This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
f Xm+l for every m E IN since for every n E IN and k E {1,...,2n}
_
(0 Xm+l)2n+k
1
2n+1
-
. Hence f sup
Q nEIN
IX
n
I
m} A
is bounded by theorem V.5.2. From (2) in theorem V.5.9 and (1) above it follows that Ym = 0 for every m E IN.
(En-1
So
F
Xn
- Xn-1,Fn)nEIN
is a
semipotential and hence an L 1 -bounded semiamart. From theorem V.5.3, for E = IR, we see that
sup
f NA
TET'
T
-
A
T-1
H
n. So I XT < vn qn1 for all T E T(n). i1,...,in A There are by construction precisely qn sets of type A = An
.
So
f XT < vn
sup
T E T(n) SZ The conclusion is now :
f XT = vn for each n E IN.
sup
Therefore
T E T(n) S2 is a semiamart.
(Xn,Fn)n E IN
M Theorem V.5.13.1
(Xn,Fn)n
:
is an amart if and only if E IN
Proof
is an amart then
If (Xn,Fn)n E IN
I X 92
gn
pn = 0.
II
n=I
f XT = 0; indeed since
lim
TET 52
= 0 for every n E IN,
this limit (which exists, since (X
n9
F )
n n E IN
is an amart) has to be zero. This implies M
limvn=
pn=0 .
II
n=1
n-Ko M
Conversely, if
pn = 0 then
1 n=1
lim
f XT = 0
sup
n- TET(n) 0
Hence (Xn'Fn)n GIN is an smart, due to theorem V.2.2.
Theorem V.5.13.2
:
lim Xn = 0 if and only if
rr
II
an = 0 or
n=
Proof : We have that P (
i 1'"'' in
U
An
...,in
II
pn = 0.
n=1
n ) =
II
ai
.
i=1
n Note that (i1,.U.,i
n
An
)n E y1 decreases and that Xt = 0 on the
V.5
204
ill ...,1
A
U
complement of
il,.... in
lim X
n-
n
0,
for t > gn. So
n
II
an = 0 implies
n=1
= 0, a.e.. 0,
Conversely, assume
II
n=1
a
n
>
0. Then lim hn = 0 if and only if
n
Now l im hn # 0 « l im Xn # 0 on n
nr
U
[
An
II
pn = 0.
n=1
ill ...,in
] which has
n=1. ill...,in
rr*w
positive measure precisely since
0
> 0
a
II
n
n=1
Example V.5.13.3 : There exists a semiamart which converges a.e. and in L 1 , but which is not an amart.
Proof
Take pn = 1, an = 2, pn = 2 for all n E IN. Then from the
preceding results
is a semiamart converging to zero and
(Xn,F ) n EIN
:
is not an amart. Furthermore for k E {1,...,gn+1 -
IXgn+k'
f hn X
II
}
n n i=1
II
n
Pi
n i=1
g
i11...,in
A n
I
1
n
ai
II
i=1
pi
and
f IX Hence lim f IX
n- S2
I
=0
gn
Q
I
= 0 also.
0
n
We remark that from the above example it follows that the Riesz-decomposition theorem for amarts does not characterise amarts! And another negative result is to follow.
V.6
205
Example V.5.13.4 : There exists an L1-bounded positive semiamart (Xn,Fn)n E IN
such that in the Riesz-decomposition theorem V.5.9, the
martingale part Yn = 0 and such that (Zn)n
E IN
= (Xn) n E IN
is not
uniformly integrable.
Proof : Take an
2' pn = pn =
I
for all n E ]N.
Then as in V.5.13.3 we
f IXnI G 1. In fact in this case we have Xk = 0 for n E IN S2 k # 2n for a certain n E IN and Xk = 2n X for k = 2n (n E IN). can see that
sup
Since it is now trivial that
In
lim 1 E IX. =0 n i=1 2
rr
1
we see that Yn = 0 for every n E IN by the construction of (Yn'Fn)n in V.5.9. Obviously (Xn)n
V.6.
V.6.1.
EIN is not uniformly integrable.
E IN
0
Notes and remarks
Our definition of uniform amart is different from the way it was first defined in Bellow [1978a], p.179. If (Q,F,P) is the
probability space and G is a sub-algebra
of F, denote by
8.E(S2,G) the set of all G-stepfunctions with values in E, and
by 8.E(I,G) = {gEBE(O,G)jjjjg(w)jj < 1, for every w E 2} In Bellow [1978a]
following
the definition of uniform amart is the
:
An adapted sequence (Xn,Fn)n if
is called a uniform amart E-=]N
.
(i)
lim
f g(XG)
L(g) exists for each g E 8E,(S2,
GET 0 (ii)
The above limit is uniform in the sense that
lim
U
nEIN
If g(XG) - L(g)I = 0
GET g E 8El1sup (Q, F6) 0
Fn)
V.6
206
Based on our definition we have that if (Xn,Fn)n
E IN
is a
uniform amart, then it is an amart and hence (i) is satisfied (see V.2.2). (ii) follows also since
:
sup1 sup lim If g(X(y) - f g(XT QET g E aE, (SZ,Fa) T E T(6) S2 S2 F
= lim
sup
sup1
QET g E aE, (0,FG)
if g(X6) - f g(E
T E T(cf)
0
XT))
0
F
< lim
fJE
sup
XT - X6I
6ET TET(o) 0
Conversely, suppose (i) and (ii) are valid. Then certainly
0 = lim
6ET
sup I
sup
g E &E, (S2, F
T E T(a)
I1 g(X6) - J g(XT
0
0 F
= lim
sup
aET TET(a)
sup 1
g E &E,(S2,Fa)
II g(E
o
XT - xQ)l
(1)
S2
Now for every sub-Q-algebra G of F and every X E LI(f,G,PIG) we have
if g(x)I =
spEl 61,G)
Ix1
(2)
Q
g E & E
Proof : We suffice to show this for stepfunctions
n x =
X
xi XA
(A1,...,An E G). For this, choose, for every e > 0 and i E {1,...,n}, elements xi E E' such that Ixil = I and such that
IxiI - C. > xi(xi)
V.6
207
Put
n g = iE1 xi XA.
Then g E gE,(0,G) and n
I g(X) = I Sl
xi(xi)
E
S2 i=1
XA.
i
dXII -E. SZ
Applying (2) with G = FG to (1) we get F
lim
sup
f I E G XT - XGN = 0
GET TET(G) SZ Hence (Xn,Fn)n
E IN
is a uniform amart in our sense. The
definition of uniform amart as given in this book is the one which is used most.
V.6.2.
As noted in section V.2 the name "amart" comes from the term "asymptotic martingale". This name "amart" was proposed by Sucheston but for quite some time the two terms were used. The second one was especially used in the works of Bellow
(see Bellow [1976] and [1977a] ), until a certain time when the battle was won in favor of the name "amart"!
V.6.3.
For an optional sampling theorem for real smarts using Tf = {all finite stopping times} we refer to Krengel and
Sucheston [1978], p.212. V.6.4.
Also for real amarts, we mention an interesting theorem of A. Bellow [1977a], theorem 2, p.284-288
Theorem (A. Bellow) : For a function G following assertions are equivalent
:
:
:
]R -+
Bt,
the
208
V.6
(i)
G satisfies (a) G is continuous
(b) lim G(t) and lim
G(t ))
t->+co
exist and are finite. (ii) For every L
1-bounded
amart (Xn,Fn)n
sequence (G(Xn),Fn)n E IN
E IN,
the adapted
is an L1-bounded amart.
For the proof, see Bellow [1977a). V.6.5.
As we mentioned earlier, the proof of corollary V.3.15 of Brunel and Sucheston is not the original one but is the proof of Edgar. The original proof uses some Ramsey theory (work of
J. Stern [19781), see Brunel and Sucheston [19771. V.6.6.
The case where the index set IN in (Xn,Fn)n
is replaced C-24
by -IN is not described in this book. We refer to Edgar and Sucheston [1976a] and Krengel and Sucheston [1978] for information on this topic (see also Gut
[1982]). Generally
speaking, adapted sequences (Xn'Fn)nE-IN are easier to handle and boundedness conditions (to get convergence) can be weakened or even deleted. The case where the index set is ]R is also not described (continuous parameter case) (see primarily Edgar and Sucheston [1976b]).
Also cartesian products IN X IN or ]R X ]R as index sets can
be studied; see f.i. Walsh [1979] , Cairoli [1970], Millet and Sucheston [1980d], Millet and Sucheston [1981b], Millet and Sucheston [1981c], Millet [1980] , Millet [1982], Edgar [1982a], Fouque and Millet [1980] , Fouque [1980a], Fouque [1980b] and
others. Although these notions are not studied in detail in this book,
the next chapter will discuss some results in this area. The purpose is to exhibit applications of the amart theory. These applications were discovered rather recently. If the reader is not interested in these more general index sets for adapted
processes, the next chapter may as well be skipped without losing continuity. In chapter VII we shall discuss the disadvantages of smarts in infinite dimensional Banach spaces
V.6
209
and we give alternative extensions of the martingale concept which behave better (i.e. which are more suited to obtain strong convergence a.e.).
V.6.7.
An abstract example of a class of adapted sequences in which smarts coincide with semiamarts is given as follows (Millet
and Sucheston [1980a])
:
Theorem V.6.7.1 (Millet and Sucheston)
:
be a
Let (Xn)n E IN
sequence in LI with disjoint supports (An)n E IN.
Put
03
A=
U An,
n=1 and put
Fn = o(X1,...)Xn,A)
Then for (Xn'Fn)n (i)
EIN
(Xn'Fn)n E IN
the following assertions are equivalent
is an smart.
(ii)
(Xn'Fn)n GIN is a semiamart.
