Inequalities for sums of adapted random fields in Banach spaces and their application to strong law of large numbers

Abstract

We obtain inequalities for sums of adapted random fields in p-uniformly smooth Banach spaces, and we extend Kolmogorov and Marcinkiewicz-Zygmund strong laws for blockwise α-martingale difference fields.

MSC:60B11, 60B12, 60F15, 60G42.

1 Introduction

The concept of blockwise-dependence was introduced by Móricz [1]. Móricz’s [1] and Gaposhkin [2] showed that some properties of sequences of independent random variables can be applied to sequences consisting of independent blocks. Huan et al. [3] extended the strong laws of large numbers to blockwise-martingale difference arrays in Banach spaces. Recently, Móricz et al. [4] introduced the concept of blockwise M-dependence for a double array of random variables and established a version of the Kolmogorov SLLN for double arrays of random variables which are blockwise M-dependent. The results of Móricz and Stadtmüller and Thalmaier [5] were generalized by Stadtmüller and Thanh [6].

The aim of this paper is to investigate inequalities for sums of random fields and the strong law of large numbers of arbitrary random fields taking values in a Banach space. In Section 2, we introduce α-strong adapted random fields, α-strong^∗ adapted random fields, blockwise α-martingale difference fields and prove some useful lemmas. In Section 3, inequalities for sums of α-strong adapted random fields and α-strong^∗ adapted random fields in p-uniformly smooth Banach spaces are given. Section 4 contains the main results including the SLLN for a such blockwise α-martingale difference field taking values in a p-uniformly smooth Banach space, in which the results of [3, 6, 7] will be generalized.

Throughout this paper, the symbol C will denote a generic constant ($0<C<\mathrm{\infty }$) which is not necessarily the same one in each appearance.

2 Preliminaries and some useful lemmas

Let $\mathbb{E}$ be a real separable Banach space. $\mathbb{E}$ is said to be p-uniformly smooth ($1\le p\le 2$) if there exists a finite positive constant C such that for all $\mathbb{E}$-valued martingales $\{S_n;1\le n\le m\}$

$$E{\parallel S_m\parallel }^p\le C\sum _{n=1}^{m}E{\parallel S_n-S_{n-1}\parallel }^p.$$

(2.1)

Clearly, every real separable Banach space is 1-uniformly smooth and the real line (the same as any Hilbert space) is 2-uniformly smooth. If a real separable Banach space is p-uniformly smooth for some $1<p\le 2$ then it is r-uniformly smooth for all $r\in [1,p)$.

Let d be a positive integer, the set of all integer d-dimensional lattice points will be denoted by ${\mathbb{Z}}^d$ and the set of all positive integer d-dimensional lattice points will be denoted by ${\mathbb{N}}^d$. For $\mathbf{m}=(m_1,\dots ,m_d)\in {\mathbb{Z}}^d$, $\mathbf{n}=(n_1,\dots ,n_d)\in {\mathbb{Z}}^d$, $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_d)\in {\mathbb{R}}^d$ denote $[\mathbf{m},\mathbf{n})={\prod }_{i=1}^d[m_i,n_i)$ is a d-dimensional rectangle, $\mathbf{m}+\mathbf{n}=(m_1+n_1,\dots ,m_d+n_d)$, $\mathbf{m}-\mathbf{n}=(m_1-n_1,\dots ,m_d-n_d)$, ${\mathbf{2}}^{\mathbf{n}}=(2^{n_1},\dots ,2^{n_d})$, $|{\mathbf{n}}^{\mathit{\alpha }}|={\prod }_{i=1}^d{n}_i^{{\alpha }_i}$, $\mathbf{1}=(1,\dots ,1)\in {\mathbb{N}}^d$. We write $\mathbf{m}⪯\mathbf{n}$ (or $\mathbf{n}⪰\mathbf{m}$) if $m_i\le n_i$, $1\le i\le d$; $\mathbf{m}\prec \mathbf{n}$ if $\mathbf{m}⪯\mathbf{n}$, and $\mathbf{m}\ne \mathbf{n}$. For $x\ge 0$, let $[x]$ denote the greatest integer less than or equal to x, we use log⁺x to denote the $log(x\vee 1)$ (the logarithms are to base 2).

For nondecreasing sequences of positive integers $\{{\alpha }_i(k),k\ge 1\}$ ($1\le i\le d$), for $\mathbf{n}\in {\mathbb{N}}^d$, let $\alpha (\mathbf{n})=({\alpha }_1(n_1),\dots ,{\alpha }_d(n_d))$.

Let $(\mathrm{\Omega },\mathcal{F},P)$ be a probability space, $\mathbb{E}$ be a real separable Banach space, and $\mathcal{B}(\mathbb{E})$ be the σ-algebra of all Borel sets in $\mathbb{E}$. Let $\{X_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ be a field of $\mathbb{E}$-valued random variables and $\{{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ be a field of nondecreasing sub-σ-algebras of ℱ with respect to the partial order ⪯ on ${\mathbb{N}}^d$ such that $X_{\mathbf{k}}$ is ${\mathcal{F}}_{\mathbf{k}}$-measurable for all $\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}$, then $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is said to be an adapted field.

Let $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ be an adapted field, we adopt the convention that ${\mathcal{F}}_{\mathbf{k}}=\{\mathrm{\varnothing },\mathrm{\Omega }\}$ if $\mathbf{k}⋡\mathbf{n}$. For $\mathbf{k}\in {\mathbb{Z}}^d$ ($\mathbf{k}⪯\mathbf{N}-\mathbf{1}$) set

$${\mathcal{F}}_{\mathbf{k}}^i=\sigma \{{\mathcal{F}}_{\mathbf{l}}:\mathbf{l}=(l_1,\dots ,l_d),l_j\le k_j(j\ne i)\text{ and }{l}_i=k_i\}$$

for all $1\le i\le d$, and

$${\mathcal{F}}_{\mathbf{k}}^{\ast }=\sigma \{{\mathcal{F}}_{\mathbf{k}}^{(i)}:1\le i\le d\}.$$

The adapted field $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is said to be α-strong adapted (or strong adapted) if $E(X_{\mathbf{k}}|{\mathcal{F}}_{\mathbf{l}-\alpha (\mathbf{l})}^{\ast })$ (or $E(X_{\mathbf{k}}|{\mathcal{F}}_{\mathbf{l}-\mathbf{1}}^{\ast })$) is ${\mathcal{F}}_{\mathbf{k}}^i$-measurable for all $\mathbf{n}⪯\mathbf{l}⪯\mathbf{k}⪯\mathbf{N}$, and $1\le i\le d$.

