THEORY PROBAB. APPL.
Vol. 43, No. 3
ON THE WEAK CONVERGENCE OF VECTOR-VALUED
CONTINUOUS RANDOM PROCESSES*
ZHONG GEN SU†
Abstract. In this paper we shall establish some results on weak convergence for vector-valued
continuous random processes, among which are vector-valued sub-Gaussian processes, vector-valued
random processes satisfying Lipschitz conditions, and random Fourier series with vector-valued coefficients. The results obtained can be considered as corresponding extensions of those in the real
setting. Our basic ideas stem from Fernique’s recent works.
Key words. weak convergence, vector-valued Gaussian process, metric entropy, type 2 space
PII. S0040585X97977057
1. Introduction and preliminary statements. Suppose that (T, d) is a
compact metric or pseudometric space, and (B, k · k) is a separable real or com0
plex Banach space with its topological dual B , h·, ·i denoting their natural bilinear
form. Let C(T, B) be the linear space of all continuous functions on (T, d) with values
in B. In particular, C(T ) represents either C(T, R) or C(T, C). Then C(T, B) is a
separable Banach space with the natural sup-norm kxk = supt∈T kx(t)k. Our aim
in this article is to discuss the weak convergence of a sequence of random processes
with sample paths in C(T, B). Fernique [4] proved the following characterization of
relatively compact subsets in C(T, B).
Proposition 1.1. Suppose that F is a separating set of real continuous functions
on B. For a subset C of C(T, B) to be relatively compact it is sufficient and necessary
that the set C satisfies the following:
(i) There is a compact set K of B such that {x(t), t ∈ T , x ∈ C} ⊂ K.
(ii) For each f ∈ F , the set f ◦ C = {f ◦ x, x ∈ C} is relatively compact in C(T ).
We endow C(T, B) with a Borel σ-field B(C(T, B)) induced by the uniform topology. Then the well-known Prokhorov theorem holds for any family of measures on
(C(T, B), B(C(T, B))) and from Proposition 1.1 we easily obtain the following.
Proposition 1.2. Let M be a family of probability measures on C(T, B). Suppose that for any positive ε there exists a compact set Kε of B such that
ª
(1.1)
µ x: ∃ t ∈ T, x(t) ∈
/ Kε 5 ε ∀ µ ∈ M.
Then M is relatively compact if and only if there exists a separating set F of real
continuous functions on B such that for any f ∈ F the family
f ◦ M = {f ◦ µ, µ ∈ M}
is relatively compact on C(T ). Here f ◦ µ(A) = µ{x ∈ C(T, B), f ◦ x ∈ A} for each
Borel subset A in C(T ).
Since it is easy for us to deal with the relative compactness of a family of probability measures on C(T ), the key to the relative compactness of a family of probability
*Received by the editors March 13, 1996. This work was supported by the Natural Science
Foundations of China and Zhejiang Province.
http://www.siam.org/journals/tvp/43-3/97705.html
† Department of Mathematics, Hangzhou University, 310028, China; Research Institute of Mathematics, Fudan University, 200034 Shanghai, China.
463
464
zhong gen su
measures on C(T, B) is to verify the first condition of Proposition 1.2. We will base our
discussions on Proposition 1.2 in two ways. One way is to use the following facts: a subset K of a separable Banach space B is relatively compact if and only if K is bounded
and for any η > 0 there is a finite-dimensional closed subspace F such that qF (x) 5 η
for all x ∈ K, where qF denotes the quotient norm, i.e., qF (x) = inf y∈F kx − yk. Thus,
we can rewrite (1.1) of Proposition 1.2 as follows.
For any ε > 0 and η > 0 there are a positive constant M and a finite-dimensional
closed subspace F such that
n
o
°
°
(1.2)
µ x: ∃ t ∈ T, °x(t)° > M < ε,
and
(1.3)
o
n
¡
µ x: ∃ t ∈ T, qF x(t) > η < ε
for any µ ∈ M.
The second way needs something more. Let C be a convex symmetric closed set
0
0
in B with its gauge NC (x) = inf{λ: x ∈ λC} and polar set C = {f ∈ B : hf, xi 5 1,
0
for all x ∈ C}. If C is convex and σ(B, B )-closed, we have C = ∩y∈C 0 A(y), where
c
c
A(y) = {x: hy, xi 5 1}. Clearly, C = ∪y∈C 0 A(y) . Thus there exists a sequence of
yn such that for any x ∈ B,
NC (x) = inf{λ > 0: x ∈ λC} = suphyn , xi.
n
Conversely, suppose that b is a closed and convex symmetric bounded subset of B;
then b is the intersection of a sequence of decreasing convex symmetric and closed
neighborhoods of origin in B, and hence there exists a sequence of increasing continuous seminorms Nk , converging simply to Nb , in B. B with such a continuous
seminorm Nk defines a separable Banach space Bk = (B, Nk ) by means of quotient
and completion. We will also work with these Banach spaces as well as (B, k · k).
