E l e c t r o n i c J o u r n a l

o f

P r o b a b

i l i t y

                  http://www.math.washington.edu/~ejpecp/

Concrete Representation of Martingales

Stephen Montgomery-Smith Department of Mathematics University of Missouri, Columbia, MO 65211

                         stephen@math.missouri.edu
                     http://math.missouri.edu/~stephen

Volume 3, paper number 15, 1998

      http://www.math.washington.edu/~ejpecp/EjpVol3/paper15.abs.html

15 pages, submitted June 4, 1998, published December 2, 1998

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 ∫_{0}^{1} d_{n}(x_{1}, …, x_{n}) dx_{n} = 0 for almost all x_{1}, …, x_{n−1}, and such that the law of (f_{n}) is the same as the law of (∑_{k=1}^{n} d_{k}(x_{1}, …, 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.

Keywords: martingale, concrete representation, tangent sequence, condition (C.I.), UMD, Skorohod representation

A.M.S. Classification (1991): 60G42, 60H05.

Research supported in part by the N.S.F. and the Research Board of the University of Missouri.

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 L_{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})) = ∑_{k=0}^{n} d_{k}(x_{1}, …, x_{k}),

where ∫_{0}^{1} d_{n}(x_{1}, …, x_{n}) dx_{n} = 0 for n ≥ 1. (Here, as in the rest of this paper, the notation (x_{n}) will refer to the sequence (x_{n})_{n∈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 Kwapień and Woyczyński [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.

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, S). (A measurable space is said to be separable if its sigma field is generated by a countable collection of sets.) We seek to find a measurable function g : [0, 1] → S that has the same law as f, that is, given any measurable set A, we have that Pr(f ∈ A) = λ(g ∈ A), or equivalently, given any measurable bounded function F : S → ℝ, we have that E(F (f)) = ∫ F (g) dλ. Here λ refers to the Lebesgue measure on [0, 1].

First, it may be shown without loss of generality that (S, S) 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 ∈ 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 ∈ 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 (Ω, F, Pr), and ([0, 1]^{ℕ}, L_{ℕ}, λ). Here L_{ℕ} refers to the Lebesgue measurable sets on [0, 1]^{ℕ}, and λ 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 L_{n} refers to the Lebesgue measurable sets on [0, 1]^{n}, and also to their canonical lifting onto [0, 1]^{ℕ}. We will let λ also refer to Lebesgue measure on [0, 1]^{n}.

Given a random variable f, and a sigma field G, we will say that f is nowhere constant with respect to G if Pr(f = g) = 0 for every G 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 G be the sigma field generated by the first coordinate. Then f is nowhere constant with respect to G if and only if for almost every x ∈ [0, 1] the function y ↦ f(x, y) is nowhere constant. Now let G be the set of all Lebesgue measurable sets on [0, 1]^{2}. In this case, f can never be nowhere constant with respect to G.

If f_{1}, …, f_{n} are random variables taking values in a measurable space (S, S), we will let σ(f_{1}, …, f_{n}) denote the minimal sigma field which contains all sets of measure zero, and for which f_{1}, …, f_{n} are measurable. Note that a simple argument shows that the σ(f_{1}, …, f_{n}) measurable functions coincide precisely with the functions that are almost everywhere equal to F (f_{1}, …, f_{n}), where F is some measurable function on S^{n}.

Throughout these proofs we will use the following idea many times. Let (Ω, F, Pr) be a probability space, and let (S, S) be a measurable space. Suppose that X : Ω → S is a measurable function such that σ(X) is the whole of F. If Y is any random variable on Ω taking values in ℝ, then there exists a measurable map ϕ : S → ℝ such that Y = ϕ ∘ X almost surely. This is easily seen by approximating Y by simple functions.

Theorem 2.1. Let (f_{n}) be a sequence of random variables taking values in a separable measurable space (S, S). Then there exists a sequence of measurable functions (g_{n} : [0, 1]^{n} → S) that has the same law as (f_{n}).

If further we have that f_{n+1} is nowhere constant with respect to σ(f_{1}, …, f_{n}) for all n ≥ 0, then we may suppose that σ(g_{1}, …, g_{n}) = L_{n}, for all n ≥ 1.

If even further we have that (S, S) 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}, …, x_{n}) is a strictly increasing function of x_{n} ∈ (0, 1) for almost all x_{1}, …, x_{n−1}.

Proof: We may suppose without loss of generality that S = ℝ, and S is the Borel subsets of ℝ. To see that we may do this, see first that we may suppose without loss of generality that S separates points in S, that is, if s ≠ t ∈ S, then there exists A ∈ S such that s ∈ A and t ∉ A. Let {C_{n}} be a countable collection of sets in S that generate S. Notice then that the sequence {C_{n}} also separates points in S, that is, if s ≠ t ∈ S, then there exists a number n such that one and only one of s or t is in C_{n}. Define a map φ : S → ℝ:

φ(s) = ∑_{m=1}^{∞} \frac{I_{(s∈C_{m})}}{3^{m}} .

