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\bigskip
\centerline{\tt http://www.math.washington.edu/\~{}ejpecp/}
\vskip 0.5 true in
\centerline{\twelvebf Concrete Representation of Martingales}
\bigskip
\centerline{\bf Stephen Montgomery-Smith}
\centerline{\it Department of Mathematics}
\centerline{\it University of Missouri, Columbia, MO 65211}
\centerline{\tt stephen@math.missouri.edu}
\centerline{\tt http://math.missouri.edu/\~{}stephen}
\bigskip
\centerline{Volume~3, paper number~15, 1998}
\centerline{\tt http://www.math.washington.edu/\~{}ejpecp/EjpVol3/paper15.abs.html}
\bigskip
\centerline{15 pages, submitted June 4, 1998, published December 2, 1998}
\bigskip
\noindent {\bf Abstract:}\ \
Let $(f_n)$ be a mean zero
vector valued martingale sequence. Then there exist
vector valued functions $(d_n)$ from $[0,1]^n$ such that
$\int_0^1 d_n(x_1,\dots,x_n)\,dx_n = 0$ for almost all
$x_1,\dots,x_{n-1}$, and such that the law of $(f_n)$
is the same as the law of
$(\sum_{k=1}^n d_k(x_1,\dots,x_k))$.
Similar results for tangent sequences and sequences satisfying condition~(C.I.)
are presented.
We also present a weaker
version of a result of McConnell that
provides a Skorohod like representation for vector valued
martingales.
\bigskip
\noindent Keywords: martingale, concrete representation, tangent sequence,
condition~(C.I.), UMD, Skorohod representation
\bigskip
\noindent A.M.S.\ Classification (1991): 60G42, 60H05.
\bigskip
\noindent Research supported in part by the N.S.F.\ and the Research Board of
the University of Missouri.
\vfill\eject
}
\pageno=2
\beginsection 1.\ \ Introduction
In this paper, we seek to give a concrete representation of martingales.
We will present theorems of the following form. Given a martingale
sequence $(f_n)$ (possibly vector valued),
there is a martingale $(g_n)$ on the probability
space $[0,1]^\N$, with respect to the filtration
$\cL_n$, the minimal sigma field for which the first $n$ coordinates
of $[0,1]^\N$ are measurable, such that the sequence $(f_n)$
has the same law as $(g_n)$. Thus any martingale may be
represented
by a martingale
$$ g_n((x_n)) = \sum_{k=0}^n d_k(x_1,\dots,x_k) ,$$
where $\int_0^1 d_n(x_1,\dots,x_n) \, dx_n = 0$ for $n \ge 1$. (Here, as
in the rest of this paper, the notation $(x_n)$ will refer to the
sequence $(x_n)_{n \in \N}$, where $\N$ refers to the positive integers.)
The value of such a result is perhaps purely psychological. However,
we will also present similar such results for tangent sequences,
and also for sequences satisfying condition (C.I.).
Tangent sequences and condition (C.I.), as defined in this paper,
were introduced by Kwapie\'n and Woyczy\'nski [KW].
The abstract definition of tangent sequences can be, perhaps, a
little hard to grasp. However, in this concrete setting, it is
clear what their meaning is. To demonstrate the psychological
advantage that this view gives, we will give a new proof
of a result of McConnell [M2] that states that tangent
sequence inequalities hold for the UMD spaces.
We will also give a weaker version of a result of McConnell [M1]
that provides a Skorohod representation theorem for
vector valued martingales, in that any martingale is a stopped
continuous time stochastic process.
This means that many martingale inequalities that
are true for continuous time stochastic processes are automatically also
true for
general martingales.
\beginsection 2.\ \ Representations of sequences of random variables
In this section, we present the basic result upon which all our other
results will depend.
First let us motivate this result by considering just one random
variable, that is a measurable
function $f$ from the underlying probability space to a separable
measurable space $(S,\cS)$.
(A measurable space is said to be {\it separable\/} if its sigma field is
generated by a countable
collection of sets.)
We seek to find a measurable function
$g:[0,1] \to S$ that has the same law as $f$, that is, given any
measurable set $A$, we have that $\Pr(f \in A) = \lambda(g \in A)$,
or equivalently, given any measurable bounded function $F:S \to \R$, we have
that $E(F(f)) = \int F(g) \, d\lambda$.
Here $\lambda$ refers to the Lebesgue measure on $[0,1]$.
First, it may be shown without loss of generality that $(S,\cS)$
is $\R$ with the Borel sets. (The argument for this may be found in
Chapter~1 of [DM], and is presented below.) Then the idea is to let
$g$ be the so called increasing rearrangement of $f$, that is,
$$ g(x) = \sup\{t \in \R : \Pr(f < t) < x \} .$$
That $g$ has the required properties is easy to show.
Next, suppose that $f$ is nowhere constant, that is, we have
that $\Pr(f = s) = 0$ for all $s \in S$. Then it may be seen that
$g$ is a strictly increasing function. In that case, it may be seen
that the minimal complete sigma field for which $g$ is measurable
is the collection of Lebesgue measurable sets.
Now let us move on to state the main result.
We will start by setting our notation.
We will work on two probability spaces: a generic one
$(\Omega,\cF,\Pr)$, and $([0,1]^\N,\cL_\N,\lambda)$.
Here
$\cL_\N$ refers to the Lebesgue measurable sets on $[0,1]^\N$, and $\lambda$
refers to Lebesgue measure on $[0,1]^\N$.
Let us make some notational abuses. The reason for this is to
make some expressions less cumbersome, while hopefully not
being too obscure.