Proposition 1.2 concerns only the relative compactness; the following Singer theorem can usually be used to identify the candidate for the limit in weak convergence
of a family of probability measures.
Theorem (see [11]). The topological dual of C(T, B) is isometrically isomorphic
to the space of all the completely additive and regular functions fe defined on the Borel
0
set e ⊂ T with values in B . The isomorphism Φ ↔ fe between the two spaces is given
by the relation
Z
­
∀ x ∈ C(T, B).
(1.4)
Φ(x) =
x(t), dft
Pn
0
As in the scalar setting , let W = { i=1 f i δti , fi ∈ B , ti ∈ T , 1 5 i 5 n, n = 1},
∗
where δt denotes the unit point mass at t, then W is sequentially w -dense in the
0
topological dual (C(T, B)) of C(T, B).
Next let us introduce some notation. Throughout we assume that (Ω, F, P) is
a complete probability space on which all random variables are defined. A random
process with values in B is a family X = (X(t), t ∈ T ) of measurable mappings from
(Ω, F, P) to (B, B) indexed by T . Two random processes X and Y are modifications
of each other if for each t ∈ T , P{X(t) = Y (t)} = 1, X = (X(t), t ∈ T ) is said to be
0
Gaussian if the family {hX(t), yi, t ∈ T , y ∈ B } is a family of zero-mean real Gaussian
random variables. Thus a B-valued Gaussian process X is uniquely determined by its
on the weak convergence of vector-valued processes
465
covariance structure EhX(t), yihX(s), zi. As an example we give the construction of a
B-valued Wiener process generated by a given probability measure µ. Let T = [0, T0 ];
0
for any t ∈ T, Rt = Rt (y, z) is a symmetric semipositive bilinear form on B . Assume
0
that for each y ∈ B , the mapping t → Rt (y, y) is increasing and right-continuous. We
also assume that there exists a Gaussian probability measure µ on B such that
Z
2
(1.5)
hx, yi dµ(x) = RT0 (y, y) ∀ y ∈ F.
Then, under these conditions, there exists a Gaussian random process W indexed by
0
T with values in B such that for all s, t ∈ T and y, z ∈ B ,
­
­
(1.6)
E W (s), y W (t), z = Rs∧t (y, z).
This assertion can be proved as follows.
For any pair (s, t) with s < t, the difference Rt −Rs is the covariance of a Gaussian
measure µs,t on B; we denote by (Gs,t , s 5 t, t ∈ T ) a family of independent Gaussian
vectors with respective laws µs,t . Given any finite number of points θ = {0 = t0 <
θ
t1 < · · · < tn = T0 } of T , we denote by µ the law of Gaussian random vectors
X
θ
Y =
Gtk −1,tk , tk 5 t, t ∈ θ .
tk 5t
0
In view of independence, for all s, t ∈ θ and y, z ∈ B we have
Z
­
­
θ
(1.7)
x(s), y x(t), z dµ (x) = Rs∧t (y, z).
θ
This shows that the family (µ ), where θ is finite subsets of T , is consistent in the
θ
sense of Kolmogorov’s consistency theorem. So the family (µ ) defines a Gaussian
T
probability µ on B with the required covariance structure. As usual, we can realize
T
T
W by the identity mapping from B into B . R
2
In particular, if Rt (y, y) = tR1 (y, y) = t hx, yi dµ(x), the B-valued Wiener
process obtained above is called homogeneous and normalized by µ.
This example and its proof are due to Fernique [4]. He has made investigations
at length into the problem of existence of continuous versions of some vector-valued
Gaussian processes.
Many authors point out that, in a compact metric or pseudometric space (T, d),
there are two important quantities measuring the size of T in the d: metric entropy and
majorizing measure, which are closely related to the continuity and weak convergence
for random processes. We denote by N (T, d, ε) the minimal number of open balls of
radius ε for d that cover T . If there exists a probability m in the space (T, d) such
that
1/2
Z η
1
dε = 0,
log
(1.8)
lim sup
m(Bd (t, ε))
η→0 t∈T 0
where Bd (t, ε) is a ball with center at t and radius ε, then m is called a continuous
majorizing measure.
Finally, for simplicity and clarity we always assume that c is a positive constant
which may vary from line to line and depend on some appropriate parameters.