(iii)
E f IXnI < n=1 S2
Proof
:
(i) - (ii)
Is obvious.
(ii) - (iii) For every n E IN,
6n
define
= first k E{1,...,n} such that Xk > 0, if it exists = n+1 if such a k does not exist = first k E{1,...,n} such that Xk < 0, if it exists = n+1 if such a k does not exist.
Now
:
:
V.6
210
n
fX 0
=
an
Xk+ I SZ\ {of 0 such that
E
n E ]N(M)
I IXnI 5 E S2
For T E T(M) we thus have I IX TI c E. So (Xn'Fn),(=-]N is an smart.
V.6.8.
In the previous remark, a class of adapted sequences in which amarts coincide with semiamarts was given. Of course example V.5.13.3 shows that this is not the case in general. A weaker but easier example, only showing the existence of an a.e. convergent semiamart that fails to be an smart, is found in
Gut [19821. Also in Gut [1982] , there is a simple example of an L1-convergent and a.e. convergent adapted sequence which
is not a semiamart. We present it here. Let p > 2. Define
UP
,
..
r 1-1 n
2
1
' n 2'
Y1(w)
otherwise for i E {I,...,n} and n E IN.
Order (j,m) < (i,n) if j < i or
if j = i and m < n. In this order, the sequence (Y°)iE{1
n EIN
n}
V.6
211
So lim Xn = 0, a.e. and in the n L1-sense. For each n E IN, define to as the index k of X kP) is denoted by (X(P')n E ]N.
n
which corresponds to the first Yn E {yn,.. ,yn} which is nonzero, if it exists; otherwise, Tn is the index k of X(P) j corresponding to Yn. Hence Tn E T for every n E IN and
I-1 Yn = nP f X(P) = f max S2 i=1,...,n 1 SZ Tn
Thus, for p E ] 2 , lim .1
1[
XTP) = +
n-,-- S2 n
Hence (XnFn)n every n E IN,
V.6.9.
E IN
where, as usual, Fn = G(X1,...,Xn) for
is not a semiamart.
So far we have not considered scalar convergence a.e. of adapted sequences (Xn,F )n E]N to an integrable function (cf. 1.3.1).
For a scalar convergence result for W smarts, see corollary IX.1.2. As a matter of fact, the result is very easy to prove here; it is postponed until chapter IX where all scalar convergence results are collected together with one proof.
212
Chapter VI
:
GENERAL DIRECTED INDEX SETS AND APPLICATIONS OF AMART THEORY
This chapter is optional and hence can be skipped without losing continuity with the rest of the text. It deals with adapted nets i.e. processes (X.,Fi)i E I where I is a directed index set filtering to the right Xi E LE is Fi measurable, for every i E I and (Fi)iE I is a stochastic basis w.r.t. I; this means that every F. is a sub-a-algebra of F and that F. C F. whenever i,j E I, i 6 j. With filtering to the right we mean that for every il,i2 E I, there exists an i3 E I such that i3 > it and i3 > i2. For i E I, denote
I(i) = {jEIIIj> i} A net (Ai)i E I of sets is called adapted w.r.t. (Fi)i E I if (XA.,Fi)i
E I
1
is an adapted net.
It is not our intention to develop the whole theory so far known concerning these adapted nets. We only intend to indicate the main trends in the theory and to give the reader a feeling of what properties can be expected and of what properties cannot. In this setting some applications of the amart theory are given showing the importance of it. In the next chapter however, shortcomings of the vector-valued amart theory are shown.
VI.1.
Convergence of adapted nets Let (X.,Fi)i E I be an adapted net. We say that (Xi'Fi)i
is a martingale if
F.
E 1X. = X. 1
1
E I
for every i E I and every j E I(i). Doob's fundamental theorem that states that real L1-bounded martingales are convergent a.e., valid when the index set is IN,
is no longer valid in this general case. For a
counterexample, see Dieudonne [1950], see also Cairoli [1970] where we see that even L]-bounded martingales indexed by IN X IN may diverge! In Krickeberg [1956], Krickeberg has put a condition on (FI)iE I in order to guarantee that Doob's theorem will be valid. It
is generally called the Vitali condition V. Before introducing condition we need to extend, when E = IR,
this
our concept of "convergence
a.e.". These extensions are classical but we include their definitions for the sake of completeness.
Definition VI.1.1
: Let (X.)i
be a net of real measurable functions.
E I
The essential supremum of (Xi)i
E I'
denoted by a sup Xi is the unique
iEI
a.e. smallest measurable function such that for every j E I, e sup X. % X., a.e.. The essential infimum of (Xi)i E I, denoted by
iEI e inf Xi, is defined as
e inf X. = -e sup (-X.)
iEI For a net (A.)i of (Ai)i
E I'
E 1
1
iEI
1
of measurable sets we define the essential supremum
denoted by e sup Ai, resp. the essential infimum of (Ai)i
denoted by e inf Ail by i E I
Xe sup A.
=
ei E I
iEI
XAi
resp.
Xe inf A.
= ei i
E I XAi
iEI which are defined a.e..
Let (Xi)iE
I
be as above. The stochastic upper limit of
(Xi)i E I' denoted by s lim sup Xi, is defined as
iEI
E I,
VI.1
214
s lim sup X. = e inf {YNlim
iEI
1
iE I
P(Y <x ) = 0}
t
and correspondingly the stochastic lower limit of (Xi)i E I, denoted by s lim inf Xi, as
iEI _ -s lim sup (-X.)
s lim inf X
iEI
1
iEI
We say that (Xi)i E I converges stochastically to X. (or converges in probability to Xe), a measurable function necessarily, if
s lim inf X. = s lim sup X. = Xo
iEI
iEI
1
X. is called the stochastic limit of (X.)i E I and denoted by
X« = s lim X. The essential upper limit of (Xi)i
iEI
E II
denoted by
e lim sup Xi is defined as i E I
e lim sup X
iEI
1
= e inf (e sup
X.)
jEI(i)
1El
The essential lower limit of (Xi)i E I, denoted by e lim inf X. is
iEI
defined as e lim inf X. _ -e lim sup (-X.)
iEI
1
iEI
We say that (X1)i E 1 converges essentially to X«, a measurable function necessarily,if
e lim sup X. = e lim inf X. = Xa
iEI X
1
iEI
1
is called the essential limit of (Xi)i E I and denoted by X. = e lim X
iEI For a net (A.)i E I of sets we only need to define the essential upper
VI.1
limit of (Ai)i
215
denoted by e lim sup Ai which is determined by
E I,
iEI
e lim sup A. = e inf (e sup
iEI
iEI jEI(i)
1
A.)
or by
Xe lim sup Ai
e urn i E Iup XAi
i E I
It is obvious that, in case I = IN,
essential convergence is convergence
a.e.. Indeed these definitions above have been given in this way so as to obtain a measurable function which resembles lim Xi. X., which might be non
iEI
measurable.
In the case of Banach space valued functions (Xi)i E 1, we say
converges stochastically (resp. essentially) to a
that
measurable function X. if s-lim sup
iEI
e-lim sup NXi - XcxJ = 0).
X. - XMj = 0 (resp.
iEI
Definition VI.1.2 : Let (Fi)i E I be a stochastic basis. We say that it satisfies the Vitali condition V if for every set A in F,for every adapted net (A.)i E =I such that A C e lim sup A. and for every C > 0, i E I
there exist finitely many indices it ... in E I and pairwise disjoint sets B . J
E F F. .
,
B . C A.
(j = 1,...,n) such that
lj
J
n
P (A\ U B.) 6 e j=1
Now we have the following well-known theorem of Krickeberg [1956].
Theorem VI.1.3 (Krickeberg) : Let (X.,Fi)i E I be a real L1-bounded
martingale such that (Fi)iE
I
satisfies condition V. Then (X.)iE
I
converges essentially.
This is an old theorem known since 1956. A new proof is contained in theorem VI.1.5.1 below. With L1-boundedness we mean of course
sup
f 1X.I
I is called a stopping time if Banach space E. A function T :
{T =i} E Fi, for every i E I. Denote by T the set of all finitely valued stopping times. For T E T, XT and FT are defined as in the case I = IN.
It is only a technical matter to see that the Vitali condition V is equivalent with the following properties (i) or (ii) (see Millet and
Sucheston [1980a]) (i)
:
For every adapted net (Ai)i E I and for every c > 0 there is a T E T such that
P (e limsup AiAAT)St
iEI
(Here A denotes the symmetric difference and A
T
[A. f1 {T = i}] )
U
=
jET(S2)
1
(ii) For every real adapted sequence (X.,Fi)i E I and every sequence (in)n E IN in I, there exists an increasing sequence (an)n E IN in T
with
an
E T(in) for every n E IN and such that
e lim sup Xi = lim Xa
iEI
n#Co
,
a.e.
n
(cf.I.3.5.4.).
VI.1.5 : With this knowledge we can see that the basic results 1.3.5.5, IV.1.4, IV.1.9 and their corollaries are valid under the Vitali condition V, if we replace "lim sup" by "e lim sup". This is already trivial for i E I
iEI
1.3.5.5. For IV.1.4 and IV.1.9 this is easy too if we can make use of theorem 11.2.4.3 adapted to our case. This is indeed the case, as is easy to prove
:
Theorem VI.1.5.1 (Millet and Sucheston [1980a]) : Let E have (RNP) and suppose that (XI,Fi)i E I is an LEI -bounded martingale such that (Fi)i E I satisfies V. Then (Xi)i E I converges essentially.