Clearly φ is injective. Further φ maps C_{n} to D_{n} ∩ φ(S), where

D_{n} = \{ ∑_{m=1}^{∞} \frac{e_{m}}{3^{m}} : e_{m} = 0 or 1, e_{n} = 0\} ,

and thus φ maps any element of S to a Borel subset of ℝ intersected with φ(S). Conversely, the preimage of D_{n} under φ is C_{n}, and hence the preimage of any Borel set under φ is in S. (This argument may be found in Chapter 1 of [DM]).

Now apply the theorem to (φ ∘ f_{n}), to obtain (g_{n}). Since the law of g_{n} is the same as φ ∘ f_{n}, the range of g_{n} lies in the range of φ(S) with probability one. Then the sequence (φ^{−1} ∘ 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} → ℝ)_{1≤n≤N} that has the same law as (f_{n})_{1≤n≤N}, and that g_{n} is Borel measurable, for all n ≤ 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 ∈ ℚ, let

p_{t} = 𝔼(I_{(f_{N+1}<t)}|σ(f_{1}, …, f_{N})).

Since p_{t} is σ(f_{1}, …, f_{N}) measurable, we may write

p_{t} = q_{t}(f_{1}, …, f_{N}) almost surely

for some Borel measurable function q_{t} : ℝ^{N} → [0, 1]. Define the Borel measurable functions r_{t} by

r_{t} = q_{t}(g_{1}, …, g_{N}).

Since the sequence (r_{t})_{t∈ℚ} has the same law as (p_{t})_{t∈ℚ}, we see that there is a Borel set B ⊂ [0, 1]^{N} of full measure such that if (x_{1}, …, x_{N}) ∈ B, then r_{t}(x_{1}, …, x_{N}) is an increasing function of t ∈ ℚ, that tends to 0 as t → −∞, and tends to 1 as t → ∞.

If (x_{1}, …, x_{N}) ∈ B and x_{N+1} ∈ (0, 1), let

g_{N+1}(x_{1}, …, x_{N+1}) = sup{t ∈ ℚ : r_{t}(x_{1}, …, x_{N}) < x_{N+1}},

and let it be zero otherwise. We see that g_{N+1} is Borel measurable, since g_{N+1} on B×(0, 1) is the supremum of countably many Borel measurable functions (s_{t})_{t∈ℚ} where

s_{t}(x_{1}, …, x_{N+1}) = \{ \begin{matrix}t & if r_{t}(x_{1}, …, x_{N}) < x_{N+1} \\ −∞ & otherwise. \end{matrix}

It is easy to see that g_{N+1} is always finite.

Let us now show that (f_{1}, …, f_{N+1}) has the same law as (g_{1}, …, g_{N+1}). It is sufficient to show that for any Borel set A ⊂ ℝ^{N}, and for t ∈ ℚ, that

Pr((f_{1}, …, f_{N}) ∈ A and f_{N+1} < t) = λ((g_{1}, …, g_{N}) ∈ A and g_{N+1} < t).

So let us begin.

λ((g_{1}, …, g_{N}) ∈ A and g_{N+1} < t) = λ({(g_{1}, …, g_{N}) ∈ A and g_{N+1} < t} ∩ B × (0, 1)) ≤ λ({(x_{1}, …, x_{N+1}) : (g_{1}(x_{1}), …, g_{N}(x_{1}, …, x_{N})) ∈ A and r_{t}(x_{1}, …, x_{N}) ≥ x_{N+1}}

∩ B × (0, 1))

= λ({(x_{1}, …, x_{N+1}) : (g_{1}(x_{1}), …, g_{N}(x_{1}, …, x_{N})) ∈ A and r_{t}(x_{1}, …, x_{N}) ≥ x_{N+1}})

= ∫_{[0,1]^{N}} ∫_{0}^{1} I_{(r_{t}≥x)}I_{((g_{1},...,g_{N})∈A)} dx dλ

= ∫_{[0,1]^{N}} r_{t}I_{((g_{1},...,g_{N})∈A)} dλ

= 𝔼(p_{t}I_{((f_{1},...,f_{N})∈A)}) = 𝔼(I_{(f_{N+1}<t)}I_{((f_{1},...,f_{N})∈A)}) = Pr((f_{1}, …, f_{N}) ∈ A and f_{N+1} < t).