The adapted field $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is said to be α-strong^∗ adapted (or strong^∗ adapted) if $\{X_{\mathbf{k}}{I}_A,{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is α-strong adapted (or strong adapted) for all $A\in \sigma (X_{\mathbf{k}})$.

Example 2.1 Let $\{X_n,{\mathcal{G}}_n:1\le n\le N\}$ be an adapted sequence of $\mathbb{E}$-valued random variables and set

$$\begin{array}{c}{X}_{\mathbf{k}}=X_k\text{if }\mathbf{k}=(k,k,\dots ,k)\text{and}{X}_{\mathbf{k}}=0\text{if }\mathbf{k}\ne (k,k,\dots ,k);\hfill \\ {\mathcal{G}}_{\mathbf{k}}={\mathcal{G}}_k\text{if }\mathbf{k}=(k,k,\dots ,k)\text{and}{\mathcal{G}}_{\mathbf{k}}=\{\mathrm{\varnothing },\mathrm{\Omega }\}\text{if }\mathbf{k}\ne (k,k,\dots ,k).\hfill \end{array}$$

Let ${\mathcal{F}}_{\mathbf{k}}=\sigma \{{\mathcal{G}}_{\mathbf{l}},\mathbf{l}⪯\mathbf{k}\}$ for all $\mathbf{k}⪰\mathbf{1}$, then $E(X_{\mathbf{k}}{I}_A|{\mathcal{F}}_{\mathbf{k}-\mathbf{1}}^{\ast })=E(X_k{I}_A|{\mathcal{G}}_{k-1})\in {\mathcal{G}}_k={\mathcal{F}}_{\mathbf{k}}^i$ if $\mathbf{k}=(k,k,\dots ,k)$, $E(X_{\mathbf{k}}{I}_A|{\mathcal{F}}_{\mathbf{k}-\mathbf{1}}^{\ast })=0$ if otherwise, for all $A\in \sigma (X_{\mathbf{k}})$ and $\mathbf{1}⪯\mathbf{k}⪯\mathbf{N}=(N,\dots ,N)$, so $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}}:\mathbf{1}⪯\mathbf{k}⪯\mathbf{N}\}$ is a strong^∗ adapted field.

Example 2.2 Let $\{Y_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ be a field of independent random variables. Put ${\mathcal{F}}_{\mathbf{k}}=\sigma (Y_{\mathbf{i}},\mathbf{i}⪯\mathbf{k})$ and $X_{\mathbf{k}}={\prod }_{\mathbf{i}⪯\mathbf{k}}{Y}_{\mathbf{i}}$, so $\{X_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is not a field of independent random variables. If $E|X_{\mathbf{k}}|<\mathrm{\infty }$ for all $\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}$, then $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is a strong^∗ adapted field.

The adapted field $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is said to be an α-martingale difference field if $E(X_{\mathbf{k}}|{\mathcal{F}}_{\mathbf{k}-\alpha (\mathbf{k})}^i)=0$ for all $\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}$ and $1\le i\le d$.

When $\alpha (\mathbf{k})=(M_1,\dots ,M_d)=\mathbf{M}$ for all $\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}$ then the adapted field $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is said to be an M-martingale difference field.

When $\alpha (\mathbf{k})=1$ for all $\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}$ then the field $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is a martingale difference field which was introduced by Huan et al. [3] in case $d=2$.

Remark 2.3

• Let $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}}:\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ be a field of martingale differences, then it is strong adapted, but it is not necessarily a strong^∗ adapted field.

• Let $\{X_{\mathbf{k}},\mathbf{k}\in {\mathbb{Z}}^d\}$ be a field of m-dependence random variables with mean 0. Put ${\mathcal{F}}_{\mathbf{k}}=\sigma (X_{\mathbf{l}},\mathbf{l}⪯\mathbf{k})$ and $\mathbf{M}=(m,\dots ,m)$, then $E(X_{\mathbf{k}}|{\mathcal{F}}_{\mathbf{k}-\mathbf{M}}^i)=E{X}_{\mathbf{k}}=0$ for all $\mathbf{k}\in {\mathbb{Z}}^d$, $1\le i\le d$. Therefore, $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{k}\in {\mathbb{Z}}^d\}$ is a field of M-martingale differences.

Example 2.4 Let $\{X_{\mathbf{k}},\mathbf{k}\in {\mathbb{Z}}^d\}$ be a field of independent random elements with mean 0. Put ${\mathcal{F}}_{\mathbf{k}}=\sigma (X_{\mathbf{l}},\mathbf{l}⪯\mathbf{k})$, then $E(X_{\mathbf{k}}|{\mathcal{F}}_{\mathbf{k}-\mathbf{1}}^i)=0$ for all $\mathbf{k}\in {\mathbb{Z}}^d$, $1\le i\le d$. Therefore, $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{k}\in {\mathbb{Z}}^d\}$ is a field of martingale differences and a strong^∗ adapted field.

Set $Y_{\mathbf{k}}={\sum }_{\mathbf{k}+\mathbf{1}-\alpha (\mathbf{k})⪯\mathbf{l}⪯\mathbf{k}}{X}_{\mathbf{l}}$, then $\{Y_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{k}\in {\mathbb{Z}}^d\}$ is a field of α-martingale differences.

Example 2.5 Let $\{Y_{\mathbf{k}},\mathbf{k}\in {\mathbb{N}}^d\}$ be a field of independent random variables with mean 0. Put ${\mathcal{F}}_{\mathbf{k}}=\sigma (Y_{\mathbf{l}},\mathbf{l}⪯\mathbf{k})$ and $X_{\mathbf{k}}={\prod }_{\mathbf{l}⪯\mathbf{k}}{Y}_{\mathbf{l}}$, so $\{X_{\mathbf{k}},\mathbf{k}\in {\mathbb{N}}^d\}$ is not a field of independent random variables. If $E|X_{\mathbf{k}}|<\mathrm{\infty }$ for all $\mathbf{k}⪰\mathbf{1}$, then $E(X_{\mathbf{k}}|{\mathcal{F}}_{\mathbf{k}-\mathbf{1}}^i)=0$, $E(X_{\mathbf{k}}{I}_A|{\mathcal{F}}_{\mathbf{k}-\mathbf{1}}^{\ast })=E(X_{\mathbf{k}}{I}_A)\in {\mathcal{F}}_{\mathbf{k}}^i$ for all $A\in \sigma (X_{\mathbf{k}})$, $\mathbf{k}⪰\mathbf{1}$, $1\le i\le d$. Therefore, $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{k}\in {\mathbb{N}}^d\}$ is a field of martingale differences and a strong^∗ adapted field.

