AN EXTENSION TO THE STRONG DOMINATION MARTINGALE INEQUALITY

STEPHEN MONTGOMERY-SMITH AND SHIH-CHI SHEN

Abstract. For each 1 < p < ∞, there exists a positive constant c_{p}, depending only on p, such that the following holds. Let (d_{k}), (e_{k}) be real-valued martingale difference sequences. If for for all bounded nonnegative predictable sequences (s_{k}) and all positive integers k we have

E[s_{k} ∨ |e_{k}|] ≤ E[s_{k} ∨ |d_{k}|]

then for all positive integers n we have

∥∑_{k=1}^{n} e_{k}∥_{p} ≤ c_{p} ∥∑_{k=1}^{n} d_{k}∥_{p} .

1. Introduction

Let (Ω, F, P) be a probability space, and let (F_{k}) be a filtration on (Ω, F, P). (We will suppose that F_{0} = {∅, Ω}.) If an adapted sequence (d_{k}) is a real-valued martingale difference sequence, Burkholder’s inequality [3] shows that for any 1 < p < ∞, if (v_{k}) is a predictable sequence bounded in absolute value by 1, then there exists a positive constant c_{p}, depending only on p, such that such that for all positive integers n

∥∑_{k=1}^{n} v_{k}d_{k}∥_{p} ≤ c_{p} ∥∑_{k=1}^{n} d_{k}∥_{p} .

Later Burkholder [4] extended this result to subordination martingales: if (d_{k}), (e_{k}) are two martingale difference sequences such that (e_{k}) is subordinate to (d_{k}), that is, for all k ≥ 1,

(1)  |e_{k}| ≤ |d_{k}|

1991 Mathematics Subject Classification. Primary 60G42; Secondary 15A51, 46B70.

Key words and phrases. Martingale inequalities, tangent sequences, decreasing rearrangement, K-functional, doubly stochastic matrices.

Both authors were supported in part by the NSF and the Research Board of the University of Missouri.

then there exists a positive constant c_{p}, depending only on p, such that for all positive integers n

(2)  ∥∑_{k=1}^{n} e_{k}∥_{p} ≤ c_{p} ∥∑_{k=1}^{n} d_{k}∥_{p} .

A different approach to this inequality was proposed by Kwapień and Woycziński [9] (see also [10]). Two adapted sequence (d_{k}) and (e_{k}) are said to be tangent if for each k ≥ 1, we have that the law of d_{k} conditionally on F_{k−1} is the same as the law of e_{k} conditionally on F_{k−1}, that is,

(3)  P (d_{k} > λ|F_{k−1}) = P (e_{k} > λ|F_{k−1})

for all real numbers λ. Answering a conjecture of Kwapień and Woycziński [9], it was proved by Hitczenko [6] and Zinn [15] that for 1 < p < ∞ that there exists a positive constant c_{p}, depending only on p, such that if (d_{k}) and (e_{k}) are martingale difference sequences and (d_{k}), (e_{k}) are tangent, then for all positive integers n we have equation (2).

Given two adapted sequences, (e_{k}) is said to be strongly dominated by (d_{k}) if for each k ≥ 1,

(4)  P (|e_{k}| > λ|F_{k−1}) ≤ P (|d_{k}| > λ|F_{k−1})

for all λ ≥ 0. It is obvious that the case of (1) and the case of (3) are contained in the cases of (4). Thus the following result of Kwapień and Woycziński [9] is a common generalization of these two results: if (d_{k}), (e_{k}) are two martingale difference sequences such that (e_{k}) is strongly dominated by (d_{k}), then there exists a positive constant c_{p}, depending only on p, such that for all positive integers n equation (2) holds.

The purpose of this paper is to use a different approach to provide another common generalization of those two results, an even a further extension to Kwapień and Woycziński’s result. Theorem 1. For each 1 < p < ∞, there exists a positive constant c_{p}, depending only on p, such that the following holds. Let (d_{k}), (e_{k}) be real-valued martingale difference sequences. If for for all bounded nonnegative predictable sequence (s_{k}) and all positive integers k we have

(5)  E[s_{k} ∨ |e_{k}|] ≤ E[s_{k} ∨ |d_{k}|]

then for all positive integers n we have equation (2). Remark (a). We have that (5) is equivalent to

(6)  E[(λ ∨ |e_{k}|)|F_{k−1}] ≤ E[(λ ∨ |d_{k}|)|F_{k−1}]

for all λ ≥ 0. This is because for any A_{k} ∈ F_{k−1} and a ≥ 0 we have that (aχ_{A_{k}^{c}} ∨ λ) is predictable, and hence

E[(aχ_{A_{k}^{c}} ∨ λ) ∨ |e_{k}| − aχ_{A_{k}^{c}}] ≤ E[(aχ_{A_{k}^{c}} ∨ λ) ∨ |d_{k}| − aχ_{A_{k}^{c}}]

