Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm

Abstract

In the present paper we identify those filtered probability spaces $(Ω,\, \mathcal{F},\, \left(\mathcal{F}_n\right),\, \mathbb{P})$ that determine already the martingale type of a Banach space $X$. We isolate intrinsic conditions on the filtration $(\mathcal{F}_n)$ of purely atomic $σ$-algebras which determine that the upper $\ell^p$ estimates \[ \|f\|_{L^p(Ω,\, X)}^p\leq C^p\left( \|\mathbb{E} f|\mathcal{F}_0\|^p_{L^p(Ω,\, X)}+\sum_{n=1}^{\infty} \|Δ_n f\|^p_{L^p(Ω,\, X)}\right),\qquad f\in L^p(Ω,X)\] imply that the Banach space $X$ is of martingale type $p$. Our paper complements \mbox{G. Pisier's} investigation \cite{Pisier1975} and continues the work by S. Geiss and second named author in \cite{Geiss2008}.

Figures (5)

• Figure 1: Here we put $A=A_1\cup A_2 \cup A_3\cup A_4$ and $N(A)=4$. Atoms are lined up according to their size from left to right.
• Figure 2: Given $A\in\mathop{\mathrm{\mathscr{A}}}\nolimits_n$, the figure depicts $A_1,A_2,A_3,A_4\in\mathop{\mathrm{\mathscr{A}}}\nolimits_{n+1}$ such that $A= A_1\cup A_2\cup A_3\cup A_4$. We have $N(A)=4$, and the functions $\{k_{A_2}$, $k_{A_3}$, $k_{A_4}\}$ (defined by \ref{['defbasis']}) form an algebraic basis of the space of martingale differences restricted to $A$.
• Figure 3: Given $A\in\mathop{\mathrm{\mathscr{A}}}\nolimits_n$, the figure depicts $A_1,A_2,A_3,A_4\in\mathop{\mathrm{\mathscr{A}}}\nolimits_{n+1}$ such that $A= A_1\cup A_2\cup A_3\cup A_4$. Then ${A}^{\tiny{\sun}}=A_2\cup A_3\cup A_4$ and ${A}^{\tiny{\diamond}}=A_1$. The figure depicts the graph of $\phi_{A}$.
• Figure 4: The picture highlights the different role of the atoms in $\mathop{\mathrm{\mathscr{A}}}\nolimits$. Let $A\in\mathop{\mathrm{\mathscr{A}}}\nolimits_n$ and $\mathop{\mathrm{\mathcal{R}}}\nolimits=\mathscr{B}$. The pairwise disjoint red coloured intervals depict ${A}^{\tiny{\sun}}$,${{A}^{\tiny{\diamond}}}^{\tiny{\sun}}$, ${{{A}^{\tiny{\diamond}}}^{\tiny{\diamond}}}^{\tiny{\sun}}$ etc. They form $G_{1}(\mathop{\mathrm{\mathcal{R}}}\nolimits, A)$. The intersection of the decreasing green coloured intervals forms the set $T_A$.
• Figure 5: This picture illustrates why functions $g_j$ have constant values on the sets from family $G_{j+1}(\mathscr{B}_i, \Omega)$. For ${A}^{\tiny{\sun}}\in G_{j+1}(\mathscr{B}_i, \Omega)$ there exists a unique $K={B}^{\tiny{\sun}}\in G_{j}(\mathscr{B}_i, \Omega)$ such that ${A}^{\tiny{\sun}}\in G_{1}(\mathscr{B}_i, K)$. Either we have ${A}^{\tiny{\sun}}={K}^{\tiny{\sun}}$ and $\mathop{\mathrm{\mathds{1}}}\nolimits_{{K}^{\tiny{\diamond}}}(\omega)=0$ for $\omega\in{A}^{\tiny{\sun}}$ or ${A}^{\tiny{\sun}}\subsetneq{K}^{\tiny{\diamond}}$ and $\mathop{\mathrm{\mathds{1}}}\nolimits_{{K}^{\tiny{\diamond}}}(\omega)=1$ for $\omega\in{A}^{\tiny{\sun}}$. Moreover for $K_0\in G_{j}(\mathscr{B}_i, \Omega)$ and $K_0\neq K$ we know that either $K\subseteq{K}^{\tiny{\diamond}}_0$ or $K\cap{K}^{\tiny{\diamond}}_0=\emptyset$. Therefore function $g_j=\sum_{{B}^{\tiny{\sun}}\in G_{j}(\mathscr{B}_i, \Omega)} x_{{B}^{\tiny{\sun}}}\mathop{\mathrm{\mathbb{P}}}\nolimits({B}^{\tiny{\sun}})\mathop{\mathrm{\mathds{1}}}\nolimits_{{B}^{\tiny{\diamond}}}$ is constant on ${A}^{\tiny{\sun}}$.

Theorems & Definitions (35)

• Theorem 1
• Theorem 2
• Remark 3
• Lemma 4
• proof
• Remark 5
• Remark 6
• Definition 7
• Remark 8
• Remark 9