466
zhong gen su
The present paper is arranged as follows. Section 2 is devoted to the weakly
relative compactness of a family of Gaussian measures in C(T, B) and some simple
remarks on invariance principle for a sequence of i.i.d. random vectors. In sections 3
and 4 our subjects will be vector-valued sub-Gaussian processes in the weak sense and
vector-valued random processes satisfying the Lipschitz condition, respectively. At
the end of this paper, random Fourier series with vector-valued coefficients are taken
into account although there is a little gap between our result and one in the scalar
setting.
2. Relative compactness of Gaussian measures on C(T, B) and invariance principle. Let T be a compact metric space and x a Gaussian random process
on T with continuous sample paths. Fernique [2] proved that the set of separable
Gaussian processes y on T such that
(2.1)
¡
2
E y(t) − y(s)
5
¡
2
E x(t) − x(s)
∀ s, t ∈ T,
and
(2.2)
2
Ey (t0 ) 5 1
for some t0 ∈ T,
is relatively compact for the weak convergence of probability measures on C(T ).
Now we extend the above result to the vector situation. If X is a Gaussian
random process with values in a separable Banach space B, we denote its covariance
2 1/2
0
by dX (f, t; g, s) = (E(hf, X(t)i − hg, X(s)i) )
for s, t ∈ T and f, g ∈ B . Gaussian
process X is completely specified by dX .
Theorem 2.1. Given a continuous Gaussian random process X on T with values
in B, let M be the set of the law µY of Gaussian random process Y on T satisfying
0
0
dY (f, t; g, s) 5 dX (f, t; g, s) for any s, t ∈ T and f, g ∈ B , and for each f ∈ B there
2
is an element a in T such that Ehf, Y (a)i 5 1. Then M is relatively compact for the
weak convergence of probability measures on C(T, B).
0
0
Proof. It is obvious that for each f ∈ B (B is, of course, a separating set of real
continuous functions on B) f ◦ M is relatively compact on C(T ). Thus by Proposition
1.2 it is enough to show that for any ε > 0 there is a compact subset K ⊂ B such that
for each µY ∈ M,
¡
µY x: ∃ t ∈ T, x(t) ∈
/ K < ε,
i.e., P{ω: ∃ t ∈ T, Y (ω, t) ∈
/ K} < ε. We need only consider E supt∈T NK (Y (t)). The
continuity of Gaussian random process X implies that for any positive ε there exists
a convex symmetric and compact set K such that
¡
E sup NK X(t) < ε.
t∈T
As we noticed in section 1, for such a convex symmetric and compact set K ⊂ B
0
there is a sequence of fn ∈ B such that NK (x) = supn=1 hfn , xi for all x ∈ B. Since
dY (f, t; g, s) 5 dX (f, t; g, s), the well-known Slepian’s lemma shows
¡
­
­
¡
E sup NK Y (t) = E sup sup fn , Y (t) 5 E sup sup fn , X(t) = E sup NK X(t) ,
t∈T
t∈T n=1
t∈T n=1
t∈T
from which we obtain the required result.
Before concluding this section, we should mention Kuelbs’ invariance principle
(cf. [8]) for a sequence of i.i.d. random vectors.
on the weak convergence of vector-valued processes
467
Theorem 2.2. Let {Xn , n = 1} be a sequence of independent and symmetric
random vectors with the common distribution µ. Suppose
S
√n ⇒ G.
n
(2.3)
Construct the partial sum process
S[nt]
nt − [nt]
X[nt]+1 ,
Xn (t) = √ + √
n
n
(2.4)
0 5 t 5 1.
Then Xn ⇒ W , W is a vector-valued Wiener process generated by µ.
Proof. Obviously, the classical Donsker invariance principle can be applied to
0
hf, Xn (t)i for each f ∈ B . Thus by Proposition 1.2 it suffices to prove that for any
ε > 0 there is a compact set K ⊂ B such that
ª
(2.5)
sup P ω: ∃ t ∈ [0, 1], Xn (t) ∈
/ K < ε.
n
√
In fact, since Sn / n ⇒ G, there must be a compact and convex symmetric set K ⊂ B
such that
S
1
sup P ω: √n ∈
/ K < ε,
n 2
n= 1
from which we deduce by using Levy’s inequality
)
(
n
o
¡
1
Sk
sup P ω: sup NK Xn (t) > 1 5 sup P ω: max NK √
>
2
n
1 5k 5 n
n=1
n= 1
t∈[0,1]
(
)
)
(
1
1
Xk
Sn
(2.6) + sup P ω: max NK √
< 4ε.
5 4 sup P ω: NK √
>
>
1 5k 5n
2
2
n
n
n= 1
n=1
Theorem 2.2 now easily follows.
Compared to Kuelbs’ direct method, Proposition 1.2 pays more attention to the
topology on any compact set K ⊂ B: the topology induced on K by the norm topology
coincides with the weak topology on K and so has more extensive applications.