For strictly increasing sequence of positive integers $\{{\omega }_i(k),k\ge 1\}$, with ${\omega }_i(1)=1$ ($1\le i\le d$), and $\mathbf{k}\in {\mathbb{N}}^d$, we set

$$\omega (\mathbf{k})=({\omega }_1(k_1),\dots ,{\omega }_d(k_d)),{\mathrm{\Delta }}_{\mathbf{k}}=[\omega (\mathbf{k}),\omega (\mathbf{k}+\mathbf{1})).$$

The adapted field $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{k}\in {\mathbb{N}}^d\}$ is said to be a blockwise-adapted field (respectively, blockwise-α-strong adapted, blockwise-α-strong^∗ adapted, blockwise-α-martingale difference field, blockwise-M-martingale difference field, blockwise-martingale difference field) with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{k}},\mathbf{k}\in {\mathbb{N}}^d\}$ if for each $\mathbf{k}\in {\mathbb{N}}^d$, $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{k}\in {\mathrm{\Delta }}_{\mathbf{k}}\}$ is an adapted field (respectively, α-strong adapted, α-strong^∗ adapted, α-martingale difference field, M-martingale difference field, martingale difference field).

Example 2.6 Let $X_{\mathbf{k}}$, ${\mathcal{F}}_{\mathbf{k}}$, $Y_{\mathbf{k}}$ as in Example 2.4. Set $Z_{\mathbf{i}}=Y_{\mathbf{i}-{\mathbf{2}}^{\mathbf{k}}+\mathbf{1}}$ and ${\mathcal{G}}_{\mathbf{i}}=\sigma (Z_{\mathbf{u}};{\mathbf{2}}^{\mathbf{k}}⪯\mathbf{u}⪯\mathbf{i})$, ${\mathbf{2}}^{\mathbf{k}}⪯\mathbf{i}\prec {\mathbf{2}}^{\mathbf{k}+\mathbf{1}}$, $\mathbf{k}⪰\mathbf{0}$ then $\{Z_{\mathbf{n}},{\mathcal{G}}_{\mathbf{n}};\mathbf{n}⪰\mathbf{1}\}$ is a blockwise-α-martingale differences and a blockwise-α-strong^∗ adapted field with respect to the blocks ${\prod }_{k=1}^d[2^k,2^{k+1})$.

To prove the main result we need the following lemmas.

Lemma 2.7 Let $\mathbb{E}$ be a real separable p-uniformly smooth Banach space for some $1\le p\le 2$. Then there exists a positive constant C such that all strong adapted random fields $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}},\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ in $\mathbb{E}$ we have

$$E\left[\underset{\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}}{max}{\parallel \sum _{\mathbf{n}⪯\mathbf{i}⪯\mathbf{k}}(X_{\mathbf{i}}-E(X_{\mathbf{i}}|{\mathcal{F}}_{\mathbf{i}-\mathbf{1}}^{\ast }))\parallel }^p\right]\le C\sum _{\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}}E{\parallel X_{\mathbf{k}}\parallel }^p.$$

(2.2)

Proof We set

$$S_{\mathbf{k}}=\sum _{\mathbf{1}⪯\mathbf{i}⪯\mathbf{k}}(X_{\mathbf{i}}-E(X_{\mathbf{i}}|{\mathcal{F}}_{\mathbf{i}-\mathbf{1}}^{\ast })).$$

Firstly, for $d=1$, note that $\{{max}_{1\le i\le k}\parallel S_i\parallel ,{\mathcal{F}}_k:n\le k\le N\}$ is a nonnegative sub-martingale. Applying Doob’s inequality and by (2.1), we have (2.2). We assume that (2.2) holds for $d-1$, we wish to show that it holds for d.

Denote $\mathbf{k}=({\mathbf{k}}^{\prime },k_d)$; $\mathbf{k}=({\mathbf{n}}^{\prime },n_d)$; $\mathbf{N}=({\mathbf{N}}^{\prime },N_d)$; with ${\mathbf{k}}^{\prime },{\mathbf{n}}^{\prime },{\mathbf{N}}^{\prime }\in {\mathbb{N}}^{d-1}$; set

$$Y_{k_d}=\underset{{\mathbf{n}}^{\prime }\le {\mathbf{k}}^{\prime }\le {\mathbf{N}}^{\prime }}{max}\parallel S_{({\mathbf{k}}^{\prime };k_d)}\parallel$$

for each $n_d\le k_d\le N_d$, we have

$$\begin{array}{rcl}E(S_{({\mathbf{k}}^{\prime };k_d)}|{\mathcal{F}}_{({\mathbf{k}}^{\prime };k_d-1)}^d)& =& E(S_{({\mathbf{k}}^{\prime };k_d-1)}|{\mathcal{F}}_{({\mathbf{k}}^{\prime };k_d-1)}^d)\\ +\sum _{{\mathbf{n}}^{\prime }\le {\mathbf{k}}^{\prime }\le {\mathbf{N}}^{\prime }}\left(E(X_{({\mathbf{k}}^{\prime };k_d)}-E(X_{({\mathbf{k}}^{\prime };k_d)}|{\mathcal{F}}_{({\mathbf{k}}^{\prime }-\mathbf{1},k_d-1)}^{\ast })|{\mathcal{F}}_{({\mathbf{k}}^{\prime };k_d-1)}^d)\right)\\ =& S_{({\mathbf{k}}^{\prime };k_d-1)};\end{array}$$

that means that for each ${\mathbf{n}}^{\prime }\le {\mathbf{k}}^{\prime }\le {\mathbf{N}}^{\prime }$ then $\{S_{({\mathbf{k}}^{\prime };k_d)};{\mathcal{F}}_{({\mathbf{k}}^{\prime };k_d)}^d:n_d\le k_d\le N_d\}$, we find that $\{Y_{k_d};{\mathcal{F}}_{({\mathbf{k}}^{\prime };k_d)}:n_d\le k_d\le N_d\}$ is a nonnegative sub-martingale sequence. Applying Doob’s inequality, we obtain

$$E\underset{\mathbf{n}⪯({\mathbf{k}}^{\prime },k_d)⪯\mathbf{N}}{max}{\parallel S_{({\mathbf{k}}^{\prime };k_d)}\parallel }^p=E\underset{n_d\le k_d\le N_d}{max}{Y}_{k_d}^p\le C\cdot E{Y}_{N_d}^p=C\cdot E\underset{{\mathbf{n}}^{\prime }\le {\mathbf{k}}^{\prime }\le {\mathbf{N}}^{\prime }}{max}{\parallel S_{({\mathbf{k}}^{\prime };N_d)}\parallel }^p.$$