When a intends to infinity, we obtain

E[(λ ∨ |e_{k}|)χ_{A_{k}}] ≤ E[(λ ∨ |d_{k}|)χ_{A_{k}}]

which is equivalent to (6). Remark (b). To see that Theorem 1 is really an extension to Kwapień and Woycziński’s result, we just simply observe that (4) is equivalent to

P ({|e_{k}| > λ} ∩ A_{k}) ≤ P ({|d_{k}| > λ} ∩ A_{k}),

and (6) is equivalent to

∫_{λ}^{∞} P ({|e_{k}| > t} ∩ A_{k})dt ≤ ∫_{λ}^{∞} P ({|d_{k}| > t} ∩ A_{k})dt

for all A_{k} ∈ F_{k−1}. Remark (c). Once we have Theorem 1, we can obtain that for κ ≥ 1, if

P (|e_{k}| > λ|F_{k−1}) ≤ κP (|d_{k}| > λ|F_{k−1}),

we have

(7)  ∥∑_{k=1}^{n} e_{k}∥_{p} ≤ κc_{p} ∥∑_{k=1}^{n} d_{k}∥_{p} .

This is because

∫_{λ}^{∞} P ({|e_{k}| > t} ∩ A_{k})dt ≤ κ ∫_{λ}^{∞} P ({|d_{k}| > t} ∩ A_{k})dt
= κ ∫ ∞_{\frac{λ}{κ}} P (\{|d_{k}| > \frac{t}{κ}\} ∩ A_{k}) d (\frac{t}{κ})

≤ ∫_{λ}^{∞} P ({κ|d_{k}| > t} ∩ A_{k})dt

Hence

E[(λ ∨ |e_{k}|)|F_{k−1}] ≤ E[(λ ∨ κ|d_{k}|)|F_{k−1}]

and equation (7) follows.

Let us give an application of Theorem 1. In fact this application is essentially equivalent to Theorem 1, and indeed will play a large role in its proof. We will consider the probability space [0, 1]^{ℕ} equipped with the product Lebesgue measure L, and consider the filtration (L_{k}), where L_{k} is the minimal σ-field for which the first k coordinate functions of [0, 1]^{ℕ} are measurable. Then two sequences (d_{k}) and (e_{k}) are tangent if

e_{k}(x_{1}, …, x_{k}) = d_{k}(x_{1}, …, x_{k−1}, ϕ_{k}(x_{1}, …, x_{k}))

where (ϕ_{k} : [0, 1]^{k} → [0, 1]) is a sequence of measurable functions such that ϕ_{k}(x_{1}, …, x_{k−1}, ·) is a measure preserving map for almost all x_{1},…, x_{k−1}.

We will consider a more general situation. Suppose we have a sequence of linear operators (T_{k}(x_{1}, …, x_{k−1})), depending measurably upon (x_{k}) ∈ [0, 1]^{ℕ}, that are bounded operators on both L_{1}([0, 1]) and L_{∞}([0, 1]) with norm 1. Then consider the condition

(8) e_{k}(x_{1}, …, x_{k−1}, ·) = [T_{k}(x_{1}, …, x_{k−1})]d_{k}(x_{1}, …, x_{k−1}, ·).

Theorem 2. For each 1 < p < ∞, there exists a positive constant c_{p}, depending only on p, such that the following holds. If (d_{k}), (e_{k}) and (T_{k}) are as above satisfying (8), then for all positive integers n we have equation (2).

We will also need the following intermediate result. For any random variable f, let f^{#} be the decreasing rearrangement of |f|, that is,

f^{#}(t) = sup{s ∈ ℝ : P (|f| < s) < t}.

Theorem 3. For each 1 < p < ∞, there exists a positive constant c_{p}, depending only on p, such that the following holds. Let (d_{k}), (e_{k}) be martingale difference sequences on [0, 1]^{ℕ} with respect to (L_{k}). Suppose that for each positive integer k

∫_{0}^{t}(e_{k}(x_{1}, …, x_{k−1}, ·))^{#}(s)ds ≤ ∫_{0}^{t}(d_{k}(x_{1}, …, x_{k−1}, ·))^{#}(s)ds

for all t ∈ [0, 1] and almost all x_{1},…,x_{k−1}. Then for all positive integers n we have equation (2).