3. Vector-valued sub-Gaussian processes. Let us first recall that a real random process x = (x(t), t ∈ T ) is called sub-Gaussian if
°
°
°x(t) − x(s)° 5 d(s, t)
φ2
2
for any s, t ∈ T , where φ2 (x) = exp |x| − 1 and k · kφ2 denotes its Orlicz norm.
Heinkel [7] showed that the continuous majorizing measure condition is sufficient for
x to satisfy the central limit theorem in C(T ). (As for the general theory on central
limit theorems (see [1]), we write CLT for central limit theorem later.)
We can similarly define a vector-valued sub-Gaussian process X = (X(t), t ∈ T )
in the strong sense, that is,
°
°
°X(t) − X(s)° 5 d(s, t) ∀ s, t ∈ T.
(3.1)
φ2
Although the continuous majorizing measure condition can still guarantee the existence of modification with continuous sample paths, the CLT does not hold for X any
longer.
468
zhong gen su
For example, let (xi , i = 1) be a sequence
P∞of elements in C(T, B) and (εi , i = 1)
a Rademacher sequence, and define X(t) = i=1 εi xi (t), for t ∈ T . By [9] we have
(3.2)
°
°
°X(s) − X(t)°
φ2
°2 !1/2
à ° ∞
°X ¡
°
°
°
5 8 E°
εi xi (s) − xi (t) °
=: d(s, t).
°
°
i=1
P∞
Only when Y (t) = i=1 gi xi (t), t ∈ T , is also a.s. continuous can X as a random
vector in C(T, B) satisfy the CLT.
Now a vector-valued random process X = (X(t), t ∈ T ) with values in B is said
0
to be sub-Gaussian in the weak sense if for each f ∈ B there is a continuous distance
df (s, t) with respect to d on T such that
°­
­
°
°
°
(3.3)
° f, X(s) − f, X(t) ° 5 df (s, t) ∀ s, t ∈ T.
φ2
Obviously, khf, X(s)i − hf, X(t)ikφ2 5 kf k kX(s) − X(t)kφ2 .
Conversely, X = (X(t), t ∈ T ) is called uniformly γ-sub-Gaussian if there is a
Gaussian vector G in B such that
­
0
(3.4)
E exp f, X(t) 5 E exphf, Gi ∀ t ∈ T, f ∈ B .
See [6] for details on γ-sub-Gaussian and sub-Gaussian vectors.
Our main result on the vector-valued sub-Gaussian process is as follows.
Theorem 3.1. Suppose that X = (X(t), t ∈ T ) is both a sub-Gaussian in the
weak sense and a uniformly γ-sub-Gaussian process. If there exists a probability measure m on (T, d) such that
Z
(3.5)
lim sup sup
η→0 f ∈B 0 t∈T
0
η
log
1
m(Bdf (t, ε))
1/2
dε = 0
R∞
−1 1/2
and supt∈T 0 (log(m(Bdf (t, ε))) ) dε is continuous with respect to the topology of
the compact convergence on B, then X is a.s. continuous on T and satisfies the CLT
in C(T, B).
0
Proof. We first note that for any s, t ∈ T and f, g ∈ B ,
°­
°­
­
°
­
°
°
°
°
°
dX (f, t; g, s) =: ° f, X(t) − g, X(s) ° 5 ° f, X(s) − g, X(s) °
φ2
φ2
°­
­
°
¡
°
°
2 1/2
+ ° f, X(s) − f, X(t) ° 5 5 Ehf − g, Gi
+ df (s, t)
φ2
(3.6)
=: dG (f, g) + df (s, t).
So, for any ε > 0, BdG (f, ε/2) × Bdf (t, ε/2) ⊂ BdX ((f, t); ε). Since G is a Gaussian
0
0
vector with values in B and B1 is σ(B, B ) compact and metrizable, by Talagrand’s
0
theorem [9] there is a probability measure µ on B1 such that
Z
(3.7)
sup
f ∈B10
0
∞
1
log
µ(BdG (f, ε))
1/2
dε 5 cE kGk.
Let π = µ × m; then
¡
ε
ε
µ BdG f,
5 π BdX (f, t); ε ,
m Bdf t,
2
2
on the weak convergence of vector-valued processes
469
and hence
1/2
1
dε
log
sup sup
π(BdX ((f, t), ε))
f ∈B10 t∈T 0
1/2
Z ∞
1
log
5 sup sup
dε
m(Bdf (t, 2ε ))
f ∈B10 t∈T 0
1/2
Z ∞
1
dε
log
+ sup
µ(BdG (f, 2ε ))
f ∈B10 0
Ã
1/2 !