VI.2
217
Proof : We only need to show the convergence in probability of (XT)T
E T
to a measurable function X., due to 1.3.5.5, trivially modified as indicated above, using the Vitali condition V. By 1.3.4 we see that this type of convergence is in fact induced by a complete metric. Hence it suffices to prove that (XT)T
E T
is Cauchy in probability, but of course
for this we may work with increasing sequences. If (Tn)n E
is an inIN
creasing sequence in T, then (XT ,FT )n E is an LE-bounded martingale. IN n n So the theorem of Ionescu-Tulcea applies.This finishes the proof too. As remarked, this proof yields also a new proof of Krickeberg's theorem VI.1.3 above.
So using VI.1.5.1 it is a relatively easy matter to reprove the results IV.1.4 and IV.1.9 in this setting, using the Vitali condition
V. Of course, since we could apply these inequalities in chapter V in order to prove convergence results of certain adapted sequences (Xn'Fn)n
E IN'
we can do the same here for adapted nets (Xi'F.)i E I where (F.)i E I satisfies V. We do not intend to write down every result that can be obtained in this way since we do not need them later on and also since it is easy to obtain them. We only mention the trivial application on uniform amarts, hence on amarts in case E = ]R.
Theorem VI.1.6 (Egghe [1980d]) : Let E have (RNP) and let (Xi'F.)i E I be an LE-bounded uniform amart such that (Fi)i E satisfies V. Then (X.)i I
E I
converges essentially to a Bochner integrable function. Here uniform amarts are defined in exactly the same way as in section V.I.
Remark VI.1.7 : Most convergence results mentioned above are valid
without V but with essential convergence replaced by stochastic convergence; see e.g. Millet and Sucheston [1980a], Millet and Sucheston
[1979a], Egghe [1980d].
VI.2.
Applications of amart convergence results We have now collected enough material to give some
applications of real amart convergence theory. They are not included here for their own sake but to indicate the interest of amarts. But it must be said that the results are very important and interesting in themselves.
VI.2
VI.2.1.
218
First application This application is concerned with adapted nets indexed by
I = Z x Z (or IN x IN). Order I in the usual way
: m = (m1,m2)
n = (nl,n2)
for nl,n2,ml,m2 E Z if ml S nl and m2 6 n2. For a stochastic basis
(Fn)n Z x Z we put 1
F(nl,n2) = 6(P F(n1.P)
2
F(nl,n2) = a(P F(p,n2))
and
U
= a(
F
Fn)
nEZ XZ
Definitions VI.2.1.1 : An adapted net (X-,Fn)n e n 1-martingale if it is a martingale and if
Z x Z
is called a
F1
EmXn X (m 1'n2) whenever m = (ml,m2) < n = (n1,n2). Analogously, (Xn'Fn)n e
Z X z
is a
2-martingale if it is a martingale and if F2 t
m
E
Xn = X(n1,m2)
whenever m < n.
The set of all finite stopping times w.r.t. (Fn)n
E Z X Z
is denoted by T1.
Its elements are called simple 1-stopping times. An adapted net (Xn,Fn)n
E Z x Z
is called a 1-amart if the net
(J XT) SZ
TET
1
converges. Analogous definitions for simple 2-stopping times and 2-amarts.
219
VI.2
We say that an adapted net (Xn,Fn)n E Z x Z is L log L-bounded if
sup
f IX-1 log
n
nEZxZ S2
IX-1
0 and choose m= (m1,m2) E Z x Z such that
220
VI.2
f IX-m - X-J 6s n
sup
(3)
nE (Z x Z) (m) S2 and
_
sup
I (D (IXm
- X - 1
) 5 c
(4)
nE (Z x Z) (m) S2
For n E (Z x Z)(m), put Y- = X- - X- and let T E TI(m); so m -< T n n m E Z x Z. If a E IN with m 5 a< m', put where m' = (m',m 1
2
1
m'
I
G a = a(U F(a,b))
Consider (Y (a,b) ,Ga )aE IN n [ml,m1] for every fixed b E ]N n [m2 ,m]2
If a < a', a,a' E IN n [m1,mi] , then, if A E Ga Y(a,,b ) =
(X(a,,b) - X(ml,m2))
A
A = A (X(a,b) -
Y(a,b)
A
since (X_,Fn)n E
is a 1-martingale. Consequently (Sa,Ga)a E IN n [ml'ml]
Z x Z
is a submartingale and
I I YTI < I I YT I< I Sm, S2
S2
1
S2
Doob's submartingale inequality implies on the submartingale (IY
(m1,b)
I,F
(ml,b) b) E INn [m2,m2]
AP (Sm, 1 > A)
A} 1
for each A > 0 that
IY
, (m1,m2)
I
VI.2
221
This inequality is in fact easily seen (cf. Neveu [1975],
IV-2-9, p.69): By the equivalent definition in 111.2.1, we have that F
Xa<EaXT a.e. from every a E T and T E T(a). Hence, for every a E T and n E IN we have on {a c n} that F
Xa c E a x n Define, for every A > 0
ax = inf {k E IN(m2)11Y(m1 ,k)I > A}
Then, on fax < m2},
< E
F°A
IY(ml,m2)I
XaA
Hence
f Since {ax<m'2} f
{ax <m2}
`m2}
S
{aA <m2} X0A
sup m2 X} = {S , >X}
,
(m1,k)
ml
> AP (a 5 m2') = AP (S, , >X), we have
x
x
aA
ml
f
AP (S ml, >A) 5 {S , >A} IY (mi'm2) ml
proving the above inequality.
So, for rl E ]0, 1[
:
I
and since
222
VI.2
W = f P(S f S > X)dX 0 m1 Q m1 ,
(using Fubini)
,
W
n + f x(f X{S ,>X}IY(m,m,)IdP]dX n
ml
S2
2
1
Sm,
=n+f
[f
SZ
n
1
>X) dP
I Y (m, " m,) I dXJ X{ S , 1 2 m1
n + f IY(ml,m2)I(log
- log n) X{Sm, > X}dP Sm,
Now use that a log+b < a log+a + e . Then
+
f Sm, < n + f S2
0
1
IY(m,,m,)I 2 1
2
1
f S em1 + Ilog nI f IY(m,mr)I
+
0
I
2
f SZSem1,
0, if e and n are chosen according to 0 < e < 6 , 0 < n < I and n + e + e I log n I < 6 ( 1) , then we see that
e
XT - xm I
I XT - xmI +
I XW - XmI
_f IyTI +f Ix-- X-1 fS
+e ml
rk for every k E IN. Hence rk < (2)k
,
for every k E IN.
Put 00
(P00
= k=0 U
(Pk
U
and write naturally Ucp00 for
B
. For every i E I, define
(1,B) E(pp
C.1 = A.\ 1
U
B
(j,B) E(p00
j
Zi = XC.
1
and F. = a( U (J(Z.)) (Zi'Fi)iE I is a potential
j j for every (j,B) E tpk. Choose T E T(i), and let T(S1) =
}. Define
VI.2
228
cp =cpkU { ( i j , { T
j
Obviously cpk C cp E (D . Now ZT =
} n Ci J XU (N \
f IZTI = P(U((p\ Qk)) S Now, since (Zi'Fi)i
I
as is easily seen. So
(pk)
(2)k
is a potential, the hypothesis ensures that
(Z.)iE I essentially converges to zero. Hence
e lim sup C. iEI
D e lim sup (A.\ Ucp )
iEI
= e lim sup A. \
iEI
Ucpp
D A\ Up So, for every e > 0, we can find a subclass of (p. which satisfies the
properties required in the definition of the Vitali condition V.
Necessity
be a potential. Let A > 0 and A= elim sup {IZij > A). Let (Z.,F.)i E 1 1 3.
iEI
For each c > 0, choose i E I such that T E T(i) implies f IZTI < E.
Define
S2
Aj
= {IZjI > a}
for j E I(i)
elsewhere
So A = e lim sup Ai. Hence, using the Vitali condition V by hypothesis,
iEI there are i1,.... in E I and B. E F. with j = 1,...,n such that
VI.2
229
for j = 1,...,n
if i
B. n B. = 0 3
j
and such that n P(A \ U
B.) 5 e
j=1
Take in+l e I(i.) for every j = 1,...,n and define
6
on B, for j = 1,...,n
= i.
J
J
elsewhere
= in+l
So a E T(i) and hence n
E> f IZTI > 0
Z
f
1
j=1 B.
IZi
j
n
> X P(U B.) j=1
A (P (A) - e)
For c small, we so have P(A) = 0 for all A > 0. So (Zi)i E I essentially converges to zero.
Theorem VI.2.3.1 is already a characterization of V in terms of amarts : potentials are amarts. A characterization of V which involves
the class of all L1-bounded amarts can easily be deduced from it
Theorem VI.2.3.2 (Astbury)
:
:
The stochastic basis satisfies V if and only
if every L1-bounded real amart essentially converges.
Proof
:
Sufficiency
Follows immediately from the previous theorem.
Necessity Again by theorem V.1.4 for E _ ]R,
extended to our case by the explanation
VI.3
230
given in VI.1.5, we see that the L1-bounded amart (Xi'Fi)i E I decomposes as
X. = Y. + Z. i
1
1
where (Y.,Fi)i E I is a martingale and (Z.,Fi)i E I is a potential. (Z.)i E I essentially converges by theorem VI.2.3.1. Now
sup
f IY.I < sup
i.EI c
1
iEI
f IX.I 1
is easily seen from the proof of this Riesz-decomposition theorem.
Hence, from theorem VI.1.5.1 and by the Vitali condition V, (Yi)iE essentially converges. Hence also (Xi)i
VI.3.
VI.3.1.
I
E I.
Notes and remarks
If we have a potential (Zi'Fi)iE I then we always have that lim f IZTI = 0 T E T S2 even if (Fi)i E I does not satisfy V. Indeed, theorem V.1.4 is
extended to general index sets in a trivial way and without using V, since in corollary IV.1.10 we only use the integral form of inequality (2) in theorem IV.1.4 instead of (2').