Similarly

λ((g_{1}, …, g_{N}) ∈ A and g_{N+1} ≤ t) ≥ Pr((f_{1}, …, f_{N}) ∈ A and f_{N+1} < t),

and so by Lebesgue’s monotone convergence theorem

λ((g_{1}, …, g_{N}) ∈ A and g_{N+1} < t) = lim_{{s↗t \atop s∈ℚ}} λ((g_{1}, …, g_{N}) ∈ A and g_{N+1} ≤ s)

≥ lim_{{s↗t \atop s∈ℚ}} Pr((f_{1}, …, f_{N}) ∈ A and f_{N+1} < s)

= Pr((f_{1}, …, f_{N}) ∈ A and f_{N+1} < t).

Thus we have shown that (f_{1}, …, f_{N+1}) has the same law as (g_{1}, …, g_{N+1}).

Now let us add the assumption that f_{n+1} is nowhere constant with respect to σ(f_{1}, …, f_{n}) for all n ≥ 0. Let us assume the inductive hypothesis that σ(g_{1}, …, g_{N}) = L_{N},

Then there exist Borel measurable functions (α_{n} : ℝ^{n} → [0, 1])_{1≤n≤N} such that for 1 ≤ n ≤ N we have

α_{n}(g_{1}(x_{1}), …, g_{N}(x_{1}, …, x_{N})) = x_{n} almost everywhere.

Let β_{n} = α_{n}(g_{1}, …, g_{N}), and let γ_{n} = α_{n}(f_{1}, …, f_{N}).

Notice that g_{N+1}(x_{1}, …, x_{N+1}) is an increasing function in x_{N+1} ∈ (0, 1) for fixed (x_{1}, …, x_{N}) ∈ B. Let us show that g_{N+1}(x_{1}, …, x_{N+1}) is a strictly increasing function in x_{N+1} ∈ (0, 1) for (x_{1}, …, x_{N}) ∈ C where C is a subset of B of full measure. For suppose otherwise. Then the following set has positive measure: the set of (x_{1}, …, x_{N+1}) such that for some rational number t ∈ [0, 1] we have that g_{N+1}(x_{1}, …, x_{N}, x_{N+1}) = g_{N+1}(x_{1}, …, x_{N}, t). But this is equal to the set

⋃_{t∈[0,1]∪ℚ}{g_{N+1} = g_{N+1}(β_{1}, …, β_{N}, t)}.

But the measure of this set is the same as the probability of the set

⋃_{t∈[0,1]∪ℚ}{f_{N+1} = g_{N+1}(γ_{1}, …, γ_{N}, t)}.

Since f_{N+1} is nowhere constant with respect to σ(f_{1}, …, f_{N}), it follows that this last set has probability zero.

Let us now show that σ(g_{1}, …, g_{N+1}) = L_{N+1}. For any a ∈ [0, 1] we have that

{g_{N+1} ≤ g_{N+1}(β_{1}, …, β_{N}, a)} = [0, 1]^{N} × [0, a] up to a set of measure zero.

Thus σ(g_{1}, …, g_{N+1}) contains all set of the form [0, 1]^{N} × [0, a]. Since it also contains σ(g_{1}, …, g_{N}) = L_{N}, the result follows.

Q.E.D.

3. Representation of martingales — 1

Theorem 3.1. Let (d_{n}) be a Bochner integrable martingale difference sequence taking values in a Banach space X. Then there exists a sequence of Bochner measurable functions (e_{n} : [0, 1]^{n} → X) such that (d_{n}) has the same law as (e_{n}), and such that for almost every x_{1}, …, x_{n} we have

∫_{0}^{1} e_{n}(x_{1}, …, x_{n}) dx_{n} = 0.

Proof: First we show that without loss of generality that we may assume that d_{n+1} is nowhere constant with respect to σ(d_{1}, …, d_{n}) for all n ≥ 0. To do this, replace Ω by Ω × [0, 1]^{ℕ}, replace X by X × ℝ (where ℝ is equipped with the sigma field of Borel sets), and replace d_{n} by (ω, (x_{n})) ↦ (d_{n}(ω), x_{n} − \frac{1}{2}). Apply the theorem to this sequence. After obtaining the resulting (e_{n}), compose these functions with the natural projection of X × ℝ to X.

Further, since (d_{n}) is Bochner integrable, we may suppose without loss of generality that X is separable.

Apply Theorem 2.1 to obtain the sequence (e_{n}). Suppose that ϕ : [0, 1]^{n−1} → ℝ is any bounded measurable function. Then there exists a bounded Borel measurable function θ : X^{n−1} → ℝ such that ϕ(x_{1}, …, x_{n−1}) = θ(e_{1}(x_{1}), …, e_{n−1}(x_{1}, …, x_{n−1})) for almost all x_{1}, …, x_{n−1}. Then

∫_{[0,1]^{n}} ϕe_{n} dλ = 𝔼(θ(d_{1}, …, d_{n−1})d_{n}),

and this is zero because 𝔼(d_{n}|σ(d_{1}, …, d_{n−1})) = 0. The result follows.

