ is the
annihilator of F .
The limit term of (3.18) can be estimated as follows:
E sup Re 1 − ei f,Xn (ΦN ) = E sup Re 1 − ei f,Xn (ΦN ) I( ΦN p 4C/ε)
f ∈F ⊥
f ∈F ⊥
+ E sup Re 1 − ei
f,Xn (ΦN )
I(
f ∈F ⊥
ΦN
p >4C/ε)
4C
ϕN p 4C/ε) + 2P ω : ϕN p >
ε
∞
&
%
4C
2
sup
E qF Xn (ϕ) ∧ 1 + 4P ω : Yj ej >
ε
ϕ p 4C/ε
j=1
!
"
ε
ε
+
sup
<2
P ω : qF Xn (ϕ) >
8
8
ϕ p 4C/ε
%
&
∞
4C
(3.19) + 4P ω : < ε.
Yj ej >
ε
j=1
2E qF Xn (ΦN ) ∧ 1 I(
Up to now, we have completed the key statement (i), i.e., Xn is uniformly tight
in Lb (S, B). Next we shall go on to prove (ii).
For this let us recall some preliminary results concerning the integral representation formula of dual Lb (S, B) of Lb (S, B) (see [9] for details). Let B(S , B) denote
the space of continuous bilinear forms on S × B and carry the topology of uniform
' denote the projective
convergence on the products of bounded sets, and let S ⊗B
tensor product of S and B. Since S and S are nuclear spaces, then we have the
canonical isomorphism
(3.20)
' ∼
S ⊗B
= Lb (S, B),
' ∼
(S ⊗B)
= B(S , B),
' carries the strong dual topology. The following lemma is
where the dual (S ⊗B)
Proposition 49.1 in [9].
Lemma 3.1. A bilinear form u on S × B is continuous if and only if there is a
weakly closed equicontinuous subset A (respectively, M ) of S (respectively, B ) and a
positive Radon measure v on the compact set A × M , with the total mass 1, such
that for all ϕ ∈ S , x ∈ B,
(3.21)
u(ϕ , x) =
A×M ϕ , ϕ x , x dv(ϕ, x ).
For any given f ∈ Lb (S, B) and TF ∈ Lb (S, B) there exist ϕF ∈ S and xF ∈ B
such that TF is the image of ϕF ⊗ xF under the canonical isomorphism mapping,
and further there is a ψf ∈ B(S, B) such that ψf (ϕF , xF ) = f (TF ). From this
and Lemma 3.1 we derive that there are a weakly closed equicontinuous subset A
(respectively, M ) of S (respectively, B ) and a positive Radon measure v on the
compact set A × M with the total mass 1 such that
f (TF ) = ψf (ϕF , xF ) =
(3.22)
=
A×M A×M ϕF , ϕ x , xF dv(ϕ, x )
x , TF (ϕ) dv(ϕ, x ).
460
Z. G. SU
Thus we have
E f 2 (TF ) = E
(3.23)
!
A×M A×M "2
x , TF (ϕ) dv(ϕ, x ) 2
x 2 E TF (ϕ) dv(ϕ, x ) c
A×M Ex , TF (ϕ)2 dv(ϕ, x )
A×M x 2 ϕ20 dv(ϕ, x ).
In addition, since A and M are weakly compact in S and B , respectively, then A
and M are weakly bounded, and hence A is bounded with respect to the semi-norms
· p , 0 p < ∞, and M is bounded with respect to the dual norm. This, together
with (3.23), implies that E f 2 (TF ) < ∞. Thus (ii) is a direct consequence of the
classical CLT.
The proof of Theorem 3.1 is now complete.
4. The CLT for measure-valued processes arising from Brownian motions. Suppose that {X, Xn , n 1} is a sequence of i.i.d. symmetric B-valued
random vectors, and {B(t), Bi (t), i 1, 0 t 1} is a sequence of i.i.d. standard
Brownian motions. Assume furthermore that they are independent of each other. Set
n
(4.1)
1 Zn (t) = √
Xi δBi (t) ,
n i=1
0 t 1.