Indeed, in theorem IV.1.4, inequality (2) is true - in the general index setting - without V as is easily seen. Inequality (2') follows from (2) by applying theorem VI.1.5.1 to the martingale; it is here that V is used.
Thus we see that this remark is also true for uniform potentials in general Banach spaces!
VI.3.2.
As proved above, L1-bounded martingales (Xi'Fi)i E I where (Fi)i E I satisfies the Vitali condition V essentially converge.
Although V is not necessary, by VI.2.3, it cannot be skipped
VI.3
231
as we mentioned earlier. If we do not suppose V but if I = IN x IN and (Xi)iE I is an L log L-bounded 1-amart then we also obtain essential convergence by theorem VI.2.1.2.
This is f.i. the case in Cairoli's result VI.2.1.3. If we want to suppose neither V nor the L log L-boundedness condition, there is another possiblity of obtaining essential convergence due to Walsh [1979] (Xn'
:
Let I = Z x Z and
be a martingale which is L1-bounded. Suppose n)n E Z x Z
in addition that for every m = (ml,m2) < n = (n1,n2) in Z x Z
EFI m A F2 m
(Xn - X(n1,m2) - X(m1,n2) + Xm) = 0
This is called a strong martingale. We have
Theorem VI.3.2.1 (Walsh) : Let (Xn,Fn)n strong real martingale. Then (Xn)n E Z
E Z x Z
x z
be an L1-bounded
converges essentially
(i.e, a.e. here).
For related results we refer the reader to Edgar [1982a].
VI.3.3.
In Millet and Sucheston [1980a] and other papers an "ordered
Vitali condition" V is studied. The definition of V is derived from that of V by requiring it < ... 6 in in definition VI.1.2. In other words by requiring in VI.1.4 (i) T to have a totally ordered range T(S2).
Such elements of T are called ordered stopping times and the set of all finite ordered stopping times is denoted by T'.
Another equivalent formulation of V is obtained by replacing T by T' in VI.1.4.(ii). For instance V is used to show that converges an adapted net (Xi'Fi)i E I such that (XT)T C -T' stochastically to X- is essentially convergent, provided
(F.)i E l satisfies V. Other Vitali conditions are studied
- see e.g. Millet and Sucheston [1980a] and [1979a], ... VI.3.4.
An application of amarts in ergodic theory was apparently presented by Korzeniowski [1978b] with the intention of proving the Banach valued ergodic theorem of Mourier by using amart
VI.3
232
convergence results. This is a nice idea but - as L. Sucheston remarked in his review [1980]
- the argument has an irreparable
error.
VI.3.5.
Another application of amarts is of a didactical nature, as was remarked by Edgar and Sucheston [1976a]. Amarts can be introduced very early in probability courses. For the real convergence theorem of amarts, we do not need conditional expectations, nor the Radon-Nikodym theorem. For the martingale convergence theorem, we do. Moreover in our approach, we proved the real amart convergence theorem based on the uniform amart convergence theorem and hence on the martingale convergence theorem. A direct proof is seen as follows (Edgar and Sucheston [1976a])
:
Let
(Xn,Fn)n E IN be a real L1-bounded amart. Then it is of class (B); this is proved in V,2.8, using martingales, but it can be
proved directly, using the lattice properties of amarts (cor.
V.2.9 - proved directly
:
see Edgar and Sucheston [1976a],
proposition 1.3 and cor.1.4; this is only a technical matter).
From lemma 11.1.5 we now have that
sup
IX
n
nEIN
:
< co ,
I
a.e.. So for X > 0 large, P(sup
IXnI > A) is arbitrarily
nEIN = -X V X A A on a set with measure arbitrarily n n close to 1, for A > 0 large. Furthermore, again using the small. So X
lattice properties of amarts, (-A V Xn A X Fn)n amart such that, where Yn
f sup
IY
S2 nEIN
n
I
E IN
is an
-A V Xn A A
0. The t -*+w
following assertions are equivalent for a Banach space E (i)
(ii)
dim E < oo
.
Every amart (Xn,Fn)n
such that sup E IN
.1
(D( Xn ) < - converges
nEIN SZ
weakly a.e.. 1
(iii) For every L E bounded amart (Xn'Fn)n
E IN
there is an M > 0 such
that for every A > 0
P (sup n EE IN
(v)
Proof
1XnII
> A) < M
The same as in (iii) but now (F.)n C ]N
:
(i) - (ii) 0(t)
Since lim inf
t -* +oo from theorem V.1.3.
is a constant sequence.
> 0, (Xn,Fn)n E7N is L1-bounded; the result follows
(1) - (iii)
Since (XnFn)n EIN is now a uniform amart, the result follows from theorem V.1.6(iii).
VII.1
244
(ii) - (iv) and (iii) - (v) Are obvious. (iv) - (i) and (v) - (i) We now use Dvoretzky's theorem,Dvoretzky [1961], stating that
R2 is finitely
representable in any infinite dimensional Banach space. We repeat here the definition of finite representability : Let E and F be two Banach spaces. F is said to be finitely representable in E if for any finite
dimensional subspace F1 of F and for every e > 0, there is an isomorphism V of F1 into E such that
(1-t)NxII < IlvxII < (l+c)IIxll for every x E F1.
We return now to the proof of (iv) - (i). We suppose that dim E _
so
that Dvoretzky's result applies in the negative way. Let F = R2. Denote by rn the least integer larger than 2n 4)(2n). In the proof of (v)
use 1(t) = t for every t E ]R
Let {fi n E IN,
collection of orthonormal vectors in F. For each n E IN, isomorphism of Fn = span {filln E IN,
(i)
i =1,...,rn} be a let Vn be an
i =1,...,rn} into E such that
IVn x1l < 211 xll
for every x E F F. Write e ni = Vn fi n. Let 0 = [0,1] and let
7rn = {Anti=1,...,rn} be a partition of S2 such that P(Ai) =
rl
. For
n
every n E IN and i = 1,...,r n, put
Yin n _ n en Yii
Let (Xm)m
E IN
X A. 1
be the sequence (Yi)n
E IN,i=l,..,,rn
ordered so that
(i,n) < (i',n') if n < n' or n = n' and i < V. So Xm = Yn implies n-1
m = Rn-1 +i, where Rn-1 =
E
ri. Take Fn constantly the Y-algebra of
i=1
the Lebesgue measurable sets. Let T E T and define
B' = Ai n {T =Rn-1 +i}
VII.1
245
Then r
XT =E
n
E neiXn
n i=1
B.
1
where the sum is over the appropriate values of T. (XM(w))m E
is IN
unbounded for every w E Q, so it does not converge weakly and of course it is also seen that
lim AP (sup Ix II n Xn E IN
Now, since for every n E IN,
> a) _
rn > 2n 4)(2n) > 2n '(2), we see that
E n< -. Let T E T(RN_1), where N E IN is fixed. Then n=1
n rn
1
1
E fne.X f0 XII n=N+1 i=1 0 B. E
T
1
1
r
n
2
E
1
f nfnX
E
n=N+1
1
i=1 0
B. 1
Since the fn are orthonormal, we see that 1
f nfnX 1
E II
i=1 0
n
12
E
=
Bn
n2 P(Bn) 2 1
i=1
1
2
rn
nrn E i=1
2
P(B') < nr n 1
Hence
1
If 0
j52
X T
En
n=N+1 V n 1
converging to zero for N
To show that sup
f 0(IXnII) < -, define
nEIN 0
VII.2
Zi = n fi X
n
246
for every n E IN and i = 1,...,rn. Then of course
A. i
IIYi(w)II ` 211Zi(w)II
for every w E [0,1] . 1
So
1
f D(IIY'll) , f (211Zk11) 1
= f Mn x n) Ak
0
_$(2n)< rn
1
2n
Hence 1
f P(jxn0) c 1
sup
0
nE IN 0
Remark VII.1.3
:
So theorem VII.1.2 is true in the special case of
$(x) = xp, where p E [1,+co).
If (Xn)n E
is uniformly bounded, the IN
result is false due to corollary V.3.4.
VII.2.
Pramarts, mils, GFT In the previous section we have sufficiently motivated the
need for an extension of the uniform amart concept in another direction than in the amart direction. So the new notions should be "very different" from amarts, at least in infinite dimensional Banach spaces. Of course since they will generalise uniform amarts, they will generalise amarts if E has finite dimension. Furthermore it will be seen that in finite dimensional Banach spaces the new notions have the same convergence
properties as amarts, while in the infinite dimensional case they have
247
VII.2
almost the convergence properties of uniform amarts, thus have much better convergence properties than amarts.
However, it must be emphasized that one cannot extend the amart concept without losing the optional sampling theorem V.2.11 or the Riesz-decomposition theorem V.2.4 (even a very weak form thereof).
Indeed
we have the following result of Edgar and Sucheston [1976c].
Theorem VII.2.1 (Edgar and Sucheston) L1 such that (Xn,a(X1'
(Xn)n E IN in
any class in L
:
Let A be the class of sequences
'Xn))n E IN
is an amart. Let B be
such that
(i)
AC8
(ii)
implies (XT )n E IN E 8 for any increasing sequence(T) n such that Tn E T(n) for each n E IN. Here T is w.r.t.
.
(Xn)n E IN E B
(6(X1,...'Xn))nEIN
X
n
=Y +Z n
where (Yn,a(Y1
I
n
...'Yn))n E IN
is a martingale and where (I Zn)n
Then A = B
Proof
:
E IN
SZ
converges. .