Q.E.D.

4. Representation of tangent sequences The following definition may be found in [KW]. Let (F_{n})_{n≥0} be an increasing sequence of sigma fields on (Ω, F, Pr), where F_{0} is the trivial sigma field. Two adapted sequences (f_{n}) and (g_{n}) taking values in a measurable space (S, S) are said to be tangent if for each n ≥ 1 we have that the law of f_{n} conditionally on F_{n−1} is the same as the law of g_{n} conditionally on F_{n−1}, that is, 𝔼(I_{(f_{n}∈A)}|F_{n−1}) = 𝔼(I_{(g_{n}∈A)}|F_{n−1}) for any A ∈ S.

Notice then that if F : S^{2n−1} → ℝ is any measurable bounded function, that

𝔼(F (f_{1}, g_{1}, …, f_{n−1}, g_{n−1}, f_{n})) = 𝔼(F (f_{1}, g_{1}, …, f_{n−1}, g_{n−1}, g_{n})).

This is easily seen by reducing it to the case where F is a finite linear combination of characteristic functions of sets of the form A_{1} × … × A_{2n−1} where A_{1}, …, A_{2n−1} ∈ S.

Theorem 4.1. Let (S, S) be a separable measurable space. Let (f_{n}) and (g_{n}) be S-valued measurable sequences adapted to (F_{n})_{n≥0} that are tangent. Then there exists a sequence (h_{n} : [0, 1]^{n} → S), and a sequence of Borel measurable functions (ϕ_{n} : [0, 1]^{n} → [0, 1]) such that the map ϕ_{n}(x_{1}, …, x_{n−1}, ·) is a measure preserving map on [0, 1] for almost every x_{1}, …, x_{n−1}, such that the law of (f_{n}, g_{n}) is the same as the law of (h_{n}, k_{n}), where

k_{n}(x_{1}, …, x_{n}) = h_{n}(x_{1}, …, x_{n−1}, ϕ_{n}(x_{1}, …, x_{n})).

Further, if S is a separable Banach space, and (f_{n}) is a martingale difference sequence with respect to F_{n}, then

∫_{0}^{1} h_{n}(x_{1}, …, x_{n}) dx_{n} = 0

for almost all x_{1}, …, x_{n−1}.

Lemma 4.2. Let (S, S, μ), (T, T , ν) and (U, U, τ) be measure spaces such that U is separable. Let ϕ : S × T → U be a measurable map such that for A ∈ S and C ∈ U we have that

μ ⊗ ν((s, t) : s ∈ A and ϕ(s, t) ∈ C) = μ(A)τ(C).

Then ϕ(s, ·) is a measure preserving map T → U for μ almost every s.

Proof: The hypothesis of the lemma can be recast as:

∫_{s∈A} ν(t : ϕ(s, t) ∈ C) dμ(s) = ∫_{A} τ(C) dμ,

from which we conclude that for every C ∈ U, we have that ν(t : ϕ(s, t) ∈ C) = τ(C) for μ almost every s.

Let C be a countable subcollection of U that generates U. Then there exists a set A ∈ S of full measure such that ν(t : ϕ(s, t) ∈ C) = τ(C) for C ∈ C and s ∈ A. Let

M = {C ∈ U : ν(t : ϕ(s, t) ∈ C) = τ(C) for s ∈ A}.

Notice that M is a sigma field that contains C, and hence M = U.

Q.E.D.

Proof of Theorem 4.1: Without loss of generality (at least until we prove the last part), let us suppose that S = ℝ with the Borel sets. To do this, we use the same argument that we gave at the beginning of the proof of Theorem 2.1.

Consider the sequence (f_{1}, g_{1}, f_{2}, g_{2}, …). By the kind of argument we presented at the beginning of Theorem 3.1, we may suppose that each element of this sequence is nowhere constant with respect to the sigma field generated by those elements in the sequence that precede it.

Applying Theorem 2.1, we create a sequence of Borel measurable functions (h̃_{1}, k̃_{1}, h̃_{2}, k̃_{2}, …) with the same law, where h̃_{n} : [0, 1]^{2n−1} → ℝ, k̃_{n} : [0, 1]^{2n} → ℝ, and h̃_{n}(x_{1}, y_{1}, …, x_{n−1}, y_{n−1}, x_{n}) is a strictly increasing function of x_{n} ∈ (0, 1) for almost every x_{1}, y_{1}, …, x_{n−1}, y_{n−1}.