2. The Discrete Type Case

In this section we will prove Theorems 2 and 3 in a special discrete situation, which we now describe. For any positive integer N, let Σ_{N} be the σ-field generated by the partition \{[\frac{i−1}{N}, \frac{i}{N}) : i = 1, 2, …, N\}. Define a filtration (F_{k}) on [0, 1]^{ℕ} by F_{k} = L_{k−1} ⊗ Σ_{N}. Suppose (d_{k}), (e_{k}) are (F_{k})-adapted. Then for each k and for each x_{1},…, x_{k−1}, we see that d_{k}(x_{1}, …, x_{k−1}, ·) and e_{k}(x_{1}, …, x_{k−1}, ·) are Σ_{N}-measurable simple functions on [0, 1). Therefore d_{k} and e_{k} can be written as Ndimensional vectors and T_{k}(x_{1}, …, x_{k−1}) can be represented by a N ×N matrix, that is,

[ e_{k}(1) ; e_{k}(2) ; ⋮ ; e_{k}(N) ] = [ \begin{matrix}a_{k}(1, 1), & … & , a_{k}(1, N) \\ a_{k}(2, 1), & … & , a_{k}(2, N) \\ ⋮ & & ⋮ \\ a_{k}(N, 1), & … & , a_{k}(N, N) \end{matrix} ] [ d_{k}(1) ; d_{k}(2) ; ⋮ ; d_{k}(N) ]

where

d_{k}(i) = d_{k}(x_{1}, …, x_{k−1}, i) = d_{k}(x_{1}, …, x_{k}) if x_{k} ∈ [\frac{i−1}{N}, \frac{i}{N})
e_{k}(i) = e_{k}(x_{1}, …, x_{k−1}, i) = e_{k}(x_{1}, …, x_{k}) if x_{k} ∈ [\frac{i−1}{N}, \frac{i}{N})

T_{k} = T_{k}(x_{1}, …, x_{k−1}) = [(a_{k}(x_{1}, …, x_{k−1}))(i, j)]_{N×N} = [a_{k}(i, j)]_{N×N}

The condition of being martingale difference sequences implies that

∑_{i=1}^{N} d_{k}(i) = ∑_{i=1}^{N} e_{k}(i) = 0

Proposition 4. Theorem 2 holds in the case that (d_{k}) and (e_{k}) are adapted to the filtration (F_{k}) described above.

In this discrete case, the boundedness of ∥T_{k}∥_{L_{1}([0,1])} and ∥T_{k}∥_{L_{∞}([0,1])} by 1 is equivalent to the condition that ∑_{j=1}^{N} |a_{k}(i, j)| ≤ 1 for all i and ∑_{i=1}^{N} |a_{k}(i, j)| ≤ 1 for all j. We claim that without loss of generality, we can assume that every row sum and column sum of T_{k} is 0, that is,

∑_{j=1}^{N} a_{k}(i, j) = ∑_{i=1}^{N} a_{k}(i, j) = 0

for all i and j. Suppose the i^{th} row sum ∑_{j=1}^{N} a_{k}(i, j) = R_{k}(i). Let T′_{k} be the liner operator defined by

T′_{k} = [a_{k}(i, j) − \frac{R_{k}(i)}{N}]_{N×N}

It is clear that every row sum of T′_{k} is 0 and

(T′_{k}d_{k})(i) = ∑_{j=1}^{N} (a_{k}(i, j) − \frac{R_{k}(i)}{N}) d_{k}(j)

= ∑_{j=1}^{N} a_{k}(i, j)d_{k}(j) − \frac{R_{k}(i)}{N} ∑_{j=1}^{N} d_{k}(j)
= e_{k}(i)

Now we can assume that every row sum of T_{k} is 0. Similarly suppose the j^{th} column sum ∑_{i=1}^{N} a_{k}(i, j) = C_{k}(j). Let T′′_{k} be the linear operator defined by

T′′_{k} = [a_{k}(i, j) − \frac{C_{k}(j)}{N}]_{N×N}

Again it is clear that every row sum and column sum of T′′_{k} is 0 and

(T′′_{k}d_{k})(i) = ∑_{j=1}^{N} (a_{k}(i, j) − \frac{C_{k}(j)}{N}) d_{k}(j)

= ∑_{j=1}^{N} a_{k}(i, j)d_{k}(j) − \frac{1}{N} ∑_{j=1}^{N} C_{k}(j)d_{k}(j)
= e_{k}(i)

since

∑_{i=1}^{N} e_{k}(i) = ∑_{j=1}^{N} C_{k}(j)d_{k}(j) = 0

After adjusting T_{k}, it is easy to check that the norms of T_{k} may be enlarged up to 4. Of course, we can pick up T_{k}/4 instead and absorb the 4 into the constant c_{p}.

A nonnegative real matrix is said to be doubly stochastic if each of its row and column sum is 1. A sub-doubly stochastic matrix means that each of its row and column sum is less than or equal to 1. Therefore we can change the assumption in Proposition 4 to be that: “for almost all x_{1},…, x_{k−1}, every row sum and column sum of the matrix from T_{k} is 0, and the matrix from |T_{k}| is sub-doubly stochastic for each positive integer k”

One of the fundamental results in the theory of doubly stochastic matrices was introduced by Birkhoff [1] (or see for example [12]). Theorem A. If M is a doubly stochastic matrix, then

M = ∑_{i=1}^{S} θ_{i}P_{i}

where P_{i} are permutation matrices, and the θ_{i} are nonnegative numbers satisfying ∑_{i=1}^{S} θ_{i} = 1.