This is a measure-valued stochastic process. As in section 3 we shall consider Zn as
a sequence of Lb (S, B)-valued continuous random processes on [0, 1]. The reader is
referred to the case having no weighted term Xi in Zn . To make our problem clearer,
we will look at two special cases.
Theorem 4.1. Suppose that B is the space of type 2 and that {X, Xi , i 1}
and {B(t), Bi (t), i 1, 0 t 1} are as above with E X2 < ∞. Define
n
(4.2)
1 Zn (t0 ) = √
Xi δBi (t0 )
n i=1
for some t0 ∈ [0, 1].
Then {Zn (t)(ϕ), ϕ ∈ S} converges weakly to a Gaussian process G = {G(ϕ), ϕ ∈ S}.
Proof. This in fact is a direct consequence of the proof of Theorem 3.1. Observe
that in the proof of Theorem 3.1 the hypotheses that E F (E)4 cµ2 (E) for some
constant c > 0 and {F (E)/µ1/2 (E), E ∈ R} is uniformly tight in B are used only
in (3.16).
However, if we let F (E) = XδB(t0 ) (E), then
E qF2 F (E) = E qF2 (X) E δB(t0 ) (E) = E qF2 (X)P ω : B(t0 , ω) ∈ E .
Obviously, for any ε > 0 there exists a finite-dimensional closed subspace F in B such
that E qF2 (X) < ε whenever E X2 < ∞. In addition, P{ω : B(t0 , ω) ∈ E} cµ(E)
for some numerical constant c > 0. These imply (3.16) and hence finish the proof of
Theorem 4.1.
Theorem 4.2. Suppose that B is the space of type 2 and that {X, Xi , i 1}
and {B(t), Bi (t), i 1, 0 t 1} are as above with E X2 < ∞. Define
(4.3)
n
1 Xi ϕ Bi (t) ,
Zn (ϕ)(t) = √
n i=1
0 t 1,
for some ϕ ∈ S.
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM MEASURES
461
Then {Zn (ϕ)(t), 0 t 1} converges weakly to a Gaussian process Gϕ = {Gϕ (t),
0 t 1} in C([0, 1], B), the set of all bounded continuous mappings from [0, 1]
into B with the uniform norm topology.
Proof. It is clear that for two finite sequences (t1 , t2 , . . . , tm ) and (f1 , f2 , . . . , fm )
the m-dimensional random vector (f1 (X) ϕ(B(t1 )), . . . , fm (X) ϕ(B(tm ))) satisfies the
classical CLT. Also, it is easily seen that the class of all cylindrical sets having the
form
#
$
$
x ∈ C [0, 1], B :
f1 , x(t1 ) , . . . , fm , x(tm ) ∈ A, A ∈ B(Rm )
generates the Borel σ-field B(C([0, 1], B)). Thus we need only turn our attention to
the relative compactness of {Zn , n 1} in C([0, 1], B). By Theorem 4.4 in [4] and
Lemma 2.2 in [6], this will be shown if
(a) {f, Zn (ϕ)(t), t ∈ [0, 1]} is relatively compact in C([0, 1]) for each f ∈ B ;
(b) for any ε > 0 there exist a positive constant M and a finite-dimensional closed
subspace F in B such that
sup P ω : sup Zn (ϕ)(t) > M < ε,
(4.4)
n
sup P ω :
(4.5)
n
0t1
sup qF
0t1
Zn (ϕ)(t) > ε
< ε.
For (a) it is enough to verify that for any ε > 0 there are δ > 0 and M > 0 such
that
%
&
n
1 sup P ω : sup √
(4.6)
f (Xi ) ϕ Bi (t) − ϕ Bi (s) > ε < ε,
n
|t−s|<δ n i=1
%
&
n
1 (4.7)
f (Xi ) ϕ Bi (t) > M < ε.
sup P ω : sup √
n
0t1 n i=1
Let us prove (4.6) and (4.7) by making use of the metric entropy techniques. In
fact, if we denote
%
&
n
f 2 (Xi ) a2 n
D1 = ω :
i=1
we have
%
&
n
1 P ω : sup √
f (Xi ) ϕ Bi (t) − ϕ Bi (s) > ε
|t−s|<δ n i=1
%
n
1 (4.8) P ω : sup √
f (Xi ) ID1 ϕ Bi (t) − ϕ Bi (s)
n i=1
|t−s|<δ
and
(4.9)
&
> ε + P(D1 )
&
n
1 P ω : sup √
f (Xi ) ϕ Bi (t) > M
0t1 n i=1
&
%
n
1 f (Xi ) ID1 ϕ Bi (t) > M + P(D1 ),
P ω : sup √
0t1 n i=1
%
where a, δ, and M are to be specified later.