Z ∞
1
log
5 c EkGk + sup sup
dε .
m(Bdf (t, ε))
f ∈B10 t∈T 0
Z
(3.8)
∞
Now we can use Fernique’s theorem [2] to obtain
°
°
­
E sup °X(t)° = E sup sup f, X(t)
t∈T f ∈B10
t∈T
Ã
5
Z
c EkGk + sup sup
f ∈B10 t∈T
∞
0
log
1
m(Bdf (t, ε))
1/2
!
dε .
Conversely, for any ε > 0 there is a convex symmetric and compact set K ⊂ B such
that
1/2
Z ∞
1
(3.9)
ENK (G) + sup sup
log
dε < ε.
m(Bdf (t, ε))
f ∈K 0 t∈T 0
For such a compact set K there exists an increasing sequence Nk of continuous
seminorms, converging simply to NK in B, and (B, Nk ) is a separable Banach space.
By repeating the previous proof for (B, Nk ) and taking the limit, we also have
¡
(3.10)
E sup NK X(t) < c ε.
t∈T
Thus, according to Theorem 4.2.1 in [4], X has a modification with continuous
sample paths since each real-valued random process f (X) is a.s. continuous on T for
0
f ∈ B1 .
To prove that X satisfies the CLT, without loss of generality we can assume
that X is symmetric and {X, Xn , n = 1} are i.i.d. Consider the normed sums Zn =
−1/2 Pn
n
i=1 Xi .
It is not hard to verify that
­
0
(3.11)
E exp f, Zn (t) 5 E exphf, Gi, f ∈ B1 , t ∈ T,
and
(3.12)
°­
­
°
°
°
° f, Zn (t) − f, Zn (s) °
φ2
5
c df (s, t).
Thus, one can similarly prove that for any ε > 0 there is a convex symmetric and
compact K ⊂ B such that
¡
(3.13)
sup E sup NK Zn (t) < ε.
n= 1
t∈T
470
zhong gen su
0
For each f ∈ B , Heinkel [7] tells us that f (X) satisfies the CLT in C(T ). Hence, we
deduce that {Zn , n = 1} is relatively compact by using Proposition 1.2 again.
As for the uniqueness of the limit, it is enough to notice the Singer theorem and
the CLT in finite-dimensional spaces. The proof of Theorem 3.1 is completed.
4. Vector-valued random Lipschitz processes. We will begin our arguments with the following two lemmas. The first lemma is Lemma 7.13 in [1] and
the second is just the formula (4.12) in [9].
Lemma 4.1. Suppose that (T, d) is a compact metric space and
Z
0
∞
¡
1/2
log N (T, d, ε)
dε < ∞.
0
0
0
Then there is another continuous pseudometric d such that d 5 d , d/d → 0, and
R∞
0
1/2
(log N (T, d , ε)) dε < ∞.
0
Lemma 4.2. Suppose that X is a sub-Gaussian vector in B. Then for any t > 0,
(4.1)
ª
P kXk > t
5
2 exp
−
t
2
2
32EkXk
.
Our object in this section is to establish the following.
Theorem 4.1. Suppose that B is the space of type 2, X = (X(t), t ∈ T ) is a
mean zero B-valued random process satisfying Lipschitz condition, i.e.,
°
°
°X(t) − X(s)° 5 M d(s, t),
(4.2)
where M is a real-valued random variable with finite variance.
R∞
1/2
If 0 (log N (T, d, ε)) dε < ∞, then X satisfies the CLT on C(T, B).
0
Proof. Our hypotheses guarantee that f (X) satisfies the CLT for each f ∈ B
and the multidimensional CLT remains true. This is a consequence of Theorem 14.2
in [9].
The key to the proof of Theorem 4.1 is proving that for each ε > 0 and η > 0
there are a positive constant a and a finite-dimensional closed subspace F such that
°
°
(
)
n
°
° 1 X
°
°
εi Xi (t)° > a < ε
(4.3)
sup P ω: sup ° √
°
t∈T ° n
n= 1
i=1
and
(4.4)
Ã
(
sup P ω: sup qF
n=1
t∈T
n
!
1 X
√
εi Xi (t)
n i=1
)
>η
< ε.
Here {X, Xn , n = 1} is a sequence of i.i.d. random processes and {εn , n = 1} is a
Rademacher sequence independent of {Xn , n = 1}. The two statements will be proved
in analogous ways using truncation.
2
In fact, since EM < ∞, for any ε > 0 there is a positive constant a0 such that
(
)
n
2
2
1X
ε
kXi (s) − Xi (t)k
EM
sup
=
1
5
< .