Lemma 5. If M is a n × n sub-doubly stochastic matrix, then there exists a 2n × 2n doubly stochastic matrix such that its upper left n × n sub-matrix is M.

Proof. Suppose that R(i) is the i^{th} row sum of M, C(j) is the j^{th} column sum and S is the sum of all entries. Let

A = [ \begin{matrix}\frac{1−R(1)}{n}, & … & , \frac{1−R(1)}{n} \\ ⋮ & & ⋮ \\ \frac{1−R(n)}{n}, & … & , \frac{1−R(n)}{n} \end{matrix} ]_{n×n}

B = [ \begin{matrix}\frac{1−C(1)}{n}, & … & , \frac{1−C(n)}{n} \\ ⋮ & & ⋮ \\ \frac{1−C(1)}{n}, & … & , \frac{1−C(n)}{n} \end{matrix} ]_{n×n}

C = Diag [ \frac{S}{n}, …, \frac{S}{n} ]_{n×n}

Then define

M′ = [ \begin{matrix}M & A \\ B & C \end{matrix} ]_{2n×2n}

It is easy to check that M′ is a doubly stochastic matrix. □

Lemma 6. If M is a sub-doubly stochastic matrix, then there exists a sub-doubly stochastic matrix N such that M + N is doubly stochastic.

Proof. Let M′ be the 2n × 2n doubly stochastic matrix such that its upper left n × n sub-matrix is M. By Theorem A,

M′ = ∑_{i=1}^{S} θ_{i}P′_{i}

where P′_{i} are 2n × 2n permutation matrices and ∑_{i=1}^{S} θ_{i} = 1. Suppose that P_{i} is the upper left n × n sub-permutation matrix of P′_{i}, then

M = ∑_{i=1}^{S} θ_{i}P_{i}

Let Q_{i} be a n × n sub-permutation matrix such that P_{i} + Q_{i} is a permutation matrix, say R_{i}. Define

N = ∑_{i=1}^{S} θ_{i}Q_{i}

thus

M + N = ∑_{i=1}^{S} θ_{i}R_{i}

which is a doubly stochastic matrix. □

Lemma 7. Let M be an n × n matrix. If every row sum and column sum of M is 0 and |M| is sub-doubly stochastic, then

M = ∑_{i=1}^{S} θ_{i}P_{i}

where P_{i} are permutation matrices, ∑_{i=1}^{S} θ_{i} = 0 and ∑_{i=1}^{S} |θ_{i}| = 1

Proof. Let

A = \frac{|M| + M}{2}

B = \frac{|M| − M}{2}

so A and B are nonnegative, and 2A and 2B are sub-doubly stochastic. By Lemma 6, there exists a sub-doubly stochastic matrix C such that 2(A + C) is a doubly stochastic. But A and B have the same row sums and column sums, and hence 2(B + C) is also a doubly stochastic. By applying Theorem A, we have

2(A + C) = ∑_{i=1}^{m} λ_{i}Q_{i}

2(B + C) = ∑_{i=1}^{m′} λ′_{i}Q′_{i}

where Q_{i}, Q′_{i} are permutation matrices, and the λ_{i}, λ′_{i} are nonnegative numbers satisfying ∑_{i=1}^{m} λ_{i} = ∑_{i=1}^{m′} λ′_{i} = 1. Then the result follows because

M = (A + C) − (B + C) = ∑_{i=1}^{m} \frac{λ_{i}}{2} Q_{i} − ∑_{i=1}^{m′} \frac{λ′_{i}}{2} Q′_{i}

□

Proof of Proposition 4. From Lemma 7, we know that for each k ≥ 1 and almost all x_{1}, …, x_{k−1}

T_{k}(x_{1}, …, x_{k−1}) = ∑_{i_{k}=1}^{S_{k}} θ_{k,i_{k}}(x_{1}, …, x_{k−1}) · P_{k,i_{k}}(x_{1}, …, x_{k−1})

where P_{k,i_{k}} are permutation matrices, ∑_{i=1}^{S_{k}} θ_{k,i_{k}} = 0, and ∑_{i=1}^{S_{k}} |θ_{k,i_{k}}| = 1. Let

(9) h_{k,i_{k}}(x_{1}, …, x_{k−1}, ·) = [P_{k,i_{k}}(x_{1}, …, x_{k−1})]d_{k}(x_{1}, …, x_{k−1}, ·).

Then

e_{k} = [∑_{i_{k}=1}^{S_{k}} θ_{k,i_{k}}P_{k,i_{k}}] d_{k}

= ∑_{i_{k}=1}^{S_{k}} |θ_{k,i_{k}}|ε_{k,i_{k}}h_{k,i_{k}}

where ε_{k,i_{k}} = sgn(θ_{k,i_{k}}).

