\documentclass[12pt]{amsart} \newtheorem{thm}{Theorem} \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{pro}[thm]{Proposition} \newtheorem{thmofothers}{Theorem} \renewcommand{\thethmofothers}{\Alph{thmofothers}} \newtheorem{rem}{Remark} \renewcommand{\therem}{(\alph{rem})} \begin{document} \title[Extension to a Martingale Inequality]{An Extension to the Strong Domination Martingale Inequality} \author{Stephen Montgomery-Smith} \address{Department of Mathematics\\ University of Missouri\\ Columbia, Missouri 65211, USA.} \email{stephen@math.missouri.edu} \urladdr{http://www.math.missouri.edu/\~{}stephen} \author{ Shih-Chi Shen } \address{Department of Mathematics\\ University of Missouri\\ Columbia, Missouri 65211, USA.} \email{mathgr75@math.missouri.edu} \subjclass{Primary 60G42; Secondary 15A51, 46B70} \keywords{Martingale inequalities, tangent sequences, decreasing rearrangement, $K$-functional, doubly stochastic matrices} \thanks{Both authors were supported in part by the NSF and the Research Board of the University of Missouri} \begin{abstract} For each $1 \lambda | \mathcal{F}_{k-1})= P(e_k > \lambda | \mathcal{F}_{k-1}) \end{equation} for all real numbers$\lambda $. Answering a conjecture of Kwapie\'n and Woyczi\'nski \cite{KW1}, it was proved by Hitczenko \cite{H1} and Zinn \cite{Z} that for$1 \lambda | \mathcal{F}_{k-1})\leq P(|d_k| > \lambda | \mathcal{F}_{k-1}) \end{equation} for all $\lambda \geq 0$. It is obvious that the case of (\ref{subordination}) and the case of (\ref{tangent}) are contained in the cases of (\ref{strong-d}). Thus the following result of Kwapie\'n and Woyczi\'nski \cite{KW1} 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 (\ref{ek<=dk}) 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 Kwapie\'n and Woyczi\'nski's result. \begin{thm} \label{t ek<=dk sk} For each $1\lambda \} \cap A_k) \leq P(\{|d_k|>\lambda \} \cap A_k),$$and (\ref{skekFk-1<=skdkFk-1}) is equivalent to$$\int_\lambda^\infty P(\{|e_k|>t \} \cap A_k)dt \leq \int_\lambda^\infty P(\{|d_k|>t \} \cap A_k)dt$$for all A_k \in \mathcal{F}_{k-1}. }\end{rem} \begin{rem} {\upshape Once we have Theorem~\ref{t ek<=dk sk}, we can obtain that for \kappa \geq 1, if$$P(|e_k| > \lambda | \mathcal{F}_{k-1})\leq \kappa P(|d_k| > \lambda | \mathcal{F}_{k-1}),$$we have \begin{equation} \label{kappa} \left\|\sum_{k=1}^n e_k \right\|_p \leq \kappa c_p \left\| \sum_{k=1}^n d_k \right\|_p. \end{equation} This is because \begin{eqnarray*} \int_\lambda^\infty P(\{|e_k|>t \} \cap A_k)dt &\leq& \kappa \int_\lambda^\infty P(\{|d_k|>t \} \cap A_k)dt\\ &=& \kappa \int_{\frac{\lambda}{\kappa}}^\infty P\left( \left\{|d_k|> \frac{t}{\kappa } \right\} \cap A_k \right)d \left(\frac{t}{\kappa}\right)\\ &\leq& \int_\lambda^\infty P(\{\kappa |d_k|>t \} \cap A_k)dt\\ \end{eqnarray*} Hence$$E[(\lambda \vee |e_k|)|\mathcal{F}_{k-1}]\leq E[(\lambda \vee \kappa |d_k|)|\mathcal{F}_{k-1}]$$and equation (\ref{kappa}) follows. }\end{rem} Let us give an application of Theorem~\ref{t ek<=dk sk}. In fact this application is essentially equivalent to Theorem~\ref{t ek<=dk sk}, and indeed will play a large role in its proof. We will consider the probability space [0,1]^\mathbb{N} equipped with the product Lebesgue measure \mathcal L, and consider the filtration (\mathcal{L}_k), where \mathcal L_k is the minimal \sigma-field for which the first k coordinate functions of [0,1]^{\mathbb{N}} are measurable. Then two sequences (d_k) and (e_k) are tangent if $e_k (x_1,\dots,x_k)=d_k(x_1,\dots,x_{k-1},\phi_k(x_1,\dots,x_k))$ where (\phi_k:[0,1]^k\rightarrow [0,1]) is a sequence of measurable functions such that \phi_k(x_1,\dots,x_{k-1},\cdot) is a measure preserving map for almost all x_1,\dots, x_{k-1}. We will consider a more general situation. Suppose we have a sequence of linear operators (T_k(x_1,\dots,x_{k-1})), depending measurably upon (x_k) \in [0,1]^{\mathbb{N}}, that are bounded operators on both L_1([0,1]) and L_\infty ([0,1]) with norm 1. Then consider the condition \begin{equation} \label{ek=Tk(dk)} e_k(x_1,\dots,x_{k-1}, \cdot )=[T_k(x_1,\dots,x_{k-1})]d_k(x_1,\dots,x_{k-1}, \cdot ). \end{equation} \begin{thm} \label{t ek<=dk T} For each 10, there exists a positive integer N and d',e' \in L_p(\Omega\times[0,1]) that are measurable with respect to \mathcal{F}\otimes\Sigma_N such that \|d-d'\|_p, \|e-e'\| \le \epsilon,$$ \int_0^t (e'(\omega,\cdot))^\# \le \int_0^t (d'(\omega,\cdot))^\# $$and$$ \int_0^1 d'(\omega,\cdot) = \int_0^1 d'(\omega,\cdot) = 0. $$\end{lem} \begin{proof} For every \epsilon > 0, pick 0 < \gamma < \min\{\epsilon / [7(\|d\|_p \vee \|e\|_p)],1/3\} . Fix \omega \in \Omega, and regard the functions as functions of only one variable x on [0,1]. Hence there exist simple functions $\bar{d}=\sum_{i=1}^S \bar{\alpha} _i\chi _{A_i}$ $\bar{e}=\sum_{i=1}^S \bar{\beta} _i\chi _{B_i}$ such that$$ \|\bar{d}- d\|_{L_p([0,1])} \le \gamma \| d \|_{L_1([0,1])}. \|\bar{e} - e\|_{L_p([0,1])} \le \gamma \| e \|_{L_1([0,1])}.$$We may suppose without loss of generality that the sets A_i and B_i are the sets of the form [r_1,s_1), where the r_i and s_i are rational numbers. Furthermore, we will suppose that A_{i_1} \cap A_{i_2} = \emptyset and B_{i_1} \cap B_{i_2} = \emptyset for i_1 \ne i_2. Let N_0=N_0(\omega) be the least common denominator of all these rational numbers. For each \omega, since d(\omega,\cdot)^\#, e(\omega,\cdot)^\# are Reimann integrable as a function of x, there is a number N_1=N_1(\omega) that is a multiple of N_0 and such that for all n\ge N_1 that$$ \| E[d(\omega,\cdot)^\#|\Sigma_n] - d(\omega,\cdot)^\#\|_{L_p([0,1])} \le \gamma \| d \|_{L_1([0,1])}. \| E[e(\omega,\cdot)^\#|\Sigma_n] - e(\omega,\cdot)^\#\|_{L_p([0,1])} \le \gamma \| e \|_{L_1([0,1])}.$$Now let d_n = d \chi_{\{N_1(\omega) \le n\}} and e_n = e \chi_{\{N_1(\omega) \le n\}}. Then d_n \to d and e_n \to e in L_p(\Omega\times[0,1]). So pick N such that$$\|d_N - d\|_{L_p(\Omega\times[0,1])} < \epsilon/7,\|e_N - e\|_{L_p(\Omega\times[0,1])} < \epsilon/7.