462
Z. G. SU
Taking a2 2Ef 2 (X)/ε, we have
%
&
n
ε
f 2 (Xi ) a2 n < .
sup P ω :
2
n
i=1
On the other hand, set β1 (ϕ) = supx∈R |dϕ(x)/dx| < ∞; then
ϕ Bi (t) − ϕ Bi (s) β1 (ϕ) Bi (t) − Bi (s),
and hence we obtain
%
&
n
1 1/2
P ω: √
f (Xi ) ID1 ϕ Bi (t) − ϕ Bi (s) > aβ1 (ϕ) |t − s| u
n
i=1
&
%
n
1 1/2
εi f (Xi ) ID1 ϕ Bi (t) − ϕ Bi (s) > aβ1 (ϕ) |t − s| u
= P √
n
i=1
&
%
n
1 Bi (t) − Bi (s) εi f (Xi ) ID1
2EX P ω : √
> au
n
|t − s|1/2 i=1
"
"
!
!
u2
u2 a2
n
.
4
exp
−
4EX exp −
2
(2/n) i=1 f 2 (Xi ) ID1
In this way, it is easy to see that
% √
n
( n)−1 i=1 f (Xi ) ID1 (ϕ(Bi (t)) − ϕ(Bi (s)))
,
aβ1 (ϕ)
&
t, s ∈ [0, 1]
is a sub-Gaussian process with the metric d(s, t) = |t − s|1/2 . By Theorem 11.6 in [6]
we know that
n
1 sup E sup √
f (Xi ) ID1 ϕ Bi (t) − ϕ Bi (s) n
|t−s|<δ n i=1
δ
1/2
log N [0, 1], d, ε
kaβ1 (ϕ)
(4.10)
dε,
0
n
1 E sup √
f (Xi ) ID1 ϕ Bi (t) 0t1 n i=1
1
1/2
log N [0, 1], d, ε
(4.11)
dε < ∞,
kaβ1 (ϕ)
0
where k is an absolute constant and N ([0, 1], d, ε) is the smallest number of balls of
radius ε in the metric d which covers [0, 1].
Now we can choose δ so small that (4.6) holds and M so large that (4.7) holds.
Thus the proof of (a) is concluded.
It remains to prove (b). The idea of its proof is similar to that used above; i.e.,
we can still apply the metric entropy techniques to some appropriate vector-valued
sub-Gaussian processes.
Set ψ2 (x) = exp x2 −1; ·ψ2 (dP) denote its Orlicz norm with respect to probability
space (Ω, F, P). We only prove (4.5) since the proof of (4.4) is similar and simpler.
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM MEASURES
By the well-known contractive principle we have
n
1 1 2
√
q
Xi ID2 ϕ Bi (t) − ϕ Bi (s)
E exp
F
2
C
n i=1
n
1 β12 (ϕ) 2
√
(4.12)
q
Xi ID2 Bi (t) − Bi (s)
E exp
F
2
C
n i=1
463
,
where C > 0 is arbitrary, a and F are to be specified below, and
%
&
n
2
2
D2 = ω :
qF (Xi ) a n .
i=1
Then
n
1 Xi ID2 ϕ Bi (t) − ϕ Bi (s)
qF √
n i=1
ψ2 (dP)
n
1 (4.13) β1 (ϕ) qF √
Xi ID2 Bi (t) − Bi (s) n i=1
c β1 (ϕ) a|t − s|1/2 .