Suppose that (X
)
n n EIN
E B. Then for any increasing sequence
(Tn)n E IN such that Tn E T(n) for every n E IN we have (XT ) nE IN E B
n
due to (ii). So, due to (iii)
X
T
=Y n
n
+Z
n
where (Y n,6(Y1,...,Yn))n
GIN
is a martingale and where (I Z
)
n n E IN
SZ
converges. Hence (f XT )n E IN converges. This of course implies that
(Xn)nEIN E A, since (Tn)nEIN is cofinal in T. The new notions we are discussing are pramarts and mils
Definitions VII.2.2 : Let E be any Banach space and (Xn'Fn)n
E IN
an
VII.2
248
adapted sequence. We say that (Xn'Fn)n E
]N
is an smart in probability
(shortly pramart) if for every e > 0, there is a0 E T such that a E T(a O)
and T E T(a) imply that F
P({j E a X - 6N > e}) < e F
i.e.
:
(IIE
a
X - a1I) goes to zero in probability for a E T, uniformly
in T E T(6). This definition is due to Millet and Sucheston [1980a1. is called a martingale in the limit (shortly mil) if
(Xn,Fn)n E IN
lim
F IIE M Xn - XmII = 0
sup
,
a.e.
m3 nE IN (m)
Theorem VII.2.3
Every pramart is a mil.
:
Proof : Using theorem 1.3.5.5 (see also corollary 1.3.5.6) we see that F
(E 6
XT - Xa) a E T
converges to zero a.e., uniformly in T E T(a). Hence, since IN is cofinal in T, (Xn,Fn)n E
is a mil. IN
As will be seen on many occasions further on, the converse of theorem VII.2.3 is not true and in fact we shall see that mils are much more general than pramarts. An abstract example of a case where mils are uniform smarts (hence pramarts) is seen as follows - see also Bellow [1981] (Xn)n E ]D1
: Let
be a sequence of independent functions with values in a Banach
space E. Put
n S
n
=
E i=1
X.
i
Fn = G(X1,...,Sn)
for every n E IN. Then the following assertions are equivalent
:
VII.2
(i)
249
is a mil.
(Sn'Fn)n E IN
(ii)
(iii)
(Sn'Fn)n E
is a uniform amart. IN
f Xn converges in E.
E
n=1 2 Indeed (i) - (iii) follows from F
EmSn - Sm=f (Sn-Sm) 2 For (iii) - (ii) define 00
Yn = Sn +
f X
E
j=n+1 S2
(Yn'Fn)n E IN
(Sn,Fn)n
E IN
is a martingale and (
E
f X.)n E
j=n+1 Q
IN
tends to zero. Hence
is a uniform amart.
Since pointwise convergence occurs in the definition of pramart, the following result should be true
Theorem VII.2.4 : Let X E LE and let
(Fn)n E
:
be a stochastic basis. IN
Then for every a E T
(EFaX)(w) =
(EFaw)X)(w)
F
Proof : Fix n E IN. We see that X{a=n} E nX is Fn measurable and that
for every A E Fn, A n {a =n} E Fa. So for every A E Fn F
F f
f XA X{a-n} E OX
XA X{a-n} E nX F
F
Hence on {a =n}, E nX = E aX
.
That pramarts and mils are indeed very different from amarts is seen by the next theorem.
Theorem VII.2.5 : The following conditions on a Banach space E are equivalent
:
VII.2
250
(i)
dim E < °°
(ii)
Every amart is a pramart.
.
(iii) Every amart is a mil.
Proof
:
(i) - (ii)
Now since the amart (Xn,Fn)n
EIN is in fact a uniform amart, we have
F
lim
sup
XT - XQII1 = 0
NE
GET TEE TO) F a Hence (IIE XT -X 0II)6 E T converges to zero in probability, uniformly in
T E T(6). So (Xn,Fn)n
is a pramart.
EIN
(ii) - (iii)
Follows from theorem VII.2.3.
(iii) - (i)
We reuse most of the example given in theorem VII.1.2 (iii) - (i), also the notation. Put rn = 2n and
2n X
=
n
E
i=1
eni X n A.
i
where for each n E IN,
the sets
n = {AI ,...,An n} 2
are partitions of
[0,1] consisting of independent sets. Put
Fn = 6(X1'" .,Xn). (Xn'Fn)n (=-IN
is an amart. This is seen by making
the same calculation as in the proof of (iii)
(i) in theorem VII.1.2.
Furthermore for every m E IN and n E IN(m) and for every w E [0,1] F
1
(E mXn)(w) = f Xn - 0. So 0
sup n E IN(m)
NEF' Xn - XmII
> IIX Mil
-
sup n E IN (m)
IIEFm
XnII >
1
0
VII.2
251
Thus we have not only shown that (Xn,Fn)n
E IN
it is not even a "game which becomes fairer with time"
is not a mil but
:
Definition VII.2.6 : An adapted sequence (Xn,Fn)n
is called a game GIN which becomes fairer with time (or game fairer with time, shortly GFT) if for every s > 0, there is m0 E IN such that m E ]N(mo) and n E IN (m) imply F
PQEmxn - X mI
>r)
0. Choose no E IN such that T E T(n0) and T' E T(T) implies F
P({IIET XT, - XTI >
(1)
and choose Ko E IN such that
P({T_ > no} \ {TK > no} ) 5 F-
(2)
0 If a E T'(K
and a' E T'((j) then
G P({IIE a Ya' - Y
F
a
> £})
I
T
6 £ + P({NE a XT
- XT
!
a F
a
> £}n
II
{Ta>n0})
T
> £ } n f T_ < n0 } )
- XT
a XT
+ P ({ II E
a
a
due to (1) and (2). So G P({hE a Ya, - YaII > £}) c £ + F
P({IIE
Tvn a
° XT
a
vn
o
-
XT
a
vn
o
II
> £}) +
VII.2
254
n
F
0 E
T
P({IE a X
i=1
- X M> C}i)
Ta,
{Tm=i})
TG
For every i < no, choose K. E IN such that
P({Too = i} A
n k>K.
= i}) < E
{T
k
(3)
n0
Choose a E T'(K) where K = max {K0,K1,...,Kn } and choose a' E T'((J). 0
Hence, on the set
n
{Tk = i} E F. we have T. = Ta, = i and so, using
k>K. theorem VII.2.4, on this set F
E
T
aX
Tat
-X
Ta
=0
Consequently
P({NE a Y(3, - Y01 > C}) 5 3e
O
.
So we also have the optional stopping theorem for pramarts (cf. remark V.1.9).
We now show that the above theorem is false for mils
: we show
that even the optional stopping theorem is false for mils and also for GFT, also that a weak form of the optional sampling theorem, not implying the optional stopping theorem, is also false for mils and for GFT :
Theorem VII.2.9 (Edgar and Sucheston [1977a]) (i)
:
The optional stopping theorems fails for mils and GFT exists a real mil (Xn'Fn)n
:
there
and a stopping time a with (=-IN
P({a =-}) > 0 such that (XnA a,FnA a)nE IN is not even a GFT. (ii)
The optional sampling theorem in a weak form fails for mils and GFT
:
there exists a real mil (Xn,F,)
sequence (Tn)n E IN
that (XT 'FT )n n n
E IN
n GIN
and an increasing
in T such that lim Tn = oo, uniformly and such nis not a GFT.
VII.2
Proof
:
255
(i) Let (An)n
E 7N
be a sequence of independent measurable sets
in a probability space (Q,F,P), such that P(A1) = 0, P(An) = 2 for n E IN(2). and Fn = G(X1,...,Xn). Using the Borel-Cantelli lemma we
Put Xn = n XA n
see that lim Xn = 0 a.e.. So due to independence we have
nF
EmX -X =JXn -X i0, a.e. n m
m
hence (Xn,Fn)nE IN is a mil. Put
a = inf {nE]NjjXn j 0} It follows that P(G = co) =
.
(1 - 2) = 2 since (An)n E
II
n=2 n independent sets. So, if we put Yn = XG
consists of IN
A n'
Gn = FG n n'
then we can show
that for w E{(j =oo} G
sup
I E m Yn - YmI(w) _ +oo
(1)
nE IN (m)
proving that (Yn,Gn)n
is not a GFT. To obtain this, let M > 0 be n > 2M. E fixed and choose m,n E IN so that n E IN(m) and so that E IN
k=m+1
Now w E {G =oo} C (Q\ A2) n ... n (Q\ Am) for each m E IN and m
n
(Q\ Ak) is an atom in G
.
So
m
k=2 G (E m Yn)(w) =
n E
G E m (Xk X{G =k})
k=m+1
since Xk(w) = 0 for k = 2,...,m. Hence G (EmY
n
n ) (w) =
E
k=m+1
G
k E m (X
A2 n ... n k-1 'Ak
(w))
256
VII.2
k-1 IT
n =
.2)
(1
k 3=2
E k_ -m+ 1
2
k
3
m
(1 - .12)
IT
j=2
j
due to independence. Hence
n I
(E Gm
Yn)(w) > 2
E
> M
k=m+1
for each M > 0, m E IN, n E IN(m). Therefore (1) is satisfied, since
Ym(w) = 0. (ii) Let (Xn'Fn)n EIN be as in (i). Put Ni = 1, and given Nk and M > 0 arbitrary, let Nk+i > Nk be the smallest element in IN such that
Nk+1
1n > M
E
n=Nk+1
Define Tk E T by
inf {nN Nk + I T2n-1, Xm < a and such that
F
E m Xm, - XmI < an
sup
(2)
m' E IN (m)
again. If such an m does not exist, put T2n = N. We have, for every
n E IN
fx S2
T2n-1
-fx T2n S2
F
N
=
f
E
(X
k
-E
N
k X) + N
E
f
F
(E k XN -
k
)< 2 n
k=1 {T2n k}
k=1 {T2n-1=k}
So
E
n=1
Define
-x
f (x S2
T2n-I
)
T2n
mo, a E T we thus have that w (I A.