Now we need to consider the probability space Ω_{1} × Ω_{2}, where Ω_{1} = Ω_{2} = [0, 1]^{ℕ}. We consider all of the previous random variables as random variables on this new probability space, depending only upon the first coordinate ω_{1}. We define a filtration (G_{k}) where G_{k} = F_{k} ⊗ L_{k+1}.

We define a predictable sequence of random variables (I_{k}) so that for each ω_{1} ∈ Ω_{1}, the random variable I_{k}(ω_{1}, ·) takes the value i with probability |θ_{k,i}(ω_{1})|. Then we see that

e_{k} = E[ε_{k,I_{k}}h_{k,I_{k}}|L ⊗ {∅, Ω_{2}}].

Hence, since conditional expectation is a contraction on L_{p}

∥∑_{k=1}^{n} e_{k}∥_{p} ≤ ∥∑_{k=1}^{n} ε_{k,I_{k}}h_{k,I_{k}} ∥_{p} .

Now we see that (ε_{k,I_{k}}) is a predictable sequence bounded by 1. Hence by Burkholder’s inequality, we see that

∥∑_{k=1}^{n} ε_{k,I_{k}}h_{k,I_{k}} ∥_{p} ≤ c_{p} ∥∑_{k=1}^{n} h_{k,I_{k}} ∥_{p} .

Next, observing (9), since P_{k,i_{k}} are permutation matrices, for each k ≥ 1, i_{k} = 1, 2, …, S_{k}, h_{k,i_{k}} is just an x_{k}-rearrangement of d_{k}. that is

h_{k,i_{k}}(x_{1}, …, x_{k−1}, j) = d_{k}(x_{1}, …, x_{k−1}, π_{k,i_{k}}(j))

for some permutation π_{k,i_{k}}. Thus for any sequence (i_{k}) we have that (h_{k,i_{k}}) and (d_{k}) are tangent sequences. But then we see that (h_{k,I_{k}}) and (d_{k}) are tangent sequences. Hence there exists a positive constant c_{p} such that

∥∑_{k=1}^{n} h_{k,I_{k}} ∥_{p} ≤ c_{p} ∥∑_{k=1}^{n} d_{k}∥_{p} .

The result follows. □

Proposition 8. Theorem 3 holds in the case that (d_{k}) and (e_{k}) are adapted to the filtration (F_{k}) described above.

This will follow immediately from the following well-known result [11]. Theorem B. f = (f_{1}, f_{2}, …, f_{N}), g = (g_{1}, g_{2}, …, g_{N}) are N-dimensional real-valued vectors. f^{#} = (f_{1}^{#}, f_{2}^{#}, …, f_{N}^{#}) is the decreasing rearrangement of |f| = (|f_{1}|, |f_{2}|, …, |f_{N}|). Then

∑_{k=1}^{n} g_{k}^{#} ≤ ∑_{k=1}^{n} f_{k}^{#}

for all n = 1, 2, …, N if and only if there exists a matrix T = [a_{ij}]_{N×N} such that T f = g, ∑_{i=1}^{N} |a_{ij}| ≤ 1 and ∑_{j=1}^{N} |a_{ij}| ≤ 1.

3. The General Case

The following theorem was proved by Crowe, Zweibel and Rosenbloom [5]. Theorem C. Suppose f, g are random variables on [0, 1], then for 1 ≤ p ≤ ∞,

∥f^{#} − g^{#}∥_{p} ≤ ∥f − g∥_{p}

Lemma 9. 1 ≤ p < ∞. (Ω, F, P) is a probability space. Let d(ω, x),e(ω, x) ∈ L_{p}(Ω×[0, 1]) be two random variables such that for almost every ω ∈ Ω that

∫_{0}^{t}(e(ω, ·))^{#} ≤ ∫_{0}^{t}(d(ω, ·))^{#}

and

∫_{0}^{1} d(ω, ·) = ∫_{0}^{1} d(ω, ·) = 0.

Then given ϵ > 0, there exists a positive integer N and d′, e′ ∈ L_{p}(Ω × [0, 1]) that are measurable with respect to F⊗Σ_{N} such that ∥d−d′∥_{p}, ∥e− e′∥ ≤ ϵ,

∫_{0}^{t}(e′(ω, ·))^{#} ≤ ∫_{0}^{t}(d′(ω, ·))^{#}

and

∫_{0}^{1} d′(ω, ·) = ∫_{0}^{1} d′(ω, ·) = 0.

Proof. For every ϵ > 0, pick 0 < γ < min{ϵ/[7(∥d∥_{p} ∨∥e∥_{p})], 1/3} . Fix ω ∈ Ω, and regard the functions as functions of only one variable x on [0, 1]. Hence there exist simple functions

d̄= ∑_{i=1}^{S} ᾱ_{i}χ_{A_{i}}

ē = ∑_{i=1}^{S} β̄_{i}χ_{B_{i}}

such that

∥d̄− d∥_{L_{p}([0,1])} ≤ γ∥d∥_{L_{1}([0,1])}.
∥ē − e∥_{L_{p}([0,1])} ≤ γ∥e∥_{L_{1}([0,1])}.

