Internal. J. Math. & Math. Scl.
VOL. !4 NO. 2 (1991) 381-384
381
SOME RESULTS ON THE SPAN OF FAMILIES OF BANACH VALUED
INDEPENDENT, RANDOM VARIABLES
ROHAN HEMASINHA
University of West Florida
Pensacola, FL 32514
(Received August 4, 1989 and in revised form March 21, 1990)
If
LI()
E
Let
ABSTRACT.
be
Banach
a
and
space,
contains an Isomorphic copy of
let (R,, P) be
LI[o,I]
then in
LEP()
a
probability
(1 < p
<
space.
), the closed
linear span of every sequence of Independent, E valued mean zero random variables has
Inftnlte codlmenslon.
(in
LEP(R))
linear
If E is reflexive or B-convex and
span of
any
of
family
Independent,
E
<
p
<
valued,
then the closed
mean
zero
random
variables Is super-reflexlve.
KEY WORDS AND PHRASES.
Banach valued random variable, Uncond[tlonal basic sequence,
Finite representabillty, Super-reflexlve banach space, B-convex banach space,
(n, 6)-tree.
1980 AMS SUBJECT CLASSIFICATION CODE.
46B20, 46E40,
60BI[,
INTRODUCTION.
Linear spans
independent real variables have been studied by
of sequences of
In [1] and [2] H.P. Rosenthal utilized their properties to define a
new class of Banach spaces that has had important applications in the structure theory
several authors.
of Banach spaces.
In this paper we consider subspaces spanned by Banach valued random
variables.
Our notation is the following. (R,
Banach space and
measurable x: R
denotes expectation.
E and
for
< p
<
(R),
r, P)
denotes a probability space.
E is a
A Banach valued random variable is Bochner
the space of E valued random
is
L=P(R)
We first state and prove a set of assertions (Lemmas 1.1 and 1.2) regarding
As a
valued random variables whose scalar counterparts are well known.
Banach
consequence, we obtain that any sequence of independent, mean zero, E valued random
This enables us to show
variables, form an unconditional Basic sequence in
L().
that in
LPE(R)
the closed linear span of any sequence of mean zero random variables has
infinite codimension (Theorem 1.5).
Furthermore, using a characterization (due to
R.C. James) of super-reflexivity by infinite trees we show that when E is reflexive or
B-convex, then in
(1 < p < (R)) the closed subspace spanned by any family of mean
LEP(R)
zero random variable is super-reflexive.
LEMMA 1.1.
Let x,y:
E be two independent, mean zero random variables.
Then
R. HEMAS INHA
382
l.et P
PROOF.
P
x
denote
y
distribution
the
functions
f dPy(y)
dPy(y) ( Ix+ylIdP x(x))r
] dPy(y) }t](x+y)dPx(X) I}r
dPy(y)l IXdPx + y ]r
f y]]rdPy(y)
(]XdPx(X)
2.
Assu
Let
{1}ig[gn
LE
variables.
_
X ,...X n
E
valued,
{i}igin
let
Sl= {il i-
+
tx i
i S
E
Set X
r.
eiXi
i
e
$2
xi
i S
Then X,Y are independent.
’:,11’+1 i’:’)
o
.e
,:SI Itxt
t/’
i-I
E
Y
iXi’
i S
e
xt
Z
and
i
$I
ixt
i 8
2
mean
random
zero
{il i
+ I}, S 2
i
--I}.
txt
.
Ixi.
2
{.llxll) t/ , <.tlx-,ll) /
.1 I1 I) ’ .1 Ix-l I)
Therefore by Lemma 1.1,
’ :":,1 I’:-11’)
pd’)/p ’ 2,:SI Ix -,-... + xnll pdP)zp"’nXnll
Banach space
I.I. A sequence (x
In
-,-...
n)
DEFINITION
is said to be a basic sequence
a
if (x n) is a Schauder base for its closed linear span
unconditional
called
O)
n
n
E
Then
respectively.
be any eholce of signs.
For any choice of signs
PROOF.
x,y
0 since B(x)
Independent,
are
of
if,
[xn].
A basic sequence (xn)
is
it
converges
series Ea x converges,
n n
characterizations of basic and unconditional basic
whenever
the
The following
sequences are well known and can be found in [3].