A.)
XSl\ U
J
A.
i=1
for a certain m E IN. For every k o
and w (I S2\
U Ai .
i=1
for every
k E a(S2). Hence, uniformly in T E T(6), if w E Q, there is an M E IN such that a E T(M) implies
VII.2
261
F
sup
TET(a)
IIE a XT(w) - XQ(w)II = 0
proving the pramart property.
This was not the case for uniform amarts
:
see V.1.6 and
also VII.1.2.
It takes more time to elaborate the following negative result of Bellow and Dvoretzky [1980a] concerning mils
:
Theorem VII.2.14 (Bellow and Dvoretzky) : There exists a real mil (Xn,Fn)n E IN
a) (Xn)n
b) (Xn)n
E IN'
E IN
such that
converges to zero a.e..
converges to zero in L1.
c) (IXnJ,Fn)n E
is not a mil. IN
So the set of L'-bounded mils is not a vector lattice.
Proof : Let (Q,F,P) be an arbitrary probability space. Divide 2 into a
partition of four sets
TT 1
= {Af, B; , A1, B1}
where P(A'j) = P(AI) = 4 and P(B1) = P(B1). Let
G1 = {A,', A1}
c1 = 4
We have the fixed numbers
k1 = # TT 1 = 4
4(7rl) = sup P(C) < C ETrI
Q(G1) = P(A;) + P(A") = 2
.
VII.2
262
Rewrite
Trl = {C1,1; C122; C133; C1'4}
.
Each C1 C. (i = 1,2,3,4) is divided into four disjoint sets
:
C1,i = A2,i I B2,i V A2,i U B2,i
where
P(A2 i) = P(A2 i)
'
2
3
P(C1'i)
P(B2"i) = P(B2,i) 11
Let
TT
= {A2 i; B2 i; A" i; B2> ill
G2 = {A2 i; AZ ill < i
k1}
23
and we have the fixed numbers
k2=*Tr2=4 k1 =42 P(C) < 2 A(Tr1)
1
(p,i) S2
G. G
E p'i-I Y
P'i
=0
for
I S i< k p-i
G
E p-l,kp-2
Y
p,1
= 0
having examined all the possible cases! So (Y But for (IYp,iI ,Gp,i)(p,i) we have, if p >
I
p,i
,G
)
p,i (p,i)
is a mil.
and I < i 5 kpI
VII.2 G E
for w E C
265
P(A'
2 c
p,i-IIYp'll(w) =
P(C
p-l,i
since G
p,i-1
)
P>)i
P
= 2
p-l,i
refines 71
p-1
to the pth-step only until
the (i-1) th-place. Also
G
E P
1
>kP-21YP'11(w) =
2c P(A' >i ) P(C
k for w E C
p-l,i
obviously. So on S2 =
= 2
p-I,i p-1
U
i-1
C
p-l,i
and for every
(p,i) '< (p',i') we have
G E P1'1Yp,>i,I = 2
Since
lira IY
(P>i)
(
P>i
= 0, a.e.,
(l Y
I,G
P>i
)
P>i (P>i
) cannot be a mil.
Compare this result with theorem V.1.6. Whether the lattice property is true for pramarts is not known at the present time, so far as I am aware. Theorem VII.2.12 is the most general a.e. convergence theorem for real adapted sequences we know so far. Indeed, we can easily show that this theorem fails for GFT. Indeed, w.r.t. constant a-algebras (Fn)n EIN a GFT is just a sequence (Xn)n E IN converging in probability.
So just take a sequence (Xn)n ESN in L', converging in probability but not converging a.e., which is LI-bounded. However, the L1-convergence theorem ("Uniformly integrable mils converge in LI-sense", which follows immediately from theorem VII.2.12) extends to GFT as shown by Subramanian [1973]; see also Bru and Heinich [1979a]. We can give the proof here but we prefer to wait until the next chapter; there a much more general result (vectorvalued), will be proved in an easier way. So we have
Theorem VII.2.15 (Subramanian) in the L1-sense.
:
: Uniformly integrable real GFT converge
266
VII.2
Now we come to the vector-valued convergence properties of pramarts and mils, which was one of the main reasons for studying these notions. It will be seen that strong convergence a.e. obtainsfor pramarts and mils under fairly reasonable conditions although the main problem remains open.
Problem VII.2.16
:
Let E have (RNP). Do LE-bounded pramarts (or mils)
converge strongly a.e.?
This problem was formulated by L. Sucheston in 1979. We present two results, one for mils (hence for pramarts also), and one for pramarts only. For another important and very new result on pramarts,
see VII.3.6. The mil result was first proved by Bellow and Dvoretzky [1980b]. The proof given here is that of Bellow and Egghe [1982]. We first need a lemma
:
Lemma VII.2.17 (Bellow and Egghe) : Let E have (RNP) and let (Xn,Fn)n be an adapted sequence, such that there is a subsequence (Xnk )k the following properties
E IN
:
(1) (Xnk)k E IN is uniformly integrable. F
nk
(2) lim sup (
kEIN
sup
X
11E
(w)N) = 0, a.e..
(w) - X
nQ
REIN(k)
nk
Then a.e. F
(3) lim sup IIXn(w) -x (w)O < 2 lim sup ( sup NE m Xn(w) - Xm(w)II) nEIN (m) mEIN m,nEE IN
Proof
:
By (1) and theorem 1.2.2.1 it follows that F
(E
Xn
-X
) k E IN Q E IN (k)
is uniformly integrable. Hence, using (2) F
l im sup (
sup
kEIN
9, EIN (k)
IIE
irk
- X II 1) = 0
Xn R
nk
E IN
with
.
267
vii .2
It is now standard to see that (AT), with T = norm-topology, in theorem IV.1.4 is satisfied; hence also inequality (3).
Theorem VII.2.18 (Bellow and Dvoretzky)
:
Let E have (RNP). Every mil
with a uniformly integrable subsequence converges strongly a.e..
Proof
:
This is trivial from lemma VII.2.17 and the definition of mil. It is also trivial that a mil can have a uniformly integrable
subsequence without making the whole sequence even LE-bounded. Indeed, take the constant (j-algebra and any a.e. convergent sequence in L1
(E _ ]R) which is not L1-bounded, but has a uniformly integrable subsequence. Another example is given by X
n
= x X
n
n+1'Yn]
where (x )
is any sequence in an arbitrary Banach space E and where (yn)n E
n nE IN 1N
is a
strictly decreasing sequence in [0,1) = Q. So (Xn'Fn)n GIN becomes a It is now easy to
pramart where Fn = a(X1,...,Xn) for every n E IN.
choose xn so that (Xn)n without (Xn)n E
GIN
has a uniformly integrable subsequence
being L1E-bounded. For uniform amarts, the existence IN
of a uniformly integrable subsequence implies that the whole sequence is uniformly integrable, due to theorem V.1.4.
The theorem above is valid for mils, hence also for pramarts. In addition, only for pramarts, the following result can be proved.
Theorem VII.2.19 (Millet and Sucheston [1980a])
:
Let E have (RNP). Then
every pramart of class (B) converges strongly a.e..
Proof
:
The main point in this proof is that pramarts do have the.
optional stopping property by theorem VII.2.8. Indeed, proceed exactly as in 11.2.4.8 : now class (B) is needed to obtain, with the notation of 11.2.4.8,
sup
E IN
Ix n I
< o0 C111
since inequality (*) there fails now. Hence we may and do suppose that our pramart (Xn'Fn)n
E IN
satisfies sup
VII.2.18 finishes the proof.
n E IN
IIXn1I E L1. But then theorem
VII.2
Remark VII.2.20
:
268
Since theorem VII.2.8 is not valid for mils, by
VII.2.9, the above proof cannot be extended to mils. So for mils we have the problem.
Problem VII.2.21 : Let E have (RNP). Do mils of class (B) converge
strongly a.e.? We do have a partial result
Theorem VII.2.22 (Edgar [19791)
:
: Let E be a subspace of a separable
dual. Then every mil of class (B) converges strongly a.e..
Proof
:
Indeed, take T = w*-topology of the separable dual Banach space E.
Then we see that, due to metrizability of T on bounded sets, closed bounded sets are T-sequentially compact. Thus we see that in theorem IV.1.9, case I, (a) and (b') are satisfied. So (AT) follows and hence
strong convergence a.e., due to theorem IV.1.4.
We remark that the assumption "subspace of a separable dual" is strictly stronger than "(RNP)" as was proved in detail in section IV.2.
Also from the inequalities proved in section IV.1, a result of Peligrad [1976]
for mils follows trivially.
Theorem VII.2.23 (Peligrad) (Xn,F )n E
:
Suppose E has (RNP) and suppose that
is an LE-bounded mil such that IN
F
lim sup (
mEIN
sup
11E M Xn -
0
(1)
nEIN (m)
Then (Xn)n ElN converges strongly a.e.. Proof
:
(1) trivially implies (AT) with T = norm topology on E in
theorem IV.1.4. Hence, convergence follows.
We proceed with the results on finitely generated mils (T is as in section IV.1)
:
Theorem VII.2.24 (Bellow and Egghe [19821) : Assume that E has (RNP) and that (Xn'Fn)n E IN is a finitely generated adapted sequence. Suppose there exists a subsequence (Xnk)k which is LE bounded and such that E IN
VII.2
269
for every m E IN and h E L%,Fm9P), the T-closure of the set
o(h) = { h Xn IlkE IN (m)} is T-sequentially compact. Then, a.e., F
1im sup 11X Cu,) -X m
m,nEIN
(w)N c 2 lim sup n mEIN
sup nEIN (m)
JE m
n
(w) -Xm (w)N
Proof : Due to the LE-boundedness and the finiteness of each Fm we see F
that (E m Xnk)k
E IN(m)
for every m E IN.
is uniformly bounded, hence uniformly integrable
So theorem IV.1.9 applies yielding the claimed
0
inequality.