Define the function

r_{n}(x_{1}, y_{1}, …, x_{n−1}, y_{n−1}, s) = sup{t ∈ ℚ ∩ (0, 1) : h̃_{n}(x_{1}, y_{1}, …, x_{n−1}, y_{n−1}, t) < s},

where we will set the first supremum equal to zero if the set is empty. Notice that r_{n} is the supremum of countably many Borel measurable functions, and hence is Borel measurable. Also, since h̃_{n}(x_{1}, y_{1}, …, y_{n−1}, x_{n−1}, x_{n}) is a strictly increasing function of x_{n} ∈ (0, 1) for almost every x_{1}, y_{1}, …, x_{n−1}, y_{n−1}, we see that

r_{n}(x_{1}, y_{1}, …, x_{n−1}, y_{n−1}, h̃_{n}(x_{1}, y_{1}, …, x_{n−1}, y_{n−1}, x_{n})) = x_{n}

for almost every x_{1}, y_{1}, …, x_{n}. Define

ϕ̃_{n}(x_{1}, y_{1}, …, x_{n}, y_{n}) = r_{n}(x_{1}, y_{1}, …, x_{n−1}, y_{n−1}, k̃_{n}(x_{1}, y_{1}, …, x_{n}, y_{n})).

Now let α_{n}, β_{n} be functions such that

x_{n} = α_{n}(h̃_{1}(x_{1}), k̃_{1}(x_{1}, y_{1}), …, h̃_{n}(x_{1}, y_{1}, …, x_{n})) almost everywhere
y_{n} = β_{n}(h̃_{1}(x_{1}), k̃_{1}(x_{1}, y_{1}), …, k̃_{n}(x_{1}, y_{1}, …, x_{n}, y_{n})) almost everywhere,

and let γ_{n}, δ_{n} be random variables defined by

γ_{n} = α_{n}(f_{1}, g_{1}, …, f_{n})
δ_{n} = β_{n}(f_{1}, g_{1}, …, f_{n}, g_{n}).

If A is any Borel subset of [0, 1]^{2n−2}, and C is a Borel subset of [0, 1], then since f_{n} and g_{n} have the same law with respect to F_{n−1},

λ((x_{1}, y_{1}, …, x_{n−1}, y_{n−1}) ∈ A and ϕ̃_{n}(x_{1}, y_{1}, …, x_{n}, y_{n}) ∈ C) = Pr((γ_{1}, δ_{1}, …, γ_{n−1}, δ_{n−1}) ∈ A and r_{n}(γ_{1}, δ_{1}, …, γ_{n−1}, δ_{n−1}, g_{n}) ∈ C) = Pr((γ_{1}, δ_{1}, …, γ_{n−1}, δ_{n−1}) ∈ A and r_{n}(γ_{1}, δ_{1}, …, γ_{n−1}, δ_{n−1}, f_{n}) ∈ C) = λ((x_{1}, y_{1}, …, x_{n−1}, y_{n−1}) ∈ A and r_{n}(x_{1}, y_{1}, …, x_{n−1}, y_{n−1}, h̃_{n}(x_{1}, y_{1}, …, x_{n})) ∈ C) = λ((x_{1}, y_{1}, …, x_{n−1}, y_{n−1}) ∈ A and x_{n} ∈ C) = λ(A)λ(C).

Thus, by Lemma 4.2, we have that ϕ̃(x_{1}, y_{1}, …, x_{n−1}, y_{n−1}, ·, ·) is a measure preserving map from [0, 1]^{2} to [0, 1] for almost every x_{1}, y_{1}, …, x_{n−1}, y_{n−1}.

Now let x ↦ (θ_{1}(x), θ_{2}(x)) be any Borel measurable measure preserving map from [0, 1] to [0, 1]^{2}. Let

h_{n}(x_{1}, …, x_{n}) = h̃_{n}(θ_{1}(x_{1}), θ_{2}(x_{1}), …, θ_{1}(x_{n}))
k_{n}(x_{1}, …, x_{n}) = k̃_{n}(θ_{1}(x_{1}), θ_{2}(x_{1}), …, θ_{1}(x_{n}), θ_{2}(x_{n})).

Notice that

k_{n}(x_{1}, …, x_{n}) = h_{n}(x_{1}, …, x_{n−1}, ϕ_{n}(x_{1}, …, x_{n})),

where

ϕ_{n}(x_{1}, …, x_{n}) = ϕ̃_{n}(θ_{1}(x_{1}), θ_{2}(x_{1}), …, θ_{1}(x_{n}), θ_{2}(x_{n})).

The last part of the theorem follows by exactly the same argument as for Theorem 3.1. Q.E.D.