We may suppose without loss of generality that the sets A_{i} and B_{i} are the sets of the form [r_{1}, s_{1}), where the r_{i} and s_{i} are rational numbers. Furthermore, we will suppose that A_{i_{1}} ∩ A_{i_{2}} = ∅ and B_{i_{1}} ∩ B_{i_{2}} = ∅ for i_{1} ≠ i_{2}.

Let N_{0} = N_{0}(ω) be the least common denominator of all these rational numbers. For each ω, since d(ω, ·)^{#}, e(ω, ·)^{#} are Reimann integrable as a function of x, there is a number N_{1} = N_{1}(ω) that is a multiple of N_{0} and such that for all n ≥ N_{1} that

∥E[d(ω, ·)^{#}|Σ_{n}] − d(ω, ·)^{#}∥_{L_{p}([0,1])} ≤ γ∥d∥_{L_{1}([0,1])}.
∥E[e(ω, ·)^{#}|Σ_{n}] − e(ω, ·)^{#}∥_{L_{p}([0,1])} ≤ γ∥e∥_{L_{1}([0,1])}.

Now let d_{n} = dχ_{{N_{1}(ω)≤n}} and e_{n} = eχ_{{N_{1}(ω)≤n}}. Then d_{n} → d and e_{n} → e in L_{p}(Ω × [0, 1]). So pick N such that

∥d_{N} − d∥_{L_{p}(Ω×[0,1])} < ϵ/7,
∥e_{N} − e∥_{L_{p}(Ω×[0,1])} < ϵ/7.

For each fixed ω ∈ \{N_{1}(ω) ≤ N\}, [\frac{i−1}{N}, \frac{i}{N}) is either contained in some A_{j} or disjoint to all A_{j}. Let α_{i} = ᾱ_{j} if [\frac{i−1}{N}, \frac{i}{N}) ⊂ A_{j} for some j, and α_{i} = 0 otherwise. Let χ_{i} = χ_{[ \frac{i−1}{N} , \frac{i}{N})}. Thus

(10)  ∥∑_{i=1}^{N} α_{i}χ_{i} − d_{N} ∥_{L_{p}([0,1])} ≤ γ∥d∥_{L_{1}([0,1])}

and

(∑_{i=1}^{N} α_{i}χ_{i})^{#} = ∑_{i=1}^{N} ε_{σ(i)}α_{σ(i)}χ_{i}

for some permutation σ, where ε_{j} = sgn(α_{j}). By Theorem C,

(11)  ∥∑_{i=1}^{N} ε_{σ(i)}α_{σ(i)}χ_{i} − d_{N}^{#}∥_{L_{p}([0,1])} ≤ γ∥d∥_{L_{1}([0,1])}

and also the analogous statement holds for e.

Now if we set

E[d_{N}^{#}|Σ_{N}] = ∑_{i=1}^{N} α′′_{i}χ_{i}

then

(12)  ∥∑_{i=1}^{N} α′′_{i}χ_{i} − d_{N}^{#}∥_{L_{p}([0,1])} ≤ γ∥d∥_{L_{1}([0,1])}

Note that in this case that

∫_{0}^{t} ∑_{i=1}^{N} α′′_{i}χ_{i} = ∫_{0}^{t} d_{N}^{#}

if t = \frac{j}{N}, j = 0, 1, 2, …, N. Then by (11) and (12),

∥∑_{i=1}^{N} α′′_{i}χ_{i} − ∑_{i=1}^{N} ε_{σ(i)}α_{σ(i)}χ_{i}∥_{L_{p}([0,1])} ≤ 2γ∥d∥_{L_{1}([0,1])}

By doing the reverse process of taking decreasing rearrangement of | ∑_{i=1}^{N} α_{i}χ_{i}|, and setting

α̂_{i} = ε_{σ^{−1}(i)}α′′_{σ^{−1}(i)}

we have

(13)  ∥∑_{i=1}^{N} α̂_{i}χ_{i} − ∑_{i=1}^{N} α_{i}χ_{i}∥_{L_{p}([0,1])} ≤ 2γ∥d∥_{L_{1}([0,1])}

From (10) and (13),

∥∑_{i=1}^{N} α̂_{i}χ_{i} − d_{N} ∥_{L_{p}([0,1])} ≤ 3γ∥d∥_{L_{1}([0,1])}

For t = \frac{j}{N}, j = 0, 1, 2, …, N, it is clear that

∫_{0}^{t} (∑_{i=1}^{N} α̂_{i}χ_{i})^{#} = ∫_{0}^{t} ∑_{i=1}^{N} α′′_{i}χ_{i} = ∫_{0}^{t} d_{N}^{#}.

Furthermore, if we set

ζ = E [∑_{i=1}^{N} α̂_{i}χ_{i}]

then

|ζ| ≤ 3γ∥d∥_{L_{1}([0,1])}.