$$For each fixed \omega \in \{N_1(\omega) \le N\}, [\frac{i-1}{N},\frac{i}{N}) is either contained in some A_j or disjoint to all A_j. Let \alpha_i=\bar{\alpha}_j if [\frac{i-1}{N},\frac{i}{N})\subset A_j for some j, and \alpha_i=0 otherwise. Let \chi _i=\chi_{[\frac{i-1}{N},\frac{i}{N})}. Thus \begin{equation} \label{approx-1} \left\|\sum_{i=1}^N \alpha _i\chi _i-d_N \right\|_{L_p([0,1])} \leq \gamma \|d\|_{L_1([0,1])} \end{equation} and $\left( \sum_{i=1}^N \alpha _i\chi _i\right)^\# = \sum_{i=1}^N \varepsilon_{\sigma (i)}\alpha _{\sigma (i)}\chi _i$ for some permutation \sigma , where \varepsilon _j=\mbox{sgn}(\alpha_j). By Theorem~\ref{t CZR}, \begin{equation} \label{approx-3} \left\| \sum_{i=1}^N \varepsilon_{\sigma (i)}\alpha _{\sigma (i)}\chi _i-d^\#_N \right\|_{L_p([0,1])} \le \gamma \|d\|_{L_1([0,1])} \end{equation} and also the analogous statement holds for e. Now if we set$$E[d^\#_N|\Sigma_N]=\sum_{i=1}^N \alpha''_i \chi_i$$then \begin{equation} \label{approx-4} \left\|\sum_{i=1}^N \alpha''_i\chi_i -d^\#_N \right\|_{L_p([0,1])} \le \gamma \|d\|_{L_1([0,1])} \end{equation} Note that in this case that $\int_0^t \sum_{i=1}^N \alpha''_i\chi _i=\int_0^t d^\#_N$ if t=\frac{j}{N}, j=0,1,2,\dots,N. Then by (\ref{approx-3}) and (\ref{approx-4}), $\left\|\sum_{i=1}^N \alpha''_i\chi_i- \sum_{i=1}^N \varepsilon_{\sigma (i)}\alpha _{\sigma (i)}\chi _i \right\|_{L_p([0,1])} \le 2 \gamma \|d\|_{L_1([0,1])}$ By doing the reverse process of taking decreasing rearrangement of |\sum_{i=1}^N \alpha _i\chi _i|, and setting $\hat \alpha_i = \varepsilon_{\sigma^{-1}(i)}\alpha''_{\sigma^{-1}(i)}$ we have \begin{equation} \label{approx-5} \left\|\sum_{i=1}^N \hat \alpha_i \chi_i - \sum_{i=1}^N \alpha _i\chi _i \right\|_{L_p([0,1])} \le 2 \gamma \| d\|_{L_1([0,1])} \end{equation} From (\ref{approx-1}) and (\ref{approx-5}), $\left\|\sum_{i=1}^N \hat \alpha_i \chi_i-d_N \right\|_{L_p([0,1])} \le 3 \gamma \|d\|_{L_1([0,1])}$ For t=\frac{j}{N}, j=0,1,2,\dots,N, it is clear that $\int_0^t \left( \sum_{i=1}^N \hat{\alpha }_i\chi _i \right)^\# = \int_0^t \sum_{i=1}^N \alpha'' _i\chi _i=\int_0^t d^\#_N.$ Furthermore, if we set $\zeta = E\left[\sum_{i=1}^N \hat{\alpha }_i\chi_i\right]$ then$$|\zeta| \le 3 \gamma\|d\|_{L_1([0,1])}.$$We can also perform this same construction for$e$, the analogues of$\hat \alpha_i$and$\zeta$being$\hat\beta_i$and$\eta$. Thus we see that for$t=\frac{j}{N}$,$j=0,1,2,\dots,N$that \begin{eqnarray} \label{int ineq seq} & &\int_0^t \left(\sum_{i=1}^N \left(\hat{\alpha}_i-\zeta \right)\chi _i \right)^\# \\ & \leq &\int_0^t\left(\sum_{i=1}^N\left(|\hat{\alpha}_i|+|\zeta |\right)\chi _i \right)^\#\nonumber\\ & \leq &\int_0^t \left( \sum_{i=1}^N \hat{\alpha}_i\chi _i \right)^\# + 3\gamma \|d\|_{L_1([0,1])} \cdot t\nonumber\\ & =& \int_0^t d^\#_N + 3\gamma \|d\|_{L_1([0,1])} \cdot t\nonumber\\ & \leq & (1+3\gamma)\int_0^t d^\#_N \nonumber \end{eqnarray} and similarly \begin{eqnarray} \label{int approx} \int_0^t \left(\sum_{i=1}^N\left(\hat{\beta}_i-\eta \right)\chi _i \right)^\# &\geq & \int_0^t e^\#_N - 3\gamma\|e\|_{L_1([0,1])} \cdot t\\ &\geq & (1-3\gamma)\int_0^t e^\#_N \nonumber \end{eqnarray} Thus, we are ready to define$d'$and$e'$. Let $d'= (1+3\gamma) \sum_{i=1}^N (\hat{\alpha }_i-\zeta )\chi_i$ $e'= (1-3\gamma) \sum_{i=1}^N (\hat{\beta }_i-\eta )\chi _i$ It is clear that$E[d']=E[e']=0 $. Combining (\ref{int ineq seq}) and (\ref{int approx}), we have for$t=\frac{j}{N}$,$j=0,1,2,\dots,N$$\int_0^t (e')^\# \leq \int_0^t e^\# _N =\int_0^t d^\# _N\leq \int_0^t (d')^\# .