5. Representation of sequences satisfying condition (C.I.) The following definition may be found in [KW]. Let (F_{n})_{n≥0} be an increasing sequence of sigma fields on (Ω, F, Pr), where F_{0} is the trivial sigma field. An adapted sequence (f_{n}) taking values in a measurable space (S, S) is said to satisfy condition (C.I.) if there exists a sigma field G ⊂ F such that the law of f_{n} conditionally on F_{n−1} is the same as the law of f_{n} conditionally on G, that is, 𝔼(I_{(f_{n}∈A)}|F_{n−1}) = 𝔼(I_{(f_{n}∈A)}|G) for any A ∈ S, and if the sequence (f_{n}) is conditionally independent with respect to G, that is, for any sequence of sets A_{n} ∈ S we have

𝔼(I_{(f_{1}∈A_{1})} ⋯ I_{(f_{n}∈A_{n})}|G) = 𝔼(I_{(f_{1}∈A_{1})}|G) ⋯ 𝔼(I_{(f_{n}∈A_{n})}|G).

It is shown in [KW] that given any sequence (f_{n}) adapted to some filtration, that after possibly enlarging the underlying probability space, that there exists a sequence (f̃_{n}) that is tangent to (f_{n}) and that satisfies condition (C.I.).

Indeed, using Theorem 2.1, we can show a technically weaker but essentially identical result as follows. Let (g_{n}) be the sequence constructed by Theorem 2.1. Enlarge the probability space [0, 1]^{ℕ} to [0, 1]^{ℕ} × [0, 1]^{ℕ}, and define the sequences

u_{n}((x_{n}), (y_{n})) = g_{n}(x_{1}, …, x_{n−1}, x_{n})
v_{n}((x_{n}), (y_{n})) = g_{n}(x_{1}, …, x_{n−1}, y_{n}).

Then (u_{n}) has the same law as (f_{n}), and (v_{n}) is tangent to (u_{n}) with respect to the filtration (L_{n} ⊗ L_{0}), and (v_{n}) satisfies condition (C.I.). (Here L_{n} is the minimal complete sigma field on [0, 1]^{ℕ} for which the first n coordinate functions are measurable. In particular, L_{0} denotes the trivial complete measure sigma field on [0, 1]^{ℕ}.)

This is the motivation behind the next result, which shows that this previous construction is the canonical representation.

Theorem 5.1. Let (S, S) be a separable measurable space. Let (f_{n}) be an S-valued measurable sequence adapted to (F_{n})_{n≥0} that satisfies condition (C.I.). Then there exists a sequence (h_{n} : [0, 1]^{n} → S) such that if we define the functions u_{n} on [0, 1]^{ℕ} × [0, 1]^{ℕ} by

u_{n}((x_{n}), (y_{n})) = h_{n}(x_{1}, …, x_{n−1}, y_{n}),

then the laws of (f_{n}), (h_{n}) and (u_{n}) are the same.

Further, if S is a separable Banach space, and (f_{n}) is a martingale difference sequence with respect to F_{n}, then

∫_{0}^{1} h_{n}(x_{1}, …, x_{n}) dx_{n} = 0

for almost all x_{1}, …, x_{n}.

Proof: As usual, we suppose without loss of generality that f_{n} is nowhere constant with respect to σ(f_{1}, …, f_{n−1}). Let (h_{n}) be the sequence generated by Theorem 2.1. Then for any A_{1}, …, A_{n} ∈ S we have that

Pr((f_{1}, …, f_{n}) ∈ A_{1} × ⋯ × A_{n})
= 𝔼(𝔼(I_{(f_{1}∈A_{1})} ⋯ I_{(f_{n}∈A_{n})}|G))
= 𝔼(𝔼(I_{(f_{1}∈A_{1})}|G) ⋯ 𝔼(I_{(f_{n}∈A_{n})}|G))
= 𝔼(𝔼(I_{(f_{1}∈A_{1})}|F_{0}) ⋯ 𝔼(I_{(f_{n}∈A_{n})}|F_{n−1})).

Now for each k = 1, …, n, let F_{k} : S^{k−1} → [0, 1] be the measurable function such that

𝔼(I_{(f_{k}∈A_{k})}|F_{k−1}) = F_{k}(f_{1}, …, f_{k−1}) almost surely.

Notice that if G : S^{n−1} → ℝ is bounded measurable, then

𝔼(I_{(f_{k}∈A_{k})}G(f_{1}, …, f_{k−1})) = 𝔼(F_{k}(f_{1}, …, f_{k−1})G(f_{1}, …, f_{k−1})),

and hence

𝔼(I_{(h_{k}∈A_{k})}G(h_{1}, …, h_{k−1})) = 𝔼(F_{k}(h_{1}, …, h_{k−1})G(h_{1}, …, h_{k−1})).

