Download TEICHIB`S STRONG LAW OF LARGE NUMBERS IN GENERAL

PROBABiWTY
AND
MATHEMATICAL STATISTICS
TEICHIB'S STRONG LAW OF LARGE NUMBERS
IN GENERALJBBNACH SPACES *
BY
BONG DAE CHOI AND SdlO HAIL SUN6 (SEOUL)
Abstract. It is shown that Teicher's version of the strong law of
large numbers for random variables, taking values in separable
Banach spaces, holds under the assumption that the weak law of
large numbers holds.
I. Introodnctiom. Let (3,)I 1)) be a real-separable Banach space. Acosta
[I], CBoi- and Sung [Z] and Kuelbs and Zinn [ 5 ] have shown that many
classical strong laws of large numbers (SLLN) hold for random variables
taking values in a general Banach space under the assumption that the weak
law of Iarge numbers (WLLN) holds; this assumption often follows from the
geometric structure on the Banach space (see [I] and [4]).
It was proved by Teicher [8] that if {X,,n 2 1) is a sequence of
independent real-valued random variables with EX, = 0,
1 EX; < m,
(EXi2/i4)
(i)
i= 2
(ii)
-
j= 1
C EX,?In2
+
0,
i=l
(iii) there exist constants ai such that
m
m
P(IXiI>ai)<oo
i= 1
and
Ca?EX$i4<w,
i=1
then SJn 4 0 as., where S , = X I + ...+X , .
Teicher's SLLN is an extension of Kolmogorov's SLLN, since Kolmogorov condition EXi/f/n2 < m, n = 1, 2, . . ., implies conditions (i)-(iii) with
q = i (for further information on Teicher's SLLN see [8]). Szynal and
2
*
tion.
This work was supported by a grant from the Korea Science and Engineering Founda-
138
B . D. C h o i and S. H. S u n g
Kuczmaszewska [7J have extended Teicher's SLLN in the case where
:X,,,
n 2 11 is a seqyence of independent Wilbert-valued random variables.
The main result 6.; this paper is a version of Teicher's SLLN for random
variables taking values in an arbitrary separable Banach space under the
assumption that WLLN holds. Note that Teicher's method for the realvalued case does ilot carry over to the Banach-valued case.
2. Main result. A key incquality in our main result is provided by the
following
LEMMA
1 (Yur~nskii[9J). Let X,, . . ., X, he independent 8-valued random
vuriclbles with E II-XiII -=z m (i = 1, . . . , n). Let Fkbe the a-field generated b y
(X,, ... , Xk),
k , . .. , n, and let .F0be the trivial ci-field. Then, for 1 6 k < n,
The foIbwing lemma is well known for real-valued case [3].
LEMMA
2. Let IX,, n 3 1 ) be a sequence of independent 8-vplued random
variables.
Then SJn + O a.s. lfl S,,
- 4 0 as. and S,Jn + 0 in probnbiliry.
P r o o f . Assume S , J2k +0 a s . and S,Jn + O in probability. ~ d r
2k < n < 2k'1 we have
lls2k11
y+ 2 k <max
,~zk+l
n G
'
llsm-s2k~~
2k
a
Hence it is enough to shaw that
l!Sm-s2k11
max
$<,<2h+
Since SJn -0
1
2k
+ O as.
in probability, there exists a k , such that
max
P(IJS,,+,
- Sn(I > 2k~ / 2 <
) 1/2
for k 2 k,.
2k < n < 2 k + l
From Skorokhod's inequality (see Stout [63, p. 102), which holds for Bvalued random variables, we have
,
- ~ , , ( ( / 7 k 4 0 as. and
the independence of (A',,). Thus (1) follows from the Borel-Cantelli lemma.
TIEOREM3. Let [ X , , ,n 2 l ] be a sequence of indeyendel~l B-valued
rrrndonr t+orinhles.Sttpposu thot
the convergence of the last series following from (IS,,,
E llXil12/n2 + 0,
(ii)
i= 1
and
(iii) there exist constants ai such that
Then S J n +0 in probubility $f S J n 4 0 n.s.
Proof. Assume that S,Jn -+ 0 in probability. From the first condition of
(iii) and E IJXi1121(11XiI]
6 4)< E11Xil12, we may assume without loss of
generality that llXill 4 a;. From Lemma 2, it is enough to show that
For each positive integer n and i (1 6 i < n), let Yn,i = E (IISnllI.Fi%i)
= =a ! X I .,. ., X i )and 9,
= (8,
a). Then
-E(IISnII IFFi- where .Fi
+
and I Yn,il < ilXill E liXill by Lemma 1.
Now we claim that E (/S,H/n -+ 0. Indeed, it is easily seen that, for fixed n,
YnSi1 1 < i < nj is a martingale difference. Thus
Therefore, from (ii), we have
Thus we conclude from jlSnll/n +O in probability and the above result
140
B. D. Choi and S. 'H. S u n g
Hence, t o prove (2) it suffices to show that
that EliSn/l/n
+O.
Consider the following simple identity:
For the simplicity of notation, Iet
...
..
n
i-1
U, =
K,,
i = 2 j= 1
K,,
n
and
=
Kfi.
.
.
i= 1
T o prove (3) it is enough to show that
(4)
U2$2'" -0 as.
and
To prove (4) it is enough to show from Borel-Cantelli lemma that, for
any E > 0,
EY.,iY.,j,
Since for j = 1, 2, ..., i-1,
2 4 i Q n ) and
are martingale differences for fixed n, we have
{ K g i 1, < i < n )
141
Teicher's law of large numbers
To prove (5) let Zn,i= Y:i - E (Y,T, I Rii
IIXi[l 4 ai, we have
EZ$
.= E Y ; ~ - E ( E ( ~F~~~~
Thus
-
..
1< i
< n.
Then, from
) )<' EY2i 6 E(IIXill + E tlXiIl)* 4 4%?EI1Xi!I2.
..
2"
Applying Borel-Cantelli lemma, we obtain
To finish the proof of (5), it suffices to show that
Since IY,,il d llXill+EIIXill, from (ii) we have
Hence the proof is completed.
Remark 4. In the preceding Theorem, WLLN is implied by the
additional conditions EX, = 0 and 3 is of Rademacher type 5 since
142
B. D. Choi and S. H. S u n g
COROLLARY
5 . The jollowing sratements are equivalent:
(i) B is of R~rdewtncher type 2.
(ii) For any sequence (Xi)of zero mean independent B-ualued random
c;nriahle.s sari,sfvi~~~q
conditions (i)-(iii) of Theorem 3, S jn +O a.s.
(iii) For. nrrg seqlrellce (Xi)
of zero mean indpendent B-valued random
variables satis3.ing
Proof. lrnplication (i) 5 (ii) follows by the above remark.
(iii) is trivial, since the condition C E jlX,,1I2/n2 < m, n = 2 , 2,
(ii)
implies conditions (i)-(lii) of Theorem 3 if we let oi = i.
(iii) =-(i) was proved by Hoffmann-Jorgensen and Pisier [4].
. .. ,
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[a] B. D. C h o i and S. H.
[3]
[4]
[5]
[6]
[7J
[8]
191
Department of Applied Mathematics
Korea Advanced Institute of Science and Technology
P.O.Box 150, Cheongryang, Seoul
Korea
Received on 8. 5. 1987