Set

$$X_{{\mathbf{k}}^{\prime }}^{d-1}=\sum _{n_d\le k_d\le N_d}{X}_{({\mathbf{k}}^{\prime };k_d)};{\mathcal{F}}_{{\mathbf{k}}^{\prime }}^{d-1}=\sigma ({\mathcal{F}}_{({\mathbf{k}}^{\prime };k_d)}^{d-1}:n_d\le k_d\le N_d).$$

We note that ${\mathcal{F}}_{({\mathbf{k}}^{\prime },k_d)}^i={({\mathcal{F}}_{{\mathbf{k}}^{\prime }}^{d-1})}^i$, ${\mathcal{F}}_{({\mathbf{k}}^{\prime },k_d)}^{\ast }={({\mathcal{F}}_{{\mathbf{k}}^{\prime }}^{d-1})}^{\ast }$, for all $n_d\le k_d\le N_d$, $1\le i\le d$, then $\{X_{{\mathbf{k}}^{\prime }}^{d-1};{\mathcal{F}}_{{\mathbf{k}}^{\prime }}^{d-1}:{\mathbf{n}}^{\prime }\le {\mathbf{k}}^{\prime }\le {\mathbf{N}}^{\prime }\}$ is a strong adapted field. Therefore, by the inductive assumption and inequality (2.1) we have

$$\begin{array}{rl}E\left[\underset{\mathbf{n}⪯({\mathbf{k}}^{\prime },k_d)⪯\mathbf{N}}{max}{\parallel S_{({\mathbf{k}}^{\prime };k_d)}\parallel }^p\right]& \le C\cdot \sum _{{\mathbf{n}}^{\prime }\le {\mathbf{k}}^{\prime }\le {\mathbf{N}}^{\prime }}E{\parallel \sum _{n_d\le k_d\le N_d}{X}_{({\mathbf{k}}^{\prime };k_d)}-E(X_{({\mathbf{k}}^{\prime };k_d)}|{\mathcal{F}}_{({\mathbf{k}}^{\prime }-\mathbf{1},k_d-1)}^{\ast })\parallel }^p\\ \le C\sum _{\mathbf{1}⪯\mathbf{k}⪯\mathbf{n}}E{\parallel X_{\mathbf{k}}-E(X_{\mathbf{k}}|{\mathcal{F}}_{\mathbf{k}-\mathbf{1}}^{\ast })\parallel }^p\le C\sum _{\mathbf{1}⪯\mathbf{k}⪯\mathbf{n}}E{\parallel X_{\mathbf{k}}\parallel }^p.\end{array}$$

 □

Remark 2.8 If $\{X_{\mathbf{k}};\mathbf{k}⪰\mathbf{1}\}$ is an $\mathbb{E}$-valued martingale difference field, from Lemma 2.7, we obtain Lemma 2.4 in [3] for $d=2$ and Corollary 3.2 in [8] for $d\ge 2$ (with $p=q$).

We note that if $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}}:\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is strong adapted, when $d=1$ then $\{X_k-E(X_k|{\mathcal{F}}_{k-1}^{\ast }),{\mathcal{F}}_k:n\le k\le N\}$ is a sequences of martingale differences, but when $d>1$ then $\{X_{\mathbf{k}}-E(X_{\mathbf{k}}|{\mathcal{F}}_{\mathbf{k}-\mathbf{1}}^{\ast }),{\mathcal{F}}_{\mathbf{k}}:\mathbf{n}⪯\mathbf{k}⪯\mathbf{N}\}$ is not necessarily a field of martingale differences, because $X_{\mathbf{k}}-E(X_{\mathbf{k}}|{\mathcal{F}}_{\mathbf{k}-\mathbf{1}}^{\ast })$ may not be ${\mathcal{F}}_{\mathbf{k}}$-measurable (see [9], Example 3.1).

Lemma 2.9 Let $1<p\le 2$, $\alpha =({\alpha }_1,\dots ,{\alpha }_d)$ where ${\alpha }_1,\dots ,{\alpha }_d$ are positive constants, $1={\alpha }_1=\cdots ={\alpha }_q<{\alpha }_{q+1}\le \cdots \le {\alpha }_d$ and X be a random variable taking values in a real separable Banach space $\mathbb{E}$. Then there exists a positive constant C such that

$$\begin{array}{rl}(\mathrm{i})& \sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{{|{\mathbf{n}}^{\alpha }|}^p}{\int }_0^{{|{\mathbf{n}}^{\alpha }|}^p}P\{{\parallel X\parallel }^p\ge t\}dt\le C\cdot (E\parallel X\parallel {log}_{+}^{q-1}\parallel X\parallel +1),\\ (\mathrm{ii})& \sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{|{\mathbf{n}}^{\alpha }|}{\int }_{|{\mathbf{n}}^{\alpha }|}^{\mathrm{\infty }}P\{\parallel X\parallel \ge t\}dt\le C\cdot E\parallel X\parallel {log}_{+}^q\parallel X\parallel .\end{array}$$

Proof Denote $d(k)={\sum }_{n_1\cdots n_q=k}1$, by Lemma 3.1 of Gut [10] we have

$$\sum _{j=k}^{\mathrm{\infty }}\frac{d(j)}{j^p}\sim C\frac{{(logk)}^{q-1}}{k^{p-1}}.$$