$ But then by linear interpolation, this follows for all$t \in [0,1]$. Now an easy argument shows that $\|d'-d\|_{L_p(\Omega \times [0,1])} \leq 6 \gamma\|d\|_{L_1(\Omega \times [0,1])}+\epsilon/7$ $\|e'-e\|_{L_p(\Omega \times [0,1])} \leq 6 \gamma\|e\|_{L_1(\Omega \times [0,1])}+\epsilon/7$ and we are done. \end{proof} \bigskip \begin{proof}[Proof of Theorem \ref{t ek<=dk sharp}] For each$1\leq k \leq n $, apply Lemma \ref{app}, there exists an integer$N_k$and functions$d'_k$,$e'_k$satisfying$\|d'_k-d_k\|_p$,$\|e'_k-e_k\|_p \leq \epsilon$such that$(d'_k)$and$(e'_k)$are adapted to$(\mathcal{L}_{k-1}\otimes \Sigma_N)$, where$N$is the least common multiple of$N_k$, keep the martingale property, and $\int_0^t (e'_k(x_1,\dots,x_{k-1},\cdot))^{\#}(s)ds \leq \int_0^t (d'_k(x_1,\dots,x_{k-1},\cdot))^{\#}(s)ds$ for all$t\in [0,1]$. By Proposition~\ref{t ek<=dk sharp disc}, there exist a positive constant$c_p$such that $\left\|\sum_{k=1}^n e'_k \right\|_p \leq c_p \left\| \sum_{k=1}^n d'_k \right\|_p$$\| d_k - d'_k \|_p \to 0$and$\| e_k - e'_k \|_p \to 0$as$\epsilon \to 0$. The result follows. \end{proof} \begin{proof}[Proof of Theorem~\ref{t ek<=dk T}] If$f$is a random variable on$(\Omega ,\mathcal{F},P)$,$1\leq p<\infty$,$0\leq t\leq 1$, we define the$K$-functional by $K(t,f;L_p,L_\infty)=\inf_{f_0+f_1=f}\{\|f_0\|_p+t\|f_1\|_\infty \}.$ J. Peetre \cite{P} has shown that $K(t,f;L_1,L_\infty) = \int_0^t f^{\#} (s)ds .$ Hence it follows that if$T$is an operator on both$L_1([0,1])$and$L_\infty([0,1])$with norm bounded by$1$, then for$t \ge 0$$\int_0^t (Tf)^{\#}(s) ds \le \int_0^t f^{\#}(s) ds .$ Thus the result follows from Theorem~\ref{t ek<=dk sharp}. \end{proof} \begin{lem} \label{l Mg<=Mf} Let$f$and$g$be real-valued random variables on$(\Omega ,\mathcal{F},P)$. Then \begin{equation} \label{Mg<=Mf} E \left[ \lambda \vee |g| \right]\leq E\left[\lambda \vee |f| \right] \end{equation} for all nonnegative number$\lambda$if and only if $\int_0^t g^{\#}(s) ds\leq \int_0^t f^{\#}(s) ds$ for all$t\in [0,1]$. \end{lem} \begin{proof} Equation~(\ref{Mg<=Mf}) is equivalent to$E \left[ \lambda \vee g^\# \right] \leq E \left[\lambda \vee f^\# \right]$. For the if'' part, let $\alpha =\sup \left\{t:f^\# (t) \geq \lambda \right\}$ $\beta =\sup \left\{t:g^\# (t)\geq \lambda \right\}.$ Then \begin{eqnarray*} E \left[ \lambda \vee f^\#\right] &=& \int_0^\alpha f^\# + (1-\alpha)\lambda \\ &=& \int_0^\beta f^\# + (1-\beta)\lambda + \int_\beta^\alpha (f^\#-\lambda) \\ &\ge& \int_0^\beta g^\# + (1-\beta)\lambda + \int_\beta^\alpha (f^\#-\lambda) \\ &=& E \left[ \lambda \vee g^\# \right] + \int_\beta^\alpha (f^\#-\lambda) . \end{eqnarray*} If$\alpha \le \beta$, then for all$x \in (\alpha,\beta)$we have$f^\#(x) \le \lambda$, and if$\beta \le \alpha$, then for all$x \in (\beta,\alpha)$we have$f^\#(x) \ge \lambda$. Either way, we see that$\int_\beta^\alpha (f^\#-\lambda) \ge 0$, and the result follows. To show the only if'', for any$\alpha \in [0,1]$, let $\lambda=f^\#(\alpha)$ $\beta =\inf \left\{t:g^\# (t)\geq \lambda \right\}.