(4.5)
sup P ω:
2 2
2
n
2
a0 d (s, t)
a0
n=1
i=1 s6=t
on the weak convergence of vector-valued processes
471
For such a fixed a0 it easily follows from Lemma 4.2 that
(°
n
° 1 X
¡
°
εi Xi (t) − Xi (s)
E X Pε ° √
° n
i=1
!°
)
à n
2
°
kXi (s) − Xi (t)k
1X
°
sup
5 1 ° > ua0 d(s, t)
×I
2 2
°
n i=1 s6=t
a0 d (s, t)
°
Ã
"
n
° 1 X
¡
°
2
5 2EX exp − u 32 Eε ° √
εi Xi (t) − Xi (s)
° n
i=1
!°2 #−1 !
à n
2
°
kXi (s) − Xi (t)k
1X
°
sup
5
1
×I
°
2 2
°
n i=1 s6=t
a0 d (s, t)
2 u
,
5 2 exp −
32c
where c is an absolute constant in the definition of the space of type 2. In other words,
°
à n
!°
n
2
°
° 1 X
¡
kXi (s) − Xi (t)k
1X
°
°
εi Xi (t) − Xi (s) I
sup
5
1
° 5 ca0 d(s, t).
°√
2 2
°
° n
n
s6=t
a0 d (s, t)
i=1
i=1
φ2
Thus we have
°
à n
n
2
° 1 X
kXi (s) − Xi (t)k
1X
°
E sup ° √
εi Xi (t) I
sup
2 2
n i=1 s6=t
t∈T ° n i=1
a0 d (s, t)
Z ∞
¡
1/2
5 ca0
dε,
log N (T, d, ε)
!°
°
°
51 °
°
0
from which there exists a positive constant a such that
°
(
n
° 1 X
°
εi Xi (t)
sup P ω: sup ° √
t∈T ° n i=1
n=1
à n
2
kXi (s) − Xi (t)k
1X
(4.6)
sup
×I
2 2
n i=1 s6=t
a0 d (s, t)
!°
)
°
ε
°
51 °=a 5 .
°
2
Now, the first statement holds.
0
The proof of the second statement is similar but uses Lemma 4.1. Let d be as in
Lemma 4.1, and define
(
Y (s, t) =
X(t) − X(s)
0
d (s, t)
0
for s 6= t,
otherwise.
0
It is easy to see that Y (s, t) is a.s. in C(T × T, B) since d/d → 0.
Therefore, for each ε > 0 and η > 0 there must be a finite-dimensional closed
subspace F such that
ª
(4.7)
P ω: ∃ s 6= t, qF (Y (s, t)) > η < ε.
472
zhong gen su
Further, by the dominated convergence theorem it is not hard to conclude that for
any ε > 0 and η > 0 there must be a finite-dimensional closed subspace F such that
(
)
n
2
1X
ε
qF (Xi (s) − Xi (t))
(4.8)
sup P ω:
sup
=1 < .
2
2
0
2
n i=1 s6=t
2
ε η (d (s, t))
n=1
Analogously,
( Ã
EX Pε qF
!
n
1 X ¡
√
εi Xi (t) − Xi (s))
n i=1
!
)
n
2
qF (Xi (s) − Xi (t))
1X
sup
5 1 > ua0 d(s, t)
×I
2 2 0
2
n i=1 s6=t
ε η (d (s, t))
Ã
Ã
Ã
n
1 X ¡
2
√
5 2EX exp − u
εi Xi (t) − Xi (s)
32Eε qF
n i=1
à n
2
q (X (s) − Xi (t))
1X
sup F 2 i2 0
×I
2
n i=1 s6=t
ε η (d (s, t))
2
u
.
5 2 exp −
32c
Ã
!!2 !−1 !
5
1
Here in the last step we use the fact that any quotient space of type 2 remains of
type 2 and has the same constant.
Therefore, we have
à n
!!
Ã
n
2
qF (Xi (s) − Xi (t))
1X
1 X
E sup qF √
εi Xi (t) I
sup
51
2 2 0
2
n i=1 s6=t
n i=1
t∈T
ε η (d (s, t))
Z ∞
Z ∞
1/2
1/2
¡
¡
0
0
(4.9)
5c
du 5 c εη
du.
log N (T, εηd , u)
log N (T, d , u)
0
0
Thus the required result (4.4) can be proved by noting Lemma 4.1. The proof of
Theorem 4.3 is concluded.
5. Random Fourier series with vector-valued coefficients. Random Fourier series with real coefficients have been studied in detail by Marcus and Pisier [10],
among others. There appear many elegant results and applications to harmonic analysis. Up to now, however, the results concerning random Fourier series with vectorvalued coefficients are still fragmentary. Let us first recall some background.