Hence, we have

$$\begin{array}{c}\sum _{\mathbf{n}}\frac{1}{{|{\mathbf{n}}^{\alpha }|}^p}{\int }_0^{{|{\mathbf{n}}^{\alpha }|}^p}P\{{\parallel X\parallel }^p\ge t\}dt\hfill \\ \le \sum _{\mathbf{n}}\frac{1}{{|{\mathbf{n}}^{\alpha }|}^p}+\sum _{\mathbf{n}}\frac{1}{{|{\mathbf{n}}^{\alpha }|}^p}{\int }_1^{{|{\mathbf{n}}^{\alpha }|}^p}P\{{\parallel X\parallel }^p\ge t\}dt\hfill \\ \le C+C\sum _{\mathbf{n}}\frac{1}{{|{\mathbf{n}}^{\alpha }|}^p}{\int }_1^{{|{\mathbf{n}}^{\alpha }|}^p}P\{{\parallel X\parallel }^p\ge t\}dt\hfill \\ \le C+C\sum _{k,n_{q+1},\dots ,n_d=1}^{\mathrm{\infty }}d(k)\frac{1}{k^p\cdot n_{q+1}^{p{\alpha }_{q+1}}\cdots n_d^{p{\alpha }_d}}\sum _{j=1}^{[k{n}_{q+1}^{{\alpha }_{q+1}}\cdots n_d^{{\alpha }_d}]}E\left({\parallel X\parallel }^{p}I(j\le \parallel X\parallel <j+1)\right)\hfill \\ \le C+C\sum _{n_{q+1},\dots ,n_d=1}^{\mathrm{\infty }}\frac{1}{n_{q+1}^{p{\alpha }_{q+1}}\cdots n_d^{p{\alpha }_d}}\sum _{k=1}^{\mathrm{\infty }}\frac{d(k)}{k^p}\sum _{j=1}^{[k{n}_{q+1}^{{\alpha }_{q+1}}\cdots n_d^{{\alpha }_d}]}{j}^{p-1}P\{\parallel X\parallel \ge j\}\hfill \\ \le C+C\sum _{n_{q+1},\dots ,n_d=1}^{\mathrm{\infty }}\frac{1}{n_{q+1}^{p{\alpha }_{q+1}}\cdots n_d^{p{\alpha }_d}}\sum _{j=1}^{[n_{q+1}^{{\alpha }_{q+1}}\cdots n_d^{{\alpha }_d}]}{j}^{p-1}P\{\parallel X\parallel \ge j\}\sum _{k=1}^{\mathrm{\infty }}\frac{d(k)}{k^p}\hfill \\ +C\sum _{n_{q+1},\dots ,n_d=1}^{\mathrm{\infty }}\frac{1}{n_{q+1}^{p{\alpha }_{q+1}}\cdots n_d^{p{\alpha }_d}}\sum _{j=[n_{q+1}^{{\alpha }_{q+1}}\cdots n_d^{{\alpha }_d}]+1}^{\mathrm{\infty }}{j}^{p-1}P\{\parallel X\parallel \ge j\}\sum _{k=[j/n_{q+1}^{{\alpha }_{q+1}}\cdots n_d^{{\alpha }_d}]}^{\mathrm{\infty }}\frac{d(k)}{k^p}\hfill \\ \le C+C\sum _{n_{q+1},\dots ,n_d=1}^{\mathrm{\infty }}\frac{1}{n_{q+1}^{{\alpha }_{q+1}}\cdots n_d^{{\alpha }_d}}\sum _{j=1}^{\mathrm{\infty }}{(logi)}^{q-1}P\{\parallel X\parallel \ge j\}\le C+CE\parallel X\parallel {log}_{+}^{q-1}\parallel X\parallel .\hfill \end{array}$$

Now we prove (ii). We have by Lemma 3 of Stadtmüller and Thalmaier [5]

$$g(j)=\sum _{1\le n_1\cdots n_q\cdot n_{q+1}^{{\alpha }_{q+1}}\cdots n_d^{{\alpha }_d}\le j}1\sim C\frac{j{(logj)}^{q-1}}{(q-1)!}\text{as }j\to \mathrm{\infty }.$$

Denote $\mathrm{\Delta }g(j)=g(j)-g(j-1)$, we get

$$\begin{array}{c}\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{|{\mathbf{n}}^{\alpha }|}{\int }_{|{\mathbf{n}}^{\alpha }|}^{\mathrm{\infty }}P\{\parallel X\parallel \ge t\}dt\hfill \\ =\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}\mathrm{\Delta }g(k){\int }_k^{\mathrm{\infty }}P\{\parallel X\parallel \ge t\}dt=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}\mathrm{\Delta }g(k)\sum _{i=k}^{\mathrm{\infty }}{\int }_i^{i+1}P\{\parallel X\parallel \ge t\}dt\hfill \\ \le \sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}\mathrm{\Delta }g(k)\sum _{j=k}^{\mathrm{\infty }}jP\{j\le \parallel X\parallel <j+1\}=\sum _{j=1}^{\mathrm{\infty }}jP\{j\le \parallel X\parallel <j+1\}\sum _{k=1}^j\frac{1}{k}\mathrm{\Delta }g(k)\hfill \\ \le C\sum _{j=1}^{\mathrm{\infty }}jP\{j\le \parallel X\parallel <j+1\}\sum _{k=1}^{j-1}(\frac{1}{k}-\frac{1}{k+1})k{(logk)}^{q-1}\hfill \\ +C\sum _{j=1}^{\mathrm{\infty }}P\{j\le \parallel X\parallel <j+1\}j{(logj)}^{q-1}\hfill \\ \le C\sum _{j=1}^{\mathrm{\infty }}jP\{j\le \parallel X\parallel <j+1\}{(logj)}^{q-1}\sum _{k=1}^j\frac{1}{k}\hfill \\ +C\sum _{j=1}^{\mathrm{\infty }}P\{j\le \parallel X\parallel <j+1\}j{(logj)}^{q-1}\hfill \\ \le C\sum _{j=1}^{\mathrm{\infty }}P\{j\le \parallel X\parallel <j+1\}j{(logj)}^q\le CE\parallel X\parallel {log}_{+}^q\parallel X\parallel .\hfill \end{array}$$

 □

Lemma 2.10 Let $\{a_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ be a nondecreasing field of positive constants such that

$$1<\underset{\mathbf{n}\prec \mathbf{m}⪯\mathbf{n}+\mathbf{1}}{lim\hspace{0.17em}inf}\frac{a_{\mathbf{m}}}{a_{\mathbf{n}}}\le \underset{\mathbf{n}\prec \mathbf{m}⪯\mathbf{n}+\mathbf{1}}{lim\hspace{0.17em}sup}\frac{a_{\mathbf{m}}}{a_{\mathbf{n}}}\le M.$$

(2.3)