ψ2 (dP)
From the above and noting Remark 11.5 in [6] we deduce
n
1 Xi ID2 ϕ Bi (t) − ϕ Bi (s)
E sup qF √
n i=1
0t1
1
1/2
log N [0, 1], d, ε
c β1 (ϕ) a
(4.14)
dε.
0
After taking a > 0 so small that
cβ1 (ϕ) a
1
0
log N [0, 1], d, ε
1/2
dε <
ε
,
2
we can choose a finite-dimensional closed subspace F in B such that E qF2 (X)/a2 <
ε/2. Moreover, we have
&
%
n
1 Xi ϕ Bi (t)
>ε
P ω : sup qF √
n i=1
0t1
&
%
n
1 (4.15)
Xi ID2 ϕ Bi (t)
> ε + P D2 < ε.
P ω : sup qF √
n i=1
0t1
Up to now we have shown (b), and hence Theorem 4.2.
At the end of this paper we shall prove the following theorem, which motivates
the present work.
Theorem 4.3. Suppose that B is the space of type 2 and that {X, Xi , i 1}
and {B(t), Bi (t), i 1, 0 t 1} are as above with E X2 < ∞. Define
(4.16)
n
1 Xi ϕ Bi (t) ,
Zn (ϕ)(t) = √
n i=1
0 t 1, ϕ ∈ S.
464
Z. G. SU
Then Zn converges weakly to a Gaussian process G in C([0, 1], Lb (S, B)), equipped
with the locally convex topology generated by a family of seminorms.
The proof of Theorem 4.3 is basically along the lines given in [7] and [10], with
some necessary changes made according to Theorem 3.1. For the reader’s convenience,
we give some key points below.
Proposition 4.1. If K is compact in C([0, 1], Lb (S, B)), then there exists a
p ∈ N such that K is also compact in C([0, 1], Lb (Sp , B)).
Proof. For each ϕ in S the set {x(ϕ), x ∈ K} is compact in C([0, 1], B); then the
following properties hold: {x(ϕ)(t), 0 t 1, x ∈ K} is relatively compact in B
and
lim sup sup x(ϕ)(t) − x(ϕ)(s) = 0.
(4.17)
δ→0 x∈K |t−s|<δ
Since supx∈K sup0t1 x(ϕ)(t) < ∞, the Banach–Steinhaus theorem shows that
there exist q ∈ N and L > 0 such that
(4.18)
sup sup x(ϕ)(t) Lϕq .
x∈K 0t1
Since S is nuclear,
there are a natural number r > q and a CONS (ej ) relative
∞
to Sr such that j=1 ej 2q < ∞, so that we have
sup sup
2
sup x(ϕ)(t) = sup sup
x∈K 0t1 ϕ
r 1
sup
x∈K 0t1 ϕ
r 1
= sup sup sup
x∈K 0t1
f ∈B1
sup sup sup
(4.19)
#
$2
sup f, x(ϕ)(t) f ∈B1
∞
#
$2
f, x(ej )(t)
j=1
∞
x∈K 0t1 f ∈B1 j=1
2
f 2 x(ej )(t) < ∞.
Since supx∈K sup|t−s|<δ x(ej )(t) − x(ej )(s)2 4L2 ej 2q , by the Lebesgue convergence theorem and (4.17) we get
lim sup sup sup x(ϕ)(t) − x(ϕ)(s)
δ→0 x∈K |t−s|<δ ϕ
= lim sup sup
r 1
sup
sup
δ→0 x∈K |t−s|<δ f ∈B1 ϕ
lim sup sup
sup
δ→0 x∈K |t−s|<δ f ∈B1
(4.20)
∞
#
r 1
∞
#
$
f, x(ϕ)(t) − x(ϕ)(s)
$2
f, x(ej )(t) − x(ej )(s)
1/2
j=1
2
lim sup sup x(ej )(t) − x(ej )(s)
j=1 δ→0 x∈K |t−s|<δ
1/2
= 0.
On the other hand, since S is nuclear, there exist a natural number p > r and a
CONS (ej ) relative to Sp such that
∞
j=1
ej 2r < ∞.