$ Then \begin{eqnarray*} \int_0^\alpha g^\# &=& \int_0^\beta g^\# + \int_\beta^\alpha (g^\#-\lambda) + \lambda(1-\beta) + \lambda(\alpha-1) \\ &=& E \left[ \lambda \vee g^\# \right] + \lambda(\alpha-1) + \int_\beta^\alpha (g^\#-\lambda) \\ &\le& E \left[ \lambda \vee f^\# \right] + \lambda(\alpha-1) + \int_\beta^\alpha (g^\#-\lambda) \\ &=& \int_0^\alpha f^\# + \int_\beta^\alpha (g^\#-\lambda) . \end{eqnarray*} Arguing as above, we see that$\int_\beta^\alpha (g^\#-\lambda) \le 0$, and again the result follows. \end{proof} Given a random variable$f$and a sigma field$\mathcal{G}$, we will say that$f$is nowhere constant with respect to$\mathcal{G}$if$P(f=g)=0$for every$\mathcal{G}$measurable function$g$. The following theorem \cite{M2} shows a concrete representation of a sequence of random variables. \begin{thmofothers} \label{t concrete} Let$(f_n)$be a sequence of random variables takeing values in a separable sigma filed$(S,\mathcal{S})$. Then there exists a sequence of measurable functions$(g_n:[0,1]^{n}\rightarrow S)$that has the same law as$(f_n)$. If further we have that$f_{n+1}$is nowhere constant with respect to$\sigma(f_1,\dots,f_n)$for all$n \geq 0$, then we may suppose that$\sigma(g_1,\dots,g_n)=\mathcal{L}_n$for all$n \geq 0$. \end{thmofothers} \begin{proof}[Proof of Theorem~\ref{t ek<=dk sk}] We will prove this theorem under the assumption (\ref{skekFk-1<=skdkFk-1}). Consider the map$D_k=(d_k,e_k,f_k):\Omega \times [0,1]^\mathbb{N} \rightarrow \mathbb{R}^3$by$(\omega,(x_k))\mapsto (d_k(\omega),e_k(\omega),x_k)$. It is clear that$D_k$is nowhere constant with respect to$\sigma(D_1,\dots,D_{k-1})$. Apply the previous theorem to get$\widetilde{D}_k=(\widetilde{d}_k,\widetilde{e}_k,\widetilde{f}_k):[0,1]^k \rightarrow \mathbb{R}^3 $such that$(\widetilde{D}_k)$has the same law as$(D_k)$and$\sigma(\widetilde{D}_1,\dots,\widetilde{D}_k)=\mathcal{L}_k$. Next, we show that for almost every$x_1, \dots,x_{k-1}$and$\lambda \ge 0$that $\int_0^1 \lambda \vee |\widetilde e_k(x_1,\dots,x_k)| \, dx_k \le \int_0^1 \lambda \vee |\widetilde d_k(x_1,\dots,x_k)| \, dx_k$ which will follow from showing that for any bounded non-negative measurable function$\phi_k : [0,1]^{k-1} \rightarrow [0,\infty)$that $E[\phi_k \vee |\widetilde{e}_k|] \leq E[\phi_k \vee |\widetilde{d}_k|] .$ But then there exists a bounded Borel measurable function$\theta_k : \mathbb{R}^{3(k-1)} \rightarrow [0,\infty)$such that$\phi=\theta(\widetilde{D}_1,\dots,\widetilde{D}_{k-1})$almost everywhere in$[0,1]^{k-1}$. Thus \begin{eqnarray*} \int_{[0,1]^k} \phi_k \vee |\widetilde{e}_k| &=&\int_{[0,1]^k} \theta(\widetilde{D}_1,\dots,\widetilde{D}_{k-1})\vee |\widetilde{e}_k| \\ &=& E[ \theta(D_1,\dots,D_{k-1}) \vee |e_k|] \\ &\le& E[ \theta(D_1,\dots,D_{k-1}) \vee |d_k|] \\ &=&\int_{[0,1]^k} \theta(\widetilde{D}_1,\dots,\widetilde{D}_{k-1})\vee |\widetilde{d}_k| \\ &=& \int_{[0,1]^k} \phi_k \vee |\widetilde{d}_k| \end{eqnarray*} Also to show that$E[\widetilde{d}_k|\mathcal{L}_{k-1}]=E[\widetilde{e}_k|\mathcal{L}_{k-1}]=0$, it is sufficient to show that for any bounded measurable function$\phi_k : [0,1]^{k-1} \rightarrow \mathbb{R}$that$E[\phi_k \widetilde{d}_k]=E[\phi_k \widetilde{e}_k]=0\$. Thus follows by a very similar argument to that above. 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