If $\{x_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a field of constants and

$$\underset{|\mathbf{n}|\to \mathrm{\infty }}{lim}{x}_{\mathbf{n}}=0,$$

then

$$\underset{|\mathbf{n}|\to \mathrm{\infty }}{lim}\frac{1}{a_{\mathbf{n}}}\underset{\mathbf{k}⪯\mathbf{n}}{sup}\left|\sum _{\mathbf{1}⪯\mathbf{j}⪯\mathbf{k}}{a}_{\mathbf{j}+\mathbf{1}}{x}_{\mathbf{j}}\right|=0.$$

Proof First, we prove that

$$\frac{1}{a_{\mathbf{n}}}\sum _{\mathbf{1}⪯\mathbf{j}⪯\mathbf{n}}{a}_{\mathbf{j}+\mathbf{1}}\le C.$$

(2.4)

By (2.3), there exist a constant $0<\delta <1$ and ${\mathbf{n}}_0$ such that for all $\mathbf{m}\succ \mathbf{n}⪰{\mathbf{n}}_0$ then $\frac{a_{\mathbf{n}}}{a_{\mathbf{m}}}\le \delta$. We have for all $\mathbf{n}⪰\mathbf{1}$

$$\frac{1}{a_{\mathbf{n}}}\sum _{\mathbf{1}⪯\mathbf{j}⪯\mathbf{n}}{a}_{\mathbf{j}+\mathbf{1}}\le M\frac{1}{a_{\mathbf{n}+\mathbf{1}}}\sum _{\mathbf{1}⪯\mathbf{j}⪯\mathbf{n}}{a}_{\mathbf{j}+\mathbf{1}}\le M(|{\mathbf{n}}_0|+\frac{1}{{(1-\delta )}^d})$$

and we have (2.4).

For every $ϵ>0$, there exists $N>0$ such that for all $|\mathbf{n}|\ge N$, $|x_{\mathbf{n}}|\le \frac{ϵ}{C}$ so that

$$\frac{1}{a_{\mathbf{n}}}\underset{\mathbf{k}⪯\mathbf{n}}{sup}\left|\sum _{\mathbf{1}⪯\mathbf{j}⪯\mathbf{k}}{a}_{\mathbf{j}+\mathbf{1}}{x}_{\mathbf{j}}\right|\le \frac{1}{a_{\mathbf{n}}}\underset{|\mathbf{k}|<\mathbf{N}}{sup}\left|\sum _{|\mathbf{j}|⪯\mathbf{k}}{a}_{\mathbf{j}+\mathbf{1}}{x}_{\mathbf{j}}\right|+ϵ.$$

The conclusion of the lemma follows upon letting $|\mathbf{n}|\to \mathrm{\infty }$ and then $ϵ\to 0$. □

3 Inequalities for sums of adapted random fields

The first theorem characterizes the p-uniformly smooth Banach spaces.

Theorem 3.1 Let $1\le p\le 2$ and $\mathbb{E}$ be a separable Banach space, then the following three statements are equivalent:

• (i) $\mathbb{E}$ is p-uniformly smooth.

• (ii) There exists a positive constant C such that for all α-strong adapted random fields $\{X_{\mathbf{k}},{\mathcal{F}}_{\mathbf{k}};\mathbf{n}⪯\mathbf{k}⪯\mathbf{m}\}$ in $\mathbb{E}$ we have

  $$\begin{array}{c}E\left[\underset{\mathbf{n}⪯\mathbf{k}⪯\mathbf{m}}{max}{\parallel \sum _{\mathbf{n}⪯\mathbf{i}⪯\mathbf{k}}{X}_{\mathbf{i}}-E(X_{\mathbf{i}}|{\mathcal{F}}_{\mathbf{i}-\alpha (\mathbf{i})}^{\ast })\parallel }^p\right]\hfill \\ \le C|{\alpha }^{p-1}(\mathbf{m})|\sum _{\mathbf{n}⪯\mathbf{k}⪯\mathbf{m}}E{\parallel X_{\mathbf{k}}\parallel }^p.\hfill \end{array}$$

  (3.1)

Proof We first prove the implication ((i) ⇒ (ii)). If $\mathbf{m}-\mathbf{n}⪯\alpha (\mathbf{m})$, (3.1) is trivial.

If $\mathbf{m}-\mathbf{n}⪰\alpha (\mathbf{m})$, note that if $\{\alpha (\mathbf{k}),\mathbf{k}⪰\mathbf{1}\}$ is a nondecreasing field of positive, for all $\mathbf{n}⪯\mathbf{i}⪯\alpha (\mathbf{m})+\mathbf{n}-\mathbf{1}$ then

$$\{X_{\mathbf{i}+(\mathbf{u}-\mathbf{1})\alpha (\mathbf{m})},{\mathcal{F}}_{\mathbf{i}+\mathbf{u}\alpha (\mathbf{m})+\alpha (\mathbf{i}+\mathbf{u}\alpha (\mathbf{m}))};\mathbf{n}⪯\mathbf{i}+(\mathbf{u}-\mathbf{1})\alpha (\mathbf{m})⪯\mathbf{m}\}$$

are α-strong adapted fields. Set

$$A_{\mathbf{i}}=\{\mathbf{k};0⪯\mathbf{k}\alpha (\mathbf{m})+\mathbf{i}⪯\mathbf{m}-\mathbf{n}\}\text{and}{Y}_{\mathbf{k}}=X_{\mathbf{k}}-E(X_{\mathbf{k}}|{\mathcal{F}}_{\mathbf{k}-\alpha (\mathbf{k})}^{\ast }).$$