Since σ(h_{1}, …, h_{k−1}) = L_{k−1}, we see that

𝔼(I_{(h_{k}∈A_{k})}|L_{k−1}) = F_{k}(h_{1}, …, h_{k−1}) almost everywhere.

Thus the sequence

(𝔼(I_{(f_{k}∈A_{k})}|F_{k−1}))_{1≤k≤n}

has the same law as the sequence

(𝔼(I_{(h_{k}∈A_{k})}|L_{k−1}))_{1≤k≤n},

and this last sequence is very easily seen to have the same law as the sequence

(𝔼(I_{(u_{k}∈A_{k})}|L_{k−1} ⊗ L_{0}))_{1≤k≤n}.

Hence

𝔼(𝔼(I_{(f_{1}∈A_{1})}|F_{0}) ⋯ 𝔼(I_{(f_{n}∈A_{n})}|F_{n−1}))
= ∫ (𝔼(I_{(u_{1}∈A_{1})}|L_{0} ⊗ L_{0}) ⋯ 𝔼I_{(u_{n}∈A_{n})}|L_{n−1} ⊗ L_{0}) dλ ⊗ λ,

and this last quantity is easily computed to be

λ ⊗ λ((u_{1}, …, u_{n}) ∈ A_{1} × ⋯ × A_{n}).

Finally, the last part of the theorem follows exactly as in Theorem 3.1.

Q.E.D.

6. Tangent sequences in UMD spaces The concept of UMD spaces was introduced by Aldous [A], and extensively explored by Burkholder [Bu]. It is from this second reference that the following definition of UMD (along with many other equivalent definitions) may be found.

A Banach space X is said to be UMD if for some 1 < p < ∞ (equivalently all 1 < p < ∞), there exists a positive constant c depending only upon X and p such that if (d_{n}) is a Bochner integrable X-valued martingale difference sequence, then for any signs ϵ_{n} = ±1, we have that

∥∑_{n=1}^{N} ϵ_{n}d_{n}∥_{L_{p}(X)} ≤ c∥∑_{n=1}^{N} d_{n}∥_{L_{p}(X)}.

This section is devoted to providing a new proof of the following result. This is essentially the same as the first part of Theorem 2.2 from the paper by McConnell [M2]. While the following proof can be rewritten so as to avoid needing the representation theorems, to do so would be to remove the motivation behind this proof.

Theorem 6.1. Let X be a UMD space. Then given 1 < p < ∞, there is a positive constant C, depending only upon X and p, such that given a filtration (F_{n})_{n≥0}, and two martingales (f_{n}) and (g_{n}) adapted to this filtration that are X-valued Bochner integrable and tangent to each other, we have that

∥∑_{n=1}^{N} g_{n}∥_{L_{p}(X)} ≤ C∥∑_{n=1}^{N} f_{n}∥_{L_{p}(X)}.

Proof: It is sufficient to prove the inequality in the case when one of the sequences satisfies condition (C.I.), since the inequality may be then obtained by comparing both sides with another sequence tangent to them both that satisfies condition (C.I.). We will prove it in the case when the sequence (g_{n}) satisfies condition (C.I.), but it will be evident that the proof also works when the sequence (f_{n}) satisfies condition (C.I.).

So applying Theorems 4.1 and 5.1, we see that there exists a sequence h_{n} : [0, 1]^{n} → X of X-valued Bochner integrable functions such that

∫_{0}^{1} h_{n}(x_{1}, …, x_{n}) dx_{n} = 0

for almost all x_{1}, …, x_{n−1}, and such that if we define the the functions d_{n}, e_{n} : [0, 1]^{ℕ} × [0, 1]^{ℕ} → X by

d_{n}((x_{n}), (y_{n})) = h_{n}(x_{1}, …, x_{n−1}, x_{n})
e_{n}((x_{n}), (y_{n})) = h_{n}(x_{1}, …, x_{n−1}, y_{n}),

then the sequence (f_{1}, g_{1}, f_{2}, g_{2}, …) has the same law as the sequence (d_{1}, e_{1}, d_{2}, e_{2}, …).

Now we define a martingale difference sequence r_{n} on [0, 1]^{ℕ}×[0, 1]^{ℕ} by r_{2n−1} = d_{n}+e_{n} and r_{2n} = d_{n} − e_{n}, where the filtration is G_{2n} = L_{n} ⊗ L_{n}, and G_{2n−1} is the collection of those sets in G_{2n} generated by sets like

{((x_{n}), (y_{n})) : (x_{1}, …, x_{n}) ∈ A × C, (y_{1}, …, y_{n}) ∈ B × C},

where A, B ∈ L_{n−1}, and C ∈ L_{1}.