Corollary VII.2.25 (Bellow and Egghe [1982]) : Let E have (RNP) and let (Xn'Fn)n E IN be a finitely generated mil such that there is a subsequence LEI -bounded and such that for every m E IN and
(Xnk)k E IN which is
h E L ($I,Fm,P), the T-closure of the set
o(h) _ {j h X
c
Dk E IN(m)}
nk
is T-sequentially compact. Then (Xn)n
E IN
converges strongly a.e.. This
is in particular the case for every finitely generated mil with an 4bounded subsequence, if E is a subspace of a separable dual Banach space.
Proof
:
This follows readily from theorem VII.2.24.
0
We close this section by introducing a new type of Banach space valued adapted sequence
:
the weak mil, and we show that with
this notion we can extend the weakly a.e. convergence theorem of Brunel and Sucheston on W smarts (theorem V.4.5).
Definition VII.2.26 (Egghe [1983])
:
We say that an adapted sequence
is a weak mil, W mil shortly, if the double sequence
(Xn'Fn)n E IN
VII.2
270
F
( mEx (E
- Xm)m E IN
nE IN (m)
converges weakly to zero a.e. for m-} - , uniformly in n E IN (m).
The relationship with W amarts is not quite clear, but due to the weak a.e. convergence theorem V.4.5 of Brunel and Sucheston, the only important case is that the adapted sequence is of class (B) and that E' is separable. In this case we can prove
Theorem VII.2.27 (Egghe [1983])
dual E'. Let (Xn'Fn)n
E IN
:
:
Let E be a Banach space with separable
be a W amart of class (B). Then (Xn,Fn)n
E IN
is
a W mil.
Proof
:
Since for every x' E E', (x'(X ),F )
is a scalar amart,
n nEIN
n
it is a uniform amart, hence a pramart and hence a mil, due to theorem VII.2.3. Put F IIE M Xn(w) - Xm(w)II
sup
Gm (w) =
n E IN (m) Since F
sup
= sup
f G
TET S2
.1
sup
11E T X
T E T S2nEIN(T)
T
sup
n
F f 11E TX T
sup
- X
T
,
TET T' ET(T) SZ
0. Suppose that for every Yn+1 is independent of Fn. Put
n E IN,
X
n
=
1 cn
then (Xn,Fn)n
n E i=1
E IN
Y. 1
is an amart if and only if it is a pramart.
For the proofs we refer to Millet and Sucheston [1980a], p.109-110.
VII.3.6.
Recently M. SYaby proved the following nice result.
Theorem VII.3.6.1 (M. Slaby [1983b])
: Let E be a weakly
sequentially complete Banach space with (RNP). Then every pramart with an LEI -bounded subsequence converges strongly a.e.,
to an integrable function.
276
VII.3
For the proof we refer to chapter VIII where the subpramart
notion is studied. This is needed in the proof of theorem VII.3.6.1.
This almost solves L. Sucheston's problem. It is not known if the weak sequential completeness of E can be deleted. However theorem VII.3.6.1 above solves the problem completely for Banach lattices since a Banach lattice with (RNP) is weakly sequentially complete since c
0
cannot be embedded in E - see
theorem 111.1.3.
Even more recently, N.E. Frangos proved.
Theorem VII.3.6.2 (N.E. Frangos [1983]) : Let E be a subspace of a separable dual Banach space. Let (Xn'Fn)n E1N be a pramart with an LE-bounded subsequence. Then (Xn)n converges
E7N
strongly a.e. to an integrable function.
Also for this proof we refer to chapter VIII again because we need subpramart convergence results. Also this result almost solves L. Sucheston's problem. It is not known if we can change "subspace of a separable dual" into "(RNP)".
VII.3.7.
One can also introduce the notion of weak pramart Definition VII.3.7.1 (Egghe [1983])
:
: An adapted sequence
is called a weak pramart (W pramart shortly) if
(Xn'Fn)n E
IN
F
(E
6
XT
- X(3)a E T T E T (a)
converges weakly to zero a.e. for a E T, uniformly in T E T(6).
Obviously, every W pramart is a W mil. However in chapter VIII we shall see that theorem VII.2.28, which is also true for W pramarts can be refined for this class : we do not have to suppose class (B); LEI-E boundedness is enough, as remarked to the
author by L. Sucheston. For the proof, see chapter VIII where the
VII.3
277
subpramart notion is used.
VII.3.8.
This seems to be a good place to mention a general strong a.e. convergence result, valid for general adapted sequences and in general Banach spaces. The result is due to Bellow and Dvoretzky [1979]
and requires some introduction. Let (Xn'Fn)n
be an arbitrary adapted sequence with values in an arbitrary Banach space E. Let Tf denote the set of all finite stopping times, i.e. T E Tf if and only if
P({T 6}. We say that
S is dense if for any c E ]0, 1[
there is n G IN such that for
any T E Tf(n) there is T' E S with
P(T' # T) 5 t For T E Tf, define, as usual
FT = {A E 6(U Fn) IIA (1 {T=n} E Fn, for every n E IN} n
co
= kEl Xk X{T=k}
and X T
We say that S satisfies the localization condition if for
every finite family (Ti)i I in S and for every finite for every i E I we
partition (Ai)i I of Q with A. E FT i
have that T E S where T is defined as
T(w) = Ti(w)
for
w E A.
for every i E I.
We say that S is abundant if S contains a set S' such that S' satisfies
:
(i)
S' is dense.
(ii)
S' satisfies the localization condition.
E IN
278
VII.3
S'(n) # c.
(iii) For every n E IN,
If S itself satisfies (i), (ii) and (iii), S is called abundant in the strict sense. These definitions can be found in Bellow and Dvoretzky [1979] where the following general result is proved
Theorem VII.3.8.1 (Bellow and Dvoretzky)
:
:
Let (XinIF )
n nEIN
be
an adapted sequence. Suppose that (Xn)n EIN is LE bounded. Consider the following three properties :
(i)
(ii)
(X)n IN
converges strongly a.e..
There is a set S C Tf which is abundant in the strict sense, such that the set {XTIIT E S} is 4-bounded and
such that F
lim
sup
E6XT-X6111 =0
6ES TES(6)
(iii) There exists a decreasing sequence (Sn)n EIN, where every Sn is dense, and every set {XTIIT E Sn} is LE bounded such that
lim
sup
n->°D 6,T E S Then (i)
IIE(XT F6
0
n (ii) and if E has (RNP), then (i) - (ii) - (iii).
Earlier, finite stopping times were used by Chow in Chow [1963]
where the following result is proved
Theorem VII.3.8.2 (Chow)
:
: Let (Xn,Fn)n
EIN
be a real sub-
martingale such that
for every T E Tf. Then lim Xn exists a.e. and is > - .
n-
VII.3
279
The same result for mils was proved recently by Yamasaki in
Yamasaki [1981]
.
Theorem VII .3.8.3 (Yamasaki)
: Let (Xn,Fn)n
E71q
be a real mil
such that
IXT < sZ
for every T E Tf. Then lim Xn exists a.e., and is > - . nFor a last extension of this type of theorem
VII.3.9.
:
see VIII.3.5.
Theorem IV.3.3.3 extends to mils, with exactly the same proof
:
Theorem VII.3.9.1 (Egghe) : Let A E F and let (X ,F )
n
n nEIN
be
a mil on A, with values in a separable dual Banach space. Suppose that
sup
f IxT11
0 there is a a0 E T such that for every a E T(ao) and T E T(a) we have F
p({w E 2IIE a XT(w) - Xa(w) > -ce (wee)
e E E+,IIell
for a certain
= 1})> 1 -e
where P denotes the outer measure.
The following easy exercise is left to the reader
:
(Xn,F n )n E-=1N
is a
sub-(super-)pramart if and only if for every e > 0, there is a a0 E T such that a E T(ao) implies
VIII.1
sup
281
P({w ESZNR(XQ - FEa X)+(w)# > E}) < C
TET(a)
(resp.
sup
( E F°
E=
XT - a)+(w)N % E}) c E)
T E T(Q)
Definition VIII.1.2
:
(X ,F
) n n nEIN is said to be a submil (resp. supermil)
if there is a null set N in 0 such that for every w E 0 \
N and for
every E > 0, there is m0 E IN such that m E ]N(mo) implies that F
E m Xn(w) - Xm(w) > -ce (5 ce)fora certain e E E+, jell = 1, uniformly
in n E IN(m). Alternatively (Xn'Fn)n
is a sub-(super-)mil if and
E IN
only if lim
VX -
sup
EFm x )+(W)N = 0, a.e.
n
m
mEIN nE]N(m)
F
(resp.
lim
sup
(E M Xn - Xm)+(w)N = 0, a.e.)
mEIN nE]N(m)
Definition VIII.1.3
:
(Xn,Fn)n E IN
is said to be a game which becomes
better (resp. worse) with time (abbreviated GBT resp. GWT) if for every e > 0 there is a m0 E IN such that for every m E IN(mo) and n E IN (m) we
have
P({w E QllE m Xn(w) - Xm(w) > -ce (5 Ee)
for a certain
e E E+, lei = 1}) > 1 -e is a GBT (resp. GWT) if and only if for
Alternatively, (Xn,Fn)n E IN
every E > 0, there is a m0 E IN such that m E ]N(mo) implies F
sup
P({wEs28II(Xm - E m Xn)+(w)II
> e}) 6 E
n E ]N (m) F
(resp.
sup n E IN (m)
P({w ESfl (E m Xn -x
+
m
)
(W) I > c}) < c)
VIII.1
282
Properties and examples VIII.1.4 We have
:
Submartingale - subpramart - submil Supermartingale
superpramart
GBT.
supermil - GWT.