A sequence (xn) is basic if and only if there exists a
PROPOSITION 1.1. (1)
unconditionally.
number k
"t
>
..... "n
0 such that for all positive integers m and n with m
n
m
, , , I1.., ".":11’ 11:., ".":11"
on,,
,:,..)
,’,
b,,,,t,
n, and all scalars
,,,,,,1,,hoe
’:"n)
"
En an xn converges
signs (),
n
is convergent.
whenever (a n is a sequence of scalars such that
n
is a sequence of independent, mean zero random variables in
LEMMA 1.3. If {X
if
unconditional
and
only
if
for
all
sequences
of
EanX
n}
LgP(f)
(1
PROOF.
zo.
o
p
<
(R)) then {Xn} is an unconditional basic sequence.
Then {anXn} is independent, mean
be any sequence of scalars.
Let {a
, . ,. ,11. xll
n)
n
that _fX is basic.
(in
L(fl))
/
, <,11
xll) ’/ e
’-
E
Furthermore, if we assume that
then for any choice of signs
{:n
and m
,:,.,11-,,, ]Sx:ll") t/’’ 2/’(,.,11 ]=m: 5x.llP) t/p.
<
anXn
,n.
is convergent
n, Lemma 1.2 gives
,
383
SPAN OF FAMILIES OF BANACH VALUED INDEPENDENT RANDOM VARIABLES
n--Zl ’"n’Xn
Therefore,
co,vr&es
shall now show Lhat in
In
L()every
p
LE().
spans a .ubpace of infinle codl.Benston.
R.
2 and E
case p
X n}
Consequently
seqJenve of
We
Is un’ondttonal.
Independent an zero random variables
In [3] there Is an elementary proof for the
Our result [s almost Im,nedlale [,
crew
of the fotlow[n fact
whose proof [ [ [4].
LEMMA 1.4.
If F is any Banach space which has
[so,norph[c copy of
an
mean
of
independent
[Xn}
L[2)
[n
of
LifO, l]
then
subspace of
p
<
.
p
[f
zero
< -,
seqe,lce
an
E ts a Banat’h space and tf
[Xnl,
then
the
[[near
span
[somorphlc
copy
closed
has infinite cod[mens[on.
It
PROOF.
LI()contains
{Xn} L()
P)Is probability space such that
Assume (fl,
LI[0,I].
unconditional basts then F does
LIfo, [].
not contain an isomorphic opy of
THEOREM l.l.
,In
easily
[s
does
so
LI(Q)
seen
LI()’E
Ll()
that [[
Since fl [s
a
contains
an
L()
probability space
Is a dense
for
L()then
tt has
Therefore, If [Xn] has finite codlmenslon [a
t
Thus
for
suffices
establish
the
to
assertion
LE().
[n
codinston
().
(),
(e).
[Xn] [s of fin[te
Suppose that for so independent sequence {Xn}
s(say). Let {Yl"’’’Ym be a base for the sbspace complentary to []. en
From the result, in section
{Y[,...,Ym,XI,...,Xn...} ts a Schauder base for
I,
{Xn}
[s
an nncondit[onal basic seqnence.
2.
The
of
notion
tree
characterized
[s an
LIfO,I].
Isomorphic copy of
infinite
erefore, the above described base
finite
properties
super
representabillty as welt as
were
reflexivity
Introduced
in
terms
by
of
the notions of
James,
R.C.
infinite
([5],
trees.
[6])
and
finite
who
also
We shall give
the
definitions and theorems used to obtain our results.
DEFINITION 2. l.
A Banach space F is said to be finitely representable in the
For every
> 0 and any finite
dimensional subspace F 0 of F there is an into isomorphism T:F
E such that
0
Bausch space E if
the following condition holds.
x
F
0
DEFINITION 2.2.
A Banach space E is said to be super-reflexive if every Banach
in E is reflexive.
representable
space finitely
An (n,6) tree (in
4 2 and let n be a positive integer.
DEFINiTiON 2.3. Let 0 <
x21 + x21+l for
such that x
a Banach space) is a finite sequence
2
admissible i, and
l}xi-x2i+tll ,
(Xl,X2,...,x 2n+l
, llxi-x2i_ll
.
i
The folloIng theorem is due to R.C. James.