We will always identify $[0,1]^n$ with the natural projection
of $[0,1]^\N$ onto the first $n$ coordinates.
Any function $g$ on $[0,1]^n$ will be identified with its
canonical lifting on $[0,1]^\N$. The notation $\cL_n$ refers
to the Lebesgue measurable sets on $[0,1]^n$, and also to their
canonical lifting
onto $[0,1]^\N$. We will let $\lambda$ also refer
to Lebesgue measure on $[0,1]^n$.
Given a random variable $f$, and a sigma field $\cG$, we
will say that $f$ is {\it nowhere constant\/} with respect
to $\cG$ if $\Pr(f = g) = 0$ for every $\cG$ measurable function
$g$. Let us illustrate this notion with respect to a measurable function
$f$ on $[0,1]^2$. It is nowhere constant with respect to the trivial
sigma field if and only if
it is nowhere constant as defined above. Let $\cG$ be the
sigma field generated by the first coordinate.
Then $f$ is nowhere constant with respect to $\cG$ if and only if
for almost every $x \in [0,1]$ the function $y \mapsto f(x,y)$
is nowhere constant. Now let $\cG$ be the set of all Lebesgue measurable
sets on $[0,1]^2$. In this case, $f$ can never be nowhere constant with respect to
$\cG$.
If $f_1,\dots,f_n$ are random variables taking values in
a measurable space $(S,\cS)$, we will let
$\sigma(f_1,\dots,f_n)$ denote the minimal sigma
field which contains all sets of measure zero,
and for which $f_1,\dots,f_n$ are measurable.
Note that a simple argument shows that
the $\sigma(f_1,\dots,f_n)$ measurable functions
coincide precisely with the functions that are
almost everywhere equal to
$F(f_1,\dots,f_n)$, where $F$ is some measurable
function on $S^n$.
Throughout these proofs we will use the following idea many times.
Let
$(\Omega,\cF,\Pr)$ be a
probability space, and let $(S,\cS)$ be a
measurable space. Suppose that $X:\Omega\to S$ is a measurable function
such that
$\sigma(X)$ is the whole
of $\cF$. If $Y$ is any random variable on $\Omega$ taking values in $\R$,
then
there exists a measurable map $\phi:S\to\R$ such that $Y = \phi \circ X$
almost surely. This is easily seen by approximating $Y$ by simple functions.
\proclaim Theorem 2.1. Let $(f_n)$ be a sequence of random variables
taking values in a separable measurable space $(S,\cS)$.
Then there exists a sequence of measurable functions
$(g_n:[0,1]^n \to S)$ that has the same law as $(f_n)$.
\moreproclaim
If further we have that $f_{n+1}$ is nowhere constant with respect to
$\sigma(f_1,\dots,f_n)$ for all $n \ge 0$,
then we may suppose that $\sigma(g_1,\dots,g_n) = \cL_n$, for all $n \ge 1$.
\moreproclaim
If even further we have that $(S,\cS)$ is $\R$ with the Borel sets, then
we may suppose $g_n$ is Borel measurable (that is, the preimages
of Borel sets are Borel sets),
and that $g_n(x_1,\dots,x_n)$ is a strictly increasing
function of $x_n \in (0,1)$ for almost all $x_1,\dots,x_{n-1}$.
\Proof:
We may suppose without loss of generality that $S = \R$, and
$\cS$ is the Borel subsets of $\R$. To see that we may do
this, see first that
we may suppose without loss of generality that $\cS$ separates
points in $S$, that is, if $s\ne t \in S$, then there exists
$A \in \cS$ such that $s \in A$ and $t \notin A$.
Let $\{\cC_n\}$ be a countable collection of sets in $\cS$
that generate $\cS$.
Notice then that the sequence $\{\cC_n\}$ also separates points in $S$,
that is, if $s\ne t \in S$, then there exists a number $n$ such that
one and only one of $s$ or $t$ is in $\cC_n$.
Define a map $\varphi:S \to \R$:
$$ \varphi(s) = \sum_{m=1}^\infty {I_{(s \in C_m)}\over 3^m} .$$
Clearly $\varphi$ is injective. Further
$\varphi$ maps $C_n$ to $D_n \cap \varphi(S)$, where
$$ D_n = \left\{ \sum_{m=1}^\infty {e_m \over 3^m} : e_m = 0 \hbox{ or }
1, \, e_n = 0 \right\} , $$
and thus $\varphi$ maps any element of $\cS$ to a Borel subset of $\R$
intersected with $\varphi(S)$. Conversely, the preimage of $D_n$ under
$\varphi$ is $C_n$, and hence the preimage of any Borel set under $\varphi$
is in $\cS$. (This argument may be found in Chapter~1 of [DM]).
Now apply the theorem to $(\varphi \circ f_n)$, to obtain $(g_n)$. Since
the law of $g_n$ is the same as $\varphi \circ f_n$, the range of
$g_n$ lies in the range of $\varphi(S)$ with probability one.
Then the sequence $(\varphi^{-1} \circ g_n)$ will have the same
law as $(f_n)$.
So let us suppose as the induction hypothesis that we have
obtained $(g_n:[0,1]^n \to \R)_{1 \le n \le N}$ that has the same law as
$(f_n)_{1 \le n \le N}$, and that $g_n$ is Borel measurable,
for all $n \le N$,
where $N$ is a non-negative integer.
(The induction is started with $N=0$ in which case the hypothesis
is vacuously true. The arguments that follow simplify greatly in this case.)
For each $t \in \Q$, let
$$ p_t = \E(I_{(f_{N+1}