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM MEASURES
465
Then it follows from (4.19) that
lim
N →∞
∞
#
$
2
sup sup x(t), ej = 0,
j=N x∈K 0t1
so that {x(t), x ∈ K} is relatively compact in Lb (Sp , B) for each 0 t 1.
Since · −r · −p , by (4.20) we have
lim sup sup sup x(ϕ)(t) − x(ϕ)(s) = 0.
δ→0 x∈K |t−s|<δ ϕ
p 1
Thus K has a compact closure in C([0, 1], Lb (Sp , B)) and K is automatically closed
in C([0, 1], Lb (Sp , B)) by the definition of the topology on C([0, 1], Lb (Sp , B)). The
proof of Proposition 4.1 is completed.
Proposition 4.1 will enable us to use Lemma 2.1. Now for each ε > 0 we try to
prove the following two claims.
Claim 1. There exist p ∈ N and M > 0 such that
ε
(4.21)
sup P ω : sup sup Zn (ϕ)(t) > M < .
4
n
0t1 ϕ p 1
Claim 2. There exists a finite-dimensional closed subspace F in B such that
ε
sup P ω : sup sup qF Zn (ϕ)(t) > ε < ,
(4.22)
4
n
0t1 ϕ q 1
where q is such that the natural embedding i : Sq → Sp is nuclear.
Claim 1 can be proved in a completely similar way to (3.5). Before the proof
of Claim 2 is given in detail, let us see how we shall verify the relative compactness
of {Zn , n 1} in C([0, 1], Lb (S, B)). Given any ε > 0, p and q are taken by both
claims. Let (ej ) be a CONS in Sq , for each ej , and choose Kj ⊂ C([0, 1], Lb (S, B))
such that
ε
sup P ω : Zn (ej ) ∈ Kj > 1 − j+1 ,
2
n
lim sup sup sup Zn (ej )(t) − Zn (ej )(s) = 0.
δ→0 x∈Kj |t−s|<δ n
Now define
K = x:
sup
sup x(ϕ)(t) M
0t1 ϕ
p 1
(
∞
( x : x(ϕ)(t), 0 t 1, ϕq 1
is relatively compact in B
Kj .
j=1
/ K} < ε and K has compact closure in C([0, 1], Lb (Sp , B))
Thus supn P{ω : Zn ∈
for some p > r.
Since the injection of C([0, 1], Lb (Sp , B)) into C([0, 1], Lb (S, B)) is continuous,
the closure of K in C([0, 1], Lb (Sp , B)) is compact in C([0, 1], Lb (S, B)). All these
arguments show that {Zn , n 1} is relatively compact in C([0, 1], Lb (S, B)).
466
Z. G. SU
We now return to the proof of Claim 2. As in Step 2 in the proof of Theorem 3.1,
it is enough to show that for any ε > 0 and 0 < C < ∞ there exist a positive
constant M and a finite-dimensional closed subspace F in B such that
(4.23) sup P ω : sup Zn (ϕ)(t) > M < ε for all ϕp C < ∞,
n
0t1
(4.24) sup P ω :
n
sup qF
0t1
Zn (ϕ)(t) > ε
<ε
for all ϕp C < ∞.
We will only prove (4.24). This statement is slightly different from (4.5). The interested reader may make some comparisons to better understand the point of these
proofs. In fact, set
d0 (s, t) = |t − s|1/2 ,
d0 (s,t) log N [0, 1], d0 , ε
d(s, t) =
1/2
0
Then for D3 = {ω :
n
2
i=1 qF (Xi ) supt=s [(Bi (t)
P ω:
sup qF
0t1
%
(4.25)
P ω : qF
Zn (ϕ)(t) > ε
1/2
dε ≈ |t − s| log |t − s|
.
− Bi (s))/d(s, t)] a2 n},
n
1 √
εi Xi ϕ Bi (t) ID3
n i=1
&
>ε
+ P(D3 ) =: I1 + I2 ,
where a > 0 and F ⊂ B will be specified later.