We can also perform this same construction for e, the analogues of α̂_{i} and ζ being β̂_{i} and η. Thus we see that for t = \frac{j}{N}, j = 0, 1, 2, …, N

EXTENSION TO A MARTINGALE INEQUALITY 13

that

∫_{0}^{t} (∑_{i=1}^{N} (α̂_{i} − ζ) χ_{i})^{#} (14)

≤ ∫_{0}^{t} (∑_{i=1}^{N} (|α̂_{i}| + |ζ|) χ_{i})^{#}
≤ ∫_{0}^{t} (∑_{i=1}^{N} α̂_{i}χ_{i})^{#} + 3γ∥d∥_{L_{1}([0,1])} · t
= ∫_{0}^{t} d_{N}^{#} + 3γ∥d∥_{L_{1}([0,1])} · t
≤ (1 + 3γ) ∫_{0}^{t} d_{N}^{#}

and similarly

(15) ∫_{0}^{t} (∑_{i=1}^{N} (β̂_{i} − η) χ_{i})^{#} ≥ ∫_{0}^{t} e_{N}^{#} − 3γ∥e∥_{L_{1}([0,1])} · t
≥ (1 − 3γ) ∫_{0}^{t} e_{N}^{#}

Thus, we are ready to define d′ and e′. Let

d′ = (1 + 3γ) ∑_{i=1}^{N}(α̂_{i} − ζ)χ_{i}

e′ = (1 − 3γ) ∑_{i=1}^{N}(β̂_{i} − η)χ_{i}

It is clear that E[d′] = E[e′] = 0. Combining (14) and (15), we have for t = \frac{j}{N}, j = 0, 1, 2, …, N

∫_{0}^{t}(e′)^{#} ≤ ∫_{0}^{t} e_{N}^{#} = ∫_{0}^{t} d_{N}^{#} ≤ ∫_{0}^{t}(d′)^{#}.

But then by linear interpolation, this follows for all t ∈ [0, 1]. Now an easy argument shows that

∥d′ − d∥_{L_{p}(Ω×[0,1])} ≤ 6γ∥d∥_{L_{1}(Ω×[0,1])} + ϵ/7

∥e′ − e∥_{L_{p}(Ω×[0,1])} ≤ 6γ∥e∥_{L_{1}(Ω×[0,1])} + ϵ/7

and we are done. □

Proof of Theorem 3. For each 1 ≤ k ≤ n, apply Lemma 9, there exists an integer N_{k} and functions d′_{k}, e′_{k} satisfying ∥d′_{k} − d_{k}∥_{p}, ∥e′_{k} − e_{k}∥_{p} ≤ ϵ such that (d′_{k}) and (e′_{k}) are adapted to (L_{k−1} ⊗ Σ_{N}), where N is the least common multiple of N_{k}, keep the martingale property, and

∫_{0}^{t}(e′_{k}(x_{1}, …, x_{k−1}, ·))^{#}(s)ds ≤ ∫_{0}^{t}(d′_{k}(x_{1}, …, x_{k−1}, ·))^{#}(s)ds

for all t ∈ [0, 1]. By Proposition 8, there exist a positive constant c_{p} such that

∥∑_{k=1}^{n} e′_{k}∥_{p} ≤ c_{p} ∥∑_{k=1}^{n} d′_{k}∥_{p}

∥d_{k} − d′_{k}∥_{p} → 0 and ∥e_{k} − e′_{k}∥_{p} → 0 as ϵ → 0. The result follows. □

Proof of Theorem 2. If f is a random variable on (Ω, F, P), 1 ≤ p < ∞, 0 ≤ t ≤ 1, we define the K-functional by

K(t, f; L_{p}, L_{∞}) = inf_{f_{0}+f_{1}=f}{∥f_{0}∥_{p} + t∥f_{1}∥_{∞}}.

J. Peetre [14] has shown that

K(t, f; L_{1}, L_{∞}) = ∫_{0}^{t} f^{#}(s)ds.

Hence it follows that if T is an operator on both L_{1}([0, 1]) and L_{∞}([0, 1]) with norm bounded by 1, then for t ≥ 0

∫_{0}^{t}(T f)^{#}(s)ds ≤ ∫_{0}^{t} f^{#}(s)ds.

Thus the result follows from Theorem 3. □

Lemma 10. Let f and g be real-valued random variables on (Ω, F, P). Then

(16)  E [λ ∨ |g|] ≤ E [λ ∨ |f|]

for all nonnegative number λ if and only if

∫_{0}^{t} g^{#}(s)ds ≤ ∫_{0}^{t} f^{#}(s)ds

for all t ∈ [0, 1].

Proof. Equation (16) is equivalent to E [λ ∨ g^{#}] ≤ E [λ ∨ f^{#}]. For the “if” part, let

α = sup {t : f^{#}(t) ≥ λ}
β = sup {t : g^{#}(t) ≥ λ} .