Let G be a compact Abel group, Γ = {γn , n = 1} a countable subset of its
character group. Suppose that {gn , n = 1} is a sequence of independent standard
Gaussian random variables, {εn , n = 1} is a Rademacher sequence, {ξn , n = 1}
2
is a sequence of independent symmetric real random variables
P∞ with 2Eξn = 1, and
inf n=1 E|ξn | > 0, {an , n = 1} are real numbers satisfying n=1 |an | < ∞. Define
three Fourier series on G as follows:
x(t) =
(5.1)
z(t) =
∞
X
n=1
∞
X
n=1
an ξn γn (t),
y(t) =
∞
X
n=1
an εn γn (t),
an gn γn (t),
on the weak convergence of vector-valued processes
473
P∞
2
2 1/2
for s, t ∈ G. Main results in [10]
and define d(s, t) = ( n=1 |an | |γn (s) − γn (t)| )
show that
°
°
°
°
°
° ∞
∞
∞
°
°X
°
°X
°
°X
°
°
°
°
°
°
an ξn γn ° ≈ E°
an gn γn ° ≈ E°
an εn γn °
(5.2)
E°
°
°
°
°
°
°
n=1
n=1
n=1
R∞
1/2
< ∞) although C(G), the set of all
(all of them are equivalent to 0 (log N (G, d, ε))
bounded continuous complex functions on G, is of no finite cotype. As an important
corollary, x satisfies the CLT on C(G) and the limit is just y.
Similar problems naturally arise when real coefficients {an , n = 1} are
P∞replaced
,
n
= 1} in a separable Banach space B. If
by
a
sequence
of
vectors
{x
n
n=1 xn ξn ,
P∞
P∞
n=1 xn gn , and
n=1 xn εn a.s. converge, respectively, we can define, respectively, by
the comparison convergence theorem for random series
X(t) =
(5.3)
Z(t) =
∞
X
n=1
∞
X
xn ξn γn (t),
Y (t) =
∞
X
xn gn γn (t),
n=1
xn εn γn (t).
n=1
Fernique [3] showed that a vector-valued stationary Gaussian process Y has a modification with continuous sample paths if and only if the mapping y → E supt∈G hy, Y (t)i
0
from B to C is continuous with respect to the topology of the compact convergence
on B. Conversely, we have the following relation (cf. [9, Corollary 13.14]):
°
!
à ° ∞
∞
°
°X
X
°
°
εn xn ° + sup E sup
εn hf, xn iγn (t)
c E°
° f ∈B10 t∈G
°
n=1
n=1
°
°
° ∞
° ∞
°
°
°X
°X
°
°
°
°
5 E sup °
εn xn γn (t)° 5 E sup °
gn xn γn (t)°
°
°
t∈G ° n=1
t∈G ° n=1
°
à ° ∞
!
∞
°X
°
X
°
°
(5.4)
5 c E°
gn xn ° + sup E sup
gn hf, xn iγn (t) .
°
° f ∈B10 t∈G
n=1
Thus it is easily seen that if
∞
X
P∞
n=1
n=1 gn xn
εn xn γn (t)
< ∞ a.s., then
and
n=1
∞
X
gn xn γn (t)
n=1
converge uniformly
P∞ a.s. simultaneously, i.e., they have the same continuity. Of course,
when Y =
n=1 xn gn γn (t) converges uniformly a.s., Z also satisfies the CLT in
C(G, B).
The problems remain open, however, for a sequence of random variables {ξn ,
n = 1}, both Gaussian and Rademacher. Next we give our results in this area.
Theorem 5.1. Suppose that B is the space of cotype q (2 5 q < ∞), 1/p +
1/q = 1. {ξn , n = 1} is a sequence of independent symmetric real random vari2
2q
ables
with Eξn = 1 and inf n E|ξ| > 0, supn E|ξn | < ∞. Further , assume that
P∞
Ek n=1 ξn xn k < ∞. Define
X(t) =
∞
X
n=1
ξn xn γn (t)
474
zhong gen su
and
Ã
(5.5)
d(s, t) = sup
f ∈B10
If
Z
∞
∞
X
hf, xn ip γn (s) − γn (t)2p
!1/(2p)
.
n=1
¡
0
1/2
log N (G, d, ε)
dε < ∞,
then X(t) is a.s. continuous on G.
P∞
In addition, if n=1 ξn xn satisfies the CLT in B, then X satisfies the CLT in
C(G, B).
P∞
q
Proof. First note that n=1 kxn k < ∞ due to the hypotheses, and for each
0
f ∈ B1
Ã
∞
X
hf, xn i2 γn (s) − γn (t)2
n=1
Ã
5
Ã
(5.6)
=
∞
X
!1/(2q) Ã
kxn k
q
n=1
∞
X
!1/(2q)
q
kxn k
!1/2
∞
X
hf, xn ip γn (s) − γn (t)2p
!1/(2p)
n=1
d(s, t)
n=1
by Hölder inequality.