This follows from theorem 1.3.5.5. The converse of these implications is not true. This is trivial for GBT I submil or GWT # supermil. Indeed, take a sequence (Xn)n
E IN'
converging in probability but not a.e. and take Fn = F = a((Xn)n for every n E IN.
E IN)
It is also trivial that subpramart * submartingale
and superpramart # supermartingale. Also submil # subpramart or equivalently supermil 4 superpramart. Indeed we can even construct a real mil which is not a subpramart
:
take (Xn,Fn)n EIN from theorem
VII.2.9(ii) and define
X' = -X n n Then (X',Fn)n EIN is a mil and not a subpramart as follows immediately is a mil which is not a superfrom the proof. Of course (Xn'Fn)n E 1N
pramart.
As we have seen in chapter VII, pramarts are mils. However, subpramarts are not mils as the following example of Millet and Sucheston [1980a]
shows.
Let (An)n EIN be independent sets such that P(An) = Fn = o(A1,...,An) and Xn = n2 XA , for every n E IN. n T E T(a) we have F
P(XQ-EGXT> E:) > 0)
< P(X E
P(Ak) -> 0
for
n;
k E IN(n) So (Xn,Fn)n
GIN
is a subpramart. However
.
1
n
2
'
For a E T(n) and
VIII.1
283
F
a.e.
lim IXn - E n Xn+lI = 1, n-)
due to independence. Hence (XnFn)n E IN
is not a mil.
A.e. convergence of submils cannot be expected as the following result of Millet and Sucheston [1980a] shows
Example VIII.1.5
:
There is a real L1-bounded submil which does not
:
converge a.e..
Proof : Let (An)n E
be independent sets such that P(An) _
for
IN
every n E IN. Define X
2 n+1
=
n
I
2
X2n
= n XA
n
Then
and Fn = a(X1,...,Xn), for every n E IN. F
lim sup (
mEIN
lim sup X
-1) = 0
- E m X )) = lim sup (X
(X
nEIN(m)
However, P(X2n = 0) -
nEIN
sup
m
n
mEIN
m
1. So
= 1, a.e.
n
lim inf Xn = 0, a.e..
nEIN Compare this with the positive result
theorem VII.2.12 for
mils. However subpramarts and superpramarts behave much better. We continue now to establish these good results concerning real sub-(super-)
pramarts. The method of proving the subpramart a.e. convergence theorem is especially interesting since it is applied later on in other important results.
First of all we establish the optional sampling theorem for sub-(super-)
284
VIII.1
pramarts. This fails for submils, due to theorem VII.2.9(ii) and its proof.
Theorem VIII.1.6 (Millet and Sucheston [1980a1) lattice and (Xn'Fn)n E (XT ,FT )k E k k
(Tk)kE IN Proof
:
IN
IN
: Let E be any Banach
an arbitrary sub-(super-)pramart. Then
is a sub-(super-)pramart for every increasing sequence
in T.
Just follow the lines of the proof of theorem VII.2.8, now using
the subpramart notion and remark that if (Xn'Fn)n E]N is a superpramart, then (-Xn,Fn)n E7N is a subpramart. The real supermartingale convergence theorem extends for real superpramarts without any additional condition.
Theorem VIII.1.7 (Millet and Sucheston [1980a])
:
Positive real super-
pramarts converge a.e..
Proof
:
a) We first show that P{lim inf X
nEIN
= } = 0
n
Let e > 0. Choose n E IN and M E IIt + such that P(Ak) < c for every
k E IN, where F
Ak= {EnXn+k-Xn>
{Xn>M} E:}
Given K E IR+ , choose k E IN such that
P({Xn+k > K}) > P(lim inf Xn = co) - E
nEIN Now F
F
E n Xn+k < M+ e K XS2 \ Ak E n X{Xn+k> K} < XS2 \ A by the definition of Ak. Hence, taking w E 0 \ Ak,
VIII.1
285
M+e>f
F
KXS2\Ak E nX{Xn+k>K} F
= K f E n (X (S2 \ Ak) n {Xn+k > K} ) since Ak is Fn-measurable. Hence
M + e > K P((P \ Ak) n
{Xn+k > K})
> K (P(Xn+k > K) - c)
> K (P(lim inf X
nEIN
n
= co) - 2e)
for any K E IR+ and e > 0. So P(lim inf X
nE]N b) From (a) and the positivity of (Xn)n
n
= ) = 0
if (Xn)n EIN does not converge a.e. then there exist a,s EIR such that P(A) > 0 where E IN'
A = {lim inf Xn < a < S < lim sup Xn} nEIN
Fix e > 0 and M E IN.
nE]N
Choose 2a set B and MI E IN(M)
and P(A A B) < 6 where 6 =
such that B E FM
Choose M2 E IN(MI) and M3 E IN(M2)
such that
P(A \ {
inf
X
MI n) < f EFT1(XD) = I XD = P(D)
Now take n = 4S L. Then
F
T
(R -(X) E
1(XA A C ) 2
F
= (B -a) E T1(XAAC ).X 2
FT 1
{E
()(AAC2)>n}
F
T
+ (R -a) E
1 (XA
AC )x 2
FT
{E
1
(XAAC)5n} 2
P(AAC2)
FT1
Here P(E
(XA A C ) > n) 5
due to (1)
n
2
< P(AAB) T1
2
8p
. e
2
FT
while (6-a) E
1(XAAC )X 2
(Q -a)n
FT
{E
1(XAAC ) 6 n} 2
We see analogously that F
aE
T
1(XC 1
\C 2 ) ($ -a) E T1(XA) - E
E T1XT 2
.
1
Using the superpramart property, choose M E IN so large that,
if
F
T
T1 E T(M) and T2 E T(T1), then E
1
XT
set of measure < E.
E, outside of a
- XT 2
1
We can now conclude that we can construct an increasing sequence (Tn)n E
IN
in T such that F
(P -a) E Tn(XA 5
n F
outside a set of measure Rn, a.e.
F
:
indeed, for every m E IN and n E IN (m)
F
EmXn >EmR'>R' n m Hence F
inf
Xn =Rm >R'm
m
E
n E IN (m) Suppose that (Xn
does not converge to zero a.e.. It follows
- Rn)n E IN
from theorem 1.3.5.5 that (Xa - Ra)a
ET does not converge to zero in
probability. Hence there is e > 0 and a sequence (an)n E
in T such IN
that
P({X6
> e}) > e
- RG
n
n F
From the fact that (E
a
XT )n E IN converges pointwise, hence in probability, n to Re for every a E T we deduce that we can choose Tn E T(an) such that F CY
P({X 6 - E n
n
x T
n
> 2}) >
2
contradicting the subpramart assumption.
Conversely, suppose there is a submartingale (Rn,Fn)n
1=3N
such that
to zero a.e.. As n - R')n E7N converges remarked before, Xn > Rn > Rn, a.e.. Hence (Xn - Rn)n E converges to
Rn < Xn, a.e. and such that (Xn
IN
zero a.e.. Hence obviously (Xa - Ra)a
T E T(a) we have that
CT
also. But for every a E T and
VIII.1
292
F
XQ-EaXT<XQ-R So (Xn,Fn)n E
IN
is a subpramart.
Lemma VIII.1.10 (Millet and Sucheston [1980a]) : Let (Xn'Fn)n real subpramart. Then for every X E IR, So (Xn'Fn)n E Proof
:
IN
(Xn v X,Fn)n E IN
E ]N
be a
is a subpramart.
is a subpramart.
For every c > 0, choose m E IN such that a E T(m) and T E T(a)
imply F P({X6 - E a XT > E})
E) > e
(Xm
n
mEIN
n
choose mn E IN such that
sup
P(
(XQ
mE{1,...,mn}
(3)
n - R6n ) > 2) > 2
Define, for each j G {1,...,mn}
A. = {w E QJj is the first index in IN for which J
sup
E{1,...,mn} Hence (A .)J.E{1,. J
..,m }
(X (w) -Ra (w)) n
=
X6 (w) -RQ (w)}
n
n
is a disjoint family of F
n -measurable sets.
an
Using the classical argument in lemma 1.3.5.4 we can arrange for a sequence
Yk,j E T(an) for each n E IN such that
F
(E a' Xj
n
Yk,j
)kE]N
(4)
VIII.1
300
decreases to Ra , a.e.. So there exists Tn E T(an) such that n F
on
XTj - Ran > 4 P(A.)) < 4 P(A.)
P(E
(5)
n
Define T
n
= TJ on A.. So T E T(a ) for every n E IN. n n J n
(6)
Now, by (3)
2 < P(
m
sup
MG {1,...,mn} m
- Rn ) >
(X6
2
n
n
n
= JE 1 P({Xan - R3 n > 2} n A.)
(by (4))
Fa
mn
< E
P({XJ
an
j=1
mn
>
e
P(A.)
(1 -
2
TJ
2
J)} n A.) J
n
Fa
P({E
E
+
n XJ
- E
P(A.)
n X3
j =l
- Ran > 2
} n Aj) 2
n
m
< E
> E} n A) +
- E Fan XJ
P({XJ
an
j=1
4
Tn
4
by (5) and (6)
J
m
n 4} nA.) +4 =P(U {X3 -EFanX3> j =l F
4) +4 n
n
Hence F
P(sup
jEIN
(Xa
On
- E
an XT ) > 4) > n
,
4
is a uniform sequence
a contradiction to the fact that (Xn,Fn)n E IN
of subpramarts. So (a) cannot be true. Hence
VIII. I lim (sup
301
m
Rn m)R
(Xn -
n- mE IN
sup
f sup
)
= 0, a.e..
(7)