Then

E [λ ∨ f^{#}] = ∫_{0}^{α} f^{#} + (1 − α)λ
= ∫_{0}^{β} f^{#} + (1 − β)λ + ∫_{β}^{α}(f^{#} − λ)

≥ ∫_{0}^{β} g^{#} + (1 − β)λ + ∫_{β}^{α}(f^{#} − λ)
= E [λ ∨ g^{#}] + ∫_{β}^{α}(f^{#} − λ).

If α ≤ β, then for all x ∈ (α, β) we have f^{#}(x) ≤ λ, and if β ≤ α, then for all x ∈ (β, α) we have f^{#}(x) ≥ λ. Either way, we see that ∫_{β}^{α}(f^{#} − λ) ≥ 0, and the result follows.

To show the “only if”, for any α ∈ [0, 1], let

λ = f^{#}(α)
β = inf {t : g^{#}(t) ≥ λ} .

Then

∫_{0}^{α} g^{#} = ∫_{0}^{β} g^{#} + ∫_{β}^{α}(g^{#} − λ) + λ(1 − β) + λ(α − 1)
= E [λ ∨ g^{#}] + λ(α − 1) + ∫_{β}^{α}(g^{#} − λ)
≤ E [λ ∨ f^{#}] + λ(α − 1) + ∫_{β}^{α}(g^{#} − λ)
= ∫_{0}^{α} f^{#} + ∫_{β}^{α}(g^{#} − λ).

Arguing as above, we see that ∫_{β}^{α}(g^{#} − λ) ≤ 0, and again the result follows. □

Given a random variable f and a sigma field G, we will say that f is nowhere constant with respect to G if P (f = g) = 0 for every G measurable function g. The following theorem [13] shows a concrete representation of a sequence of random variables. Theorem D. Let (f_{n}) be a sequence of random variables takeing values in a separable sigma filed (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 ≥ 0.

Proof of Theorem 1. We will prove this theorem under the assumption (6). Consider the map D_{k} = (d_{k}, e_{k}, f_{k}) : Ω × [0, 1]^{ℕ} → ℝ^{3} by (ω, (x_{k})) ↦ (d_{k}(ω), e_{k}(ω), x_{k}). It is clear that D_{k} is nowhere constant with respect to σ(D_{1}, …, D_{k−1}). Apply the previous theorem to get D̃_{k} = (d̃_{k}, ẽ_{k}, f̃_{k}) : [0, 1]^{k} → ℝ^{3} such that (D̃_{k}) has the same law as (D_{k}) and σ(D̃_{1}, …, D̃_{k}) = L_{k}.

Next, we show that for almost every x_{1}, …, x_{k−1} and λ ≥ 0 that

∫_{0}^{1} λ ∨ |ẽ_{k}(x_{1}, …, x_{k})| dx_{k} ≤ ∫_{0}^{1} λ ∨ |d̃_{k}(x_{1}, …, x_{k})| dx_{k}

which will follow from showing that for any bounded non-negative measurable function ϕ_{k} : [0, 1]^{k−1} → [0, ∞) that

E[ϕ_{k} ∨ |ẽ_{k}|] ≤ E[ϕ_{k} ∨ |d̃_{k}|].

But then there exists a bounded Borel measurable function θ_{k} : ℝ^{3(k−1)} → [0, ∞) such that ϕ = θ(D̃_{1}, …, D̃_{k−1}) almost everywhere in [0, 1]^{k−1}. Thus

∫_{[0,1]^{k}} ϕ_{k} ∨ |ẽ_{k}| = ∫_{[0,1]^{k}} θ(D̃_{1}, …, D̃_{k−1}) ∨ |ẽ_{k}|
= E[θ(D_{1}, …, D_{k−1}) ∨ |e_{k}|]
≤ E[θ(D_{1}, …, D_{k−1}) ∨ |d_{k}|]
= ∫_{[0,1]^{k}} θ(D̃_{1}, …, D̃_{k−1}) ∨ |d̃_{k}|
= ∫_{[0,1]^{k}} ϕ_{k} ∨ |d̃_{k}|

Also to show that E[d̃_{k}|L_{k−1}] = E[ẽ_{k}|L_{k−1}] = 0, it is sufficient to show that for any bounded measurable function ϕ_{k} : [0, 1]^{k−1} → ℝ that E[ϕ_{k}d̃_{k}] = E[ϕ_{k}ẽ_{k}] = 0. Thus follows by a very similar argument to that above.

The result then follows from Lemma 10 and Theorem 3. □

Acknowledgments. We would like to mention the help of Jim Reeds and David Boyd in obtaining the argument about doubly stochastic matrices and other useful remarks. Also, we would like to express our thanks to Mitch Taibleson for bringing this problem to our attention.

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Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA.

E-mail address: stephen@math.missouri.edu
URL: http://www.math.missouri.edu/~stephen

Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA.

E-mail address: mathgr75@math.missouri.edu