Then we have

$$\begin{array}{c}E\left(\underset{\mathbf{n}⪯\mathbf{k}⪯\mathbf{m}}{max}{\parallel \sum _{\mathbf{n}⪯\mathbf{i}⪯\mathbf{k}}{X}_{\mathbf{i}}-E(X_{\mathbf{i}}|{\mathcal{F}}_{\mathbf{i}-\alpha (\mathbf{i})}^{\ast })\parallel }^p\right)\hfill \\ \le E{\left(\sum _{\mathbf{0}⪯\mathbf{i}⪯\alpha (\mathbf{m})-\mathbf{1}}\underset{\mathbf{k}\in A_{\mathbf{i}}}{max}\parallel \sum _{\mathbf{0}⪯\mathbf{u}⪯\mathbf{k}}{Y}_{\mathbf{u}\alpha (\mathbf{m})+\mathbf{i}+\mathbf{n}}\parallel \right)}^p\hfill \\ \le {\alpha }^{p-1}(\mathbf{m})\left(\sum _{\mathbf{0}⪯\mathbf{i}⪯\alpha (\mathbf{m})-\mathbf{1}}E\underset{\mathbf{k}\in A_{\mathbf{i}}}{max}{\parallel \sum _{\mathbf{0}⪯\mathbf{u}⪯\mathbf{k}}{Y}_{\mathbf{u}\alpha (\mathbf{m})+\mathbf{i}+\mathbf{n}}\parallel }^p\right)\hfill \\ \le C{\alpha }^{p-1}(\mathbf{m})\sum _{\mathbf{0}⪯\mathbf{i}⪯\alpha (\mathbf{m})-\mathbf{1}}\left(\sum _{\mathbf{k}\in A_{\mathbf{i}}}E{\parallel Y_{\mathbf{k}\alpha (\mathbf{m})+\mathbf{i}+\mathbf{n}}\parallel }^p\right)\hfill \\ =C{\alpha }^{p-1}(\mathbf{m})\sum _{\mathbf{n}⪯\mathbf{i}⪯\mathbf{m}}E{\parallel X_{\mathbf{i}}\parallel }^p\text{by (2.2)},\hfill \end{array}$$

(3.2)

again establishing (3.1). If ${\bigvee }_{i=1}^d(m_i-n_i<{\alpha }_i(m_i))$, the proof is similar to (3.2).

((ii) ⇒ (i)) Let $\{X_n,{\mathcal{G}}_n,n\ge 1\}$ be a martingale difference sequence in $\mathbb{E}$.

For ${\mathbf{n}}^{\prime }\succ \mathbf{1}$, $n_d\ge 1$, put

$$\begin{array}{c}{X}_{({\mathbf{n}}^{\prime };n_d)}=X_{n_d}\text{if }{\mathbf{n}}^{\prime }=\mathbf{1}\text{and}{X}_{({\mathbf{n}}^{\prime };n_d)}=0\text{if }{\mathbf{n}}^{\prime }\succ \mathbf{1},\hfill \\ {\mathcal{F}}_{({\mathbf{n}}^{\prime };n_d)}={\mathcal{G}}_{{\mathbf{n}}^{\prime }}.\hfill \end{array}$$

Then $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is an α-strong adapted field with $\alpha (\mathbf{n})=1$ by (ii); we have (2.1) and then $\mathbb{E}$ is p-uniformly smooth. □

From now on, let $\{{\mathrm{\Gamma }}_i(k),k\ge 1\}$ be strictly increasing sequences of positive integers with ${\mathrm{\Gamma }}_i(1)=1$ ($1\le i\le d$). For $\mathbf{m}\in {\mathbb{N}}^d$, $\mathbf{n}\in {\mathbb{N}}^{\ast d}$, we introduce the following notations:

$$\begin{array}{c}\mathrm{\Gamma }(\mathbf{n})=({\mathrm{\Gamma }}_1(n_1),\dots ,{\mathrm{\Gamma }}_d(n_d)),{\mathrm{\Delta }}^{\mathbf{m}}=[\mathrm{\Gamma }(\mathbf{m}),\mathrm{\Gamma }(\mathbf{m}+\mathbf{1})),{\mathrm{\Delta }}_{\mathbf{n}}^{\mathbf{m}}={\mathrm{\Delta }}_{\mathbf{n}}\cap {\mathrm{\Delta }}^{\mathbf{m}},\hfill \\ I_{\mathbf{m}}=\{\mathbf{n}:{\mathrm{\Delta }}_{\mathbf{n}}^{\mathbf{m}}\ne \mathrm{\varnothing }\},c_{\mathbf{m}}=card{I}_{\mathbf{m}},\hfill \\ {\phi }_1(\mathbf{n})=\sum _{\mathbf{k}⪰\mathbf{1}}^{\mathrm{\infty }}{c}_{\mathbf{k}}{I}_{{\mathrm{\Delta }}^{\mathbf{k}}}(\mathbf{n}),{\varphi }_1(\mathbf{n})=\underset{\mathbf{k}⪯\mathbf{n}}{max}{\phi }_1(\mathbf{k}),\hfill \\ {\phi }_2(\mathbf{n})=\sum _{\mathbf{k}⪰\mathbf{1}}\left|\alpha (\mathrm{\Gamma }(\mathbf{k}+\mathbf{1}))\right|\cdot c_{\mathbf{k}}{I}_{{\mathrm{\Delta }}^{\mathbf{k}}}(\mathbf{n}),{\varphi }_2(\mathbf{n})=\underset{\mathbf{k}⪯\mathbf{n}}{max}{\phi }_2(\mathbf{k}),\hfill \end{array}$$

where $I_{{\mathrm{\Delta }}^{(\mathbf{k})}}$ denotes the indicator function of the set ${\mathrm{\Delta }}^{(\mathbf{k})};\mathbf{k}\in {\mathbb{N}}^d$.

Let $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}};\mathbf{n}\in {\mathbb{N}}^d\}$ be a blockwise-α-adapted field taking values in the Banach space $\mathbb{E}$ with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{n}};\mathbf{n}\in {\mathbb{N}}^d\}$, we put

$$T_{\mathbf{n}}=\underset{\mathbf{k}\in {\mathrm{\Delta }}^{\mathbf{n}}}{sup}\parallel \sum _{\mathrm{\Gamma }(\mathbf{n})⪯\mathbf{i}⪯\mathbf{k}}{X}_{\mathbf{i}}-E(X_{\mathbf{i}}|{\mathcal{F}}_{i-\alpha (i)}^{\ast })\parallel .$$

Theorem 3.2 Let $\mathbb{E}$ be a p-uniformly smooth Banach space ($1\le p\le 2$). Then there exists a positive constant C such that for all strong blockwise-α-strong adapted random fields $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}};\mathbf{n}⪰\mathbf{1}\}$ in $\mathbb{E}$ with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{k}},\mathbf{k}⪰\mathbf{1}\}$ and every nondecreasing field of positive constants $\{a_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ we have

$$\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{a_{\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})}^p{\phi }_2^{p-1}(\mathrm{\Gamma }(\mathbf{n}))}\mathbb{E}{T}_{\mathbf{n}}^p\le C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{a_{\mathbf{n}}^p}E{\parallel X_{\mathbf{n}}\parallel }^p.$$

(3.3)