Since
Z
∞
0
¡
1/2
log N (G, d, ε)
dε < ∞,
0
then f (X) is a.s. continuous on G and satisfies the CLT in C(G) for each f ∈ B1 .
Next define
Ã
df (s, t) =
∞
X
2
ξn hf,
2 2
xn i γn (s) − γn (t)
!1/2
.
n=1
Then using (5.4) and noting B is of cotype q space, we have
°
° ∞
°
°X
°
°
E sup °
ξn xn γn (t)°
°
t∈G ° n=1
°
à ° ∞
!
∞
°X
°
X
°
°
5 c E°
ξn xn ° + Eξ sup Eε sup
ξn εn hf, xn iγn (t)
°
°
t∈G n=1
f ∈B10
n=1
°
à ° ∞
!
Z ∞
°X
°
1/2
¡
°
°
5 c E°
ξn xn ° + Eξ sup
dε
log N (G, df , ε)
°
°
f ∈B10 0
n=1
°
à ° ∞
!
Z ∞
°X
°
1/2
°
°
(5.7)
5 c E°
ξn xn ° + Eξ
dε .
log N G, sup df (s, t), ε
°
°
f ∈B10
0
n=1
on the weak convergence of vector-valued processes
475
Clearly, supf ∈B10 df (s, t) is a translation invariant metric; from Lemma 3.11 in [11] it
follows that
Z ∞
1/2
dε
log N G, sup df (s, t), ε
Eξ
0
5
Ã
Ã
c Eξ sup
f ∈B10
Z
+
Ã
5
Ã
c Eξ
∞
+
c
n=1
∞
+
1/2
!1/(2q)
q
2q
kxn k |ξn |
Ã
Ã
log N G, Eξ
!1/(2q)
kxn k
∞
¡
0
Ã
sup
Ã
n=1
(5.8)
xn i|
Ã
q
Z
2
log N G, Eξ sup df (s, t), ε
0
∞
X
!1/2
2
ξn |hf,
f ∈B10
∞
X
Z
5
∞
X
0
n=1
ÃÃ
f ∈B10
f ∈B10
∞
X
sup
f ∈B10
n=1
1/2
log N (G, d, ε)
n=1
2q
kxn k |ξn |
!!1/2
d, ε
!
dε
!1/(2p)
p
|hf, xn i|
!
dε .
∞
0
n>N
!1/(2p)
!1/(2q)
q
Similarly,
°
°
∞
°
°X
°
°
ξn xn γn (t)°
E sup °
°
°
t∈G
n>N
°
°
!1/(2q)
Ã
à ∞
∞
°X
°
X
°
°
q
5 c E°
ξn xn ° +
kxn k
°
°
n>N
n>N
Ã
à ∞
!1/(2p) Z
X
p
(5.9)
× sup
|hf, xn i|
+
f ∈B10
dε
∞
X
hf, xn ip
n=1
∞
X
!
¡
1/2
log N (G, dN , ε)
!!
dε
,
P∞
p
2p 1/(2p)
.
where dN (s, t) = supf ∈B10 ( n>N | hf, xn i| |γn (s) − γn (t)| )
Thus X(t) is a.s. continuous on G.
P∞
If n=1 ξn xn satisfies the CLT in B, in order to prove that X satisfies the CLT
−1/2 Pm P∞
i
i
in C(G, B) it suffices to consider Zm = m
i=1
n=1 ξn xn γn , where {ξn , n = 1},
i = 1, 2, . . . , is a sequence of independent sequences with the same distributions as
{ξn , n = 1}. By some simple calculations,
2
m
1 X
i
ξn = 1,
E √
m
i=1
2q
m
1 X
i
sup sup E √
ξn < ∞.
m
m
n
i=1
Therefore we obtain, as above,
(5.10)
sup E sup kZm k < ∞
m=1
t∈G
476
and
(5.11)
zhong gen su
°
°
m X
∞
°
° 1 X
°
°
i
ξn xn γn (t)° = 0.
lim sup E sup ° √
°
°
m
N →∞ m=1
t∈G
i=1 n>N
We can now complete the proof of Theorem 5.1 by using (1.2) and (1.3) of Proposition 1.2.
Acknowledgments. This paper is part of the author’s Ph. D. dissertation at
Fudan University. He would like to express his gratitude to Professors Z. Y. Lin and
C. R. Lu for their encouragement and help. The author also wishes to thank Professor
X. Fernique for providing the author with his manuscripts.
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