TSEOREH 2.1. A Banach space E is super-reflexive if and only if for each
N such that the unit ball of E does not contain an (n, ) tree.
there exists n
>
0
We also utilize the folloelng definitions and theorems.
A Banach space E is said to be B-convex i
DEFINITION 2.4.
is not finltely
representable in E.
THEOREM 2.2. Let E be a Banach space with unconditional base.
Then the following
are equlvalent:
(1)
E is B-convex
R. HEMASINHA
384
([i)
E is relexiw
(iii)
E i SUl)er-ref[eivc
A proof of
theore,n
thiq
lect,lre notes of
appears In the
Woyczynski [7].
It tq know.n
that the property of super-reflexivity is stronger than the property of B-,-onveity.
.o,
B-convexlty doe not [ply anti i not implied by ref|exivlty.
{ 1ow Late ,1lid pro’#e olr reSl[t.
ssum
TffE)REM 2.].
<
|n
,
<
p
L().
<
p
LPE ()
[f,]
[s
L
not
[f]
L
B-conve
is
spaces
(reflexive)
super-reflexlve.
Let
B-conve Banach space.
mean,
variables
random
zero
Is super-reflexive.
espectlvely
B-convex
also
is
B-conve
of
E
if
Independent,
of
fa,nlly
a
that
known
is
it
,
<
subspaces
that
be
c L
a ref[exlve or
E is either
Then its closed linear span,
PROOF.
for
{fA}
and
:_hat
are
(respectively
reflexive)
reflexive).
Further
Then by the negation of
closed
Suppose
(reflexive).
B-convex
then
theorem 2.1,
there
for eacl n there t an (n,5) tree contained in the unit ball
G be the closed linear span of the union of these (n,5) trees.
Let
of [fX],k
L"
We claim that there [s a
Then G is separable [nce the above union is countable.
>
0 such
that
countable set
{fn}n
is 6
Indeed,
dense in G.
{fX}X
G
since
For each
the /A such tha
C
subfamily
)}
is
Yn’
L
[fn In"
such that
separable, we may choose a sequence
there Is sequence
Zu)-- Ynsuch that
as ]c-oo.
{IA}A
Y.
{fA}A and G[f")].,}. By
{zn)}
k" of
finite
{Yn}n
e N
Thus for each Yn, here is
Now U U
)].
G whl.ch ts
of
linear comb[natlons
a
countable
n=l
{f")}.,}
subfamily of
the results of Section 1,
is an
uconditional bsic seq.c.cc.
the
has
unco.ditio.al
subspacc
Therefore,
basis. Since this subspace is B-convex (reflexive) it is, in view of Theorem 2.2
also super-reflexive. But the unit ball of
contai.s the .nit iI of G -hich
in turn contains (.,6) tress for a11
[f")].,}
[/")]},.
..
REFERENCES
R.C. Some Self Dual Properties of Normed Ltnear Spaces, Symposium on
Infinite Dimensional Topology, Annals. of blath. Studies 69 (1972), 159-175.
I.
.)AMES,
2.
ROSENTIAL, H.P.
3.
BEAUZAMY,
4.
ROSENTHAL, H.P. On the Span in L P of Sequences of Independent Random Variables
II, Sixth Berkeley Symposium on blath/Stat and Probability, Volume II, 149-167.
5.
GELBAUbl, B.R.
333-343.
6.
JAMES, R.C.
LP(P
On the Subspaces of
> 2) Spanned by Sequences of Independent
Random Variables, Israel J. Math. 8 (1970), 273-303.
B. Introduction Banach Spaces and
Mathematics Studies, 68, 2rid Edition (1985).
Their
Geometry,
North
Holland
Independence of Events and Random Variables, Z. Wahr. 36 (1976),
Superreflexlve Spaces with Bases, Pac. Journ. Math. 41(2), (19/2),
409-419.
7.
SINGER, I. Bases in Banach Spaces, Volume I, A Series of Comprehensive Studies
Math, 154, Sprlnger-Verlag, (1970).
8.
WOYCZINSKI, W. Geometry and Martingales in Banach Spaces, II, Independent
Increments, Probability in Banach Spaces, 26-517. Advances in Probability and
Related Topics, Volume 4. Marcel Dekker, (1978).
in