Then we see that ∑_{n=1}^{N} d_{n} = \frac{1}{2} ∑_{n=1}^{2N} r_{n} and ∑_{n=1}^{N} e_{n} = \frac{1}{2} ∑_{n=1}^{2N}(−1)^{n+1}r_{n}, and thus the result follows immediately from the fact that X is a UMD space.

Q.E.D.

7. Representations of martingales — 2 In this section, we present a version of a result of McConnell [M1]. This result provides a Skorohod like representation for vector valued martingales. Our result is weaker than that of McConnell in that his process is adapted to a filtration generated by a one-dimensional Brownian motion. The process presented here is adapted to a two dimensional Brownian motion.

Theorem 7.1. Let (d_{n}) be a Bochner integrable martingale difference sequence taking values in a Banach space X. Then there exists a continuous time X-valued stochastic process (F_{t})_{t≥0} with continuous sample paths such that (d_{n}) has the same law as (F_{n} − F_{n−1}). Furthermore, the process (F_{t}) is adapted to a two dimensional brownian motion.

Proof: Let D = {z ∈ ℂ : |z| < 1}, D̄ = {z ∈ ℂ : |z| ≤ 1} and ∂D = {z ∈ ℂ : |z| = 1}, and give ∂D normalized Lebesgue measure d|z|/2π. Since ∂D is measure equivalent to [0, 1], we may apply Theorem 3.1 to obtain a sequence (e_{n} : ∂D^{n} → X) that has the same law as (f_{n}), and such that

∫_{∂D} e_{n}(z_{1}, …, z_{n}) \frac{d|z_{n}|}{2π} = 0

for almost every z_{1}, …, z_{n−1}.

For each z_{1}, …, z_{n−1}, extend the function z_{n} ↦ e_{n}(z_{1}, …, z_{n}) to its harmonic extension on D̄. We will identify e_{n} with this extension.

Now let (b_{t}^{(n)})_{n∈ℕ} be a sequence of independent Brownian motions into ℂ with origin at zero. Let τ_{n} be the stopping time τ_{n} = inf{t : |b_{t}^{(n)}| ≥ 1}. Notice that τ_{n} is finite almost surely.

By the Itô calculus, we have that

e_{n}(b_{τ_{1}}^{(1)}, …, b_{min{t,τ_{n}}}^{(n)}) = ∫_{0}^{min{t,τ_{n}}} ∇_{n}e_{n}(b_{τ_{1}}^{(1)}, …, b_{τ_{n−1}}^{(n−1)}, b_{s}^{(n)}) · db_{s}^{(n)},

where ∇_{n} refers to taking the gradient with respect to the nth coordinate. Define a process

G_{n,t} = ( ∑_{m=1}^{n−1} e_{m}(b_{τ_{1}}^{(1)}, …, b_{τ_{m}}^{(m)})) + e_{n}(b_{τ_{1}}^{(1)}, …, b_{min{t,τ_{n}}}^{(n)}).

Let ϕ : [0, 1) → [0, ∞) be defined by ϕ(t) = t/(1 − t), and let

F_{t} = G_{[t+1],ϕ(t−[t])},

where [t] represents the largest integer less than t.

To see that (F_{t}) may be adapted to a two-dimensional brownian motion, let b̃_{t} be a brownian motion taking values in ℂ, and set

b_{t}^{(n)} = ∫_{n−1}^{n−1+ϕ^{−1}(t)} \sqrt{ϕ′(s)} db̃_{s}.

Q.E.D.

Acknowledgments I would like to mention the help of Gerald Edgar and P. Fitzsimmons in obtaining the argument at the beginning of Theorem 2.1, where it is shown that without loss of generality that (S, S) is ℝ with the Borel sets, and for bringing the reference [DM] to my attention. I also extend thanks to Donald Burkholder, who brought the reference [M1] to may attention. Finally, I would like to thank Terry McConnell and an anonymous referee for many useful remarks.

References

[A] D.J. Aldous, Unconditional bases and martingales in L_{p}(F), Math. Proc. Cambridge Philos. Soc. 85, (1979), 117–123.

[Bu] D.L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9, (1981), 997–1011.

[DM] C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North-Holland Mathematics Studies 29, North-Holland, Amsterdam-New York-Oxford, 1978.

[KW] S. Kwapień and W.A. Woyczyński, Tangent sequences of random variables, in Almost Everywhere Convergence, G.A. Edgar and L. Sucheston, Eds., Academic Press, 1989, pp. 237–265.

[M1] T.R. McConnell, A Skorohod-like representation in infinite dimensions, Probability in Banach spaces, V (Medford, Mass., 1984), 359–368, Lecture Notes in Math., 1153, Springer, Berlin-New York, 1985.

[M2] T.R. McConnell, Decoupling and stochastic integration in UMD Banach spaces, Probab. Math. Statist. 10, (1989), 283–295.