To estimate I1 , observe that
n
1 2
εi Xi ID3 ϕ Bi (t) − ϕ(Bi (s)
Eε qF √
n i=1
n
2
1 2
qF (Xi ) ϕ Bi (t) − ϕ(Bi (s)
ID3
n i=1
! Bi (t)
"2
n
1 2
c
q (Xi )
ϕ (x) dx ID3
n i=1 F
Bi (s)
)
*
n
2
1 2
c
qF (Xi )
ϕ (x) dx Bi (t) − Bi (s) ID3
n i=1
R
c
(4.26)
c a d(s, t)ϕ 20 c a d(s, t),
where we have used ϕ 0 ϕ1 and ϕp C.
Thus it is not hard to get
n
1 Xi ID3 ϕ Bi (t) − ϕ Bi (s)
qF √
n i=1
Since
1
(log N ([0, 1], d, ε))1/2 dε
0
c a d(s, t).
ψ2 (dP)
< ∞, then by taking a (depending on C and
ON CENTRAL LIMIT THEOREMS FOR VECTOR RANDOM MEASURES
ε > 0) small enough, we have
467
n
1 √
E sup qF
Xi ID3 ϕ Bi (t)
n i=1
0t1
1
1/2
ε2
log N [0, 1], d, ε
ca
dε < .
2
0
(4.27)
This implies I1 < ε/2.
For the estimation of I2 , we give the following result concerning Brownian motion.
Lemma 4.1. E supt=s,0t,s1 [(B(t) − B(s))/d(s, t)] < ∞.
Proof. Let ψ2 (x) = exp x2 − 1 and · ψ2 (dP) denote the Orlicz norm of a
random variable with respect to probability space (Ω, F, P) and · ψ(dµ×dµ) the
Orlicz norm of a measurable function with respect to Lebesgue measure space
([0, 1] × [0, 1], B([0, 1]) × B([0, 1]), µ × µ). According to the main results in [3], we
have
sup
t=s
where
|B(t) − B(s)|
cY (ω),
d(s, t)
B(t) − B(s) Y (ω) = I(d0 (s,t)=0)
.
d0 (s, t) ψ2 (dµ×dµ)
So it is enough to show E Y (ω) < ∞. In fact, set
B(t) − B(s) < ∞;
M = sup d0 (s, t) t=s
ψ2 (dP)
then
P ω : Y (ω) > u
P ω:
P ω:
2−u
2
/M 2
"
|B(t) − B(s)|2
I
dµ(t)
dµ(s)
>
2
(d0 (s,t)=0)
u2 d20 (s, t)
[0,1]×[0,1]
"
!
|B(t) − B(s)|2
u2 /M 2
I
exp
dµ(t)
dµ(s)
>
2
(d0 (s,t)=0)
M 2 d20 (s, t)
[0,1]×[0,1]
!
exp
,
from which the required result holds. Furthermore, Y (ω) is exponentially square
integrable, although we will not use this fact.
Now I2 can be estimated as follows:
n
E i=1 qF2 (Xi ) sups=t (|B(t) − B(s)|/d(s, t))
I2 a2 n
1
|B(t)
− B(s)|
(4.28)
.
= 2 E qF2 (X) E sup
a
d(s, t)
s=t
Since E X2 < ∞ and E sups=t (|B(t) − B(s)|/d(s, t)) < ∞, for any ε > 0, after a
is determined to satisfy I1 < ε/2, we are able to choose a finite-dimensional closed
subspace F in B such that I2 < ε.
468
Z. G. SU
Summarizing the preceding arguments, we have shown that {Zn , n 1} is relatively compact in C([0, 1], Lb (S, B)).
Finally, it remains to verify that for each f in C ([0, 1], Lb (S, B)), f, Zn converges weakly to a normal random variable as in Theorems 3.1 and 4.2. The proof of
Theorem 4.3 is complete.
Acknowledgments. This paper is one part of the author’s Ph. D. dissertation
submitted to Fudan University at Shanghai. The author would like to express his
gratitude to Professors Z. Y. Lin and C. R. Lu for their guidance and encouragement. The author also thanks Professor X. Fernique who kindly communicated him
preprints. Many thanks are due to the anonymous referee for a careful reading and
valuable suggestions.
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