Proof For $\mathbf{m}\in {\mathbb{N}}^d$, $\mathbf{n}\in I_{\mathbf{m}}$, we set

$$\begin{array}{c}{r}_{\mathbf{n}}^{\mathbf{m}}(i)=min\{j:j\in [{\omega }_i(n_i),{\omega }_i(n_i+1))\cap [{\mathrm{\Gamma }}_i(m_i),{\mathrm{\Gamma }}_i(m_i+1)\left)\right\},\hfill \\ r_{\mathbf{n}}^{\mathbf{m}}=(r_{\mathbf{n}}^{\mathbf{m}}(1),\dots ,r_{\mathbf{n}}^{\mathbf{m}}(d)),T_{\mathbf{n}}^{\mathbf{m}}=\underset{u\in {\mathrm{\Delta }}_{\mathbf{n}}^{\mathbf{m}}}{max}\parallel \sum _{r_{\mathbf{n}}^{\mathbf{m}}⪯\mathbf{i}⪯\mathbf{u}}{X}_{\mathbf{i}}\parallel .\hfill \end{array}$$

We have $T_{\mathbf{m}}\le {\sum }_{\mathbf{n}\in I_{\mathbf{m}}}{T}_{\mathbf{m}}^{\mathbf{n}}$ for $\mathbf{m}\in {\mathbb{N}}^d$. Applying the $C_r$ inequality, we have

$$\begin{array}{rcl}\frac{E{(T_{\mathbf{m}})}^p}{a_{\mathrm{\Gamma }(\mathbf{m}+\mathbf{1})}^p{\phi }_2^{p-1}(\mathrm{\Gamma }(\mathbf{m}))}& \le & \frac{c_{\mathbf{m}}^{p-1}}{a_{\mathrm{\Gamma }(\mathbf{m}+\mathbf{1})}^p{\phi }_2{(\mathrm{\Gamma }(\mathbf{m}))}^{p-1}}\sum _{\mathbf{n}\in I_{\mathbf{m}}}E{\left(T_{\mathbf{n}}^{\mathbf{m}}\right)}^p\\ \le & \frac{c_{\mathbf{n}}^{p-1}\cdot C\cdot |{\alpha }^{p-1}(\mathrm{\Gamma }(\mathbf{m}+\mathbf{1}))|}{{(a_{\mathrm{\Gamma }(\mathbf{m}+\mathbf{1})})}^p{({\phi }_2(\mathrm{\Gamma }(\mathbf{m})))}^{p-1}}\sum _{\mathbf{k}\in I_{\mathbf{m}}}\sum _{\mathbf{i}\in {\mathrm{\Delta }}_{\mathbf{n}}^{\mathbf{m}}}E{\parallel X_{\mathbf{i}}\parallel }^p\text{(by Lemma 2.7)}\\ =& \frac{C}{a_{\mathrm{\Gamma }(\mathbf{m}+\mathbf{1})}^p}\sum _{\mathbf{i}\in {\mathrm{\Delta }}^{\mathbf{m}}}E{\parallel X_{\mathbf{i}}\parallel }^p\le C\sum _{\mathbf{i}\in {\mathrm{\Delta }}^{\mathbf{m}}}\frac{1}{a_i^p}E{\parallel X_{\mathbf{i}}\parallel }^p.\end{array}$$

 □

When $\alpha (\mathbf{i})=\mathbf{M}$ for all $\mathbf{i}⪰\mathbf{1}$, with a note that ${\phi }_2(\mathbf{m})=|\mathbf{M}|{\phi }_1(\mathbf{m})$ for all $\mathbf{m}⪰\mathbf{1}$, we have the following corollary.

Corollary 3.3 Let $\mathbb{E}$ be a p-uniformly smooth Banach space ($1\le p\le 2$). Then there exists a positive constant C such that for all strong blockwise-M-adapted random fields $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}}:\mathbf{n}⪰\mathbf{1}\}$ in $\mathbb{E}$ with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{k}},\mathbf{k}⪰\mathbf{1}\}$ and every nondecreasing field of positive constants $\{a_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ we have

$$\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{a_{\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})}^p{\phi }_1^{p-1}(\mathrm{\Gamma }(\mathbf{n}))}\mathbb{E}{T}_{\mathbf{n}}^p\le C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{a_{\mathbf{n}}^p}E{\parallel X_{\mathbf{n}}\parallel }^p.$$

(3.4)

Theorem 3.4 $\mathbb{E}$ is a p-uniformly smooth Banach space ($1\le p\le 2$), $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a blockwise α-strong^∗-adapted random field in $\mathbb{E}$ with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{k}},\mathbf{k}⪰\mathbf{1}\}$. $\{{\psi }_{\mathbf{n}}(x)\in {\mathbb{R}}^{+}\}$ is a field of a positive Borel function which has the following property:

$$C_{\mathbf{n}}\frac{u^{{\lambda }_{\mathbf{n}}}}{v^{{\lambda }_{\mathbf{n}}}}\le \frac{{\psi }_{\mathbf{n}}(u)}{{\psi }_{\mathbf{n}}(v)}\le D_{\mathbf{n}}\frac{u^{{\mu }_{\mathbf{n}}}}{v^{{\mu }_{\mathbf{n}}}}\mathit{\text{for all }}u\ge v>0,$$

(3.5)

where $C_{\mathbf{n}}\ge 1$, $D_{\mathbf{n}}\ge 1$, ${\lambda }_{\mathbf{n}}\ge 1$, $0<{\mu }_{\mathbf{n}}\le p$. $\{a_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a nondecreasing field of positive constants satisfying (2.3). Then there exists a positive constant C such that for all $ϵ>0$, we have

$$\sum _{\mathbf{n}⪰\mathbf{1}}P(T_{\mathbf{n}}\ge ϵa_{\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})}{\phi }_2^{(p-1)/p}(\mathrm{\Gamma }(\mathbf{n})))\le C\sum _{\mathbf{n}⪰\mathbf{1}}{A}_{\mathbf{n}}\frac{E{\psi }_{\mathbf{n}}(\parallel X_{\mathbf{n}}\parallel )}{{\psi }_{\mathbf{n}}(a_{\mathbf{n}})},$$

(3.6)

where $A_{\mathbf{n}}=max\{\frac{1}{C_{\mathbf{n}}},D_{\mathbf{n}}\}$.

Proof For each $\mathbf{n}⪰\mathbf{1}$, set

$$\begin{array}{c}{Y}_{\mathbf{n}}=X_{\mathbf{n}}I(\parallel X_{\mathbf{n}}\parallel \le a_{\mathbf{n}}),Z_{\mathbf{n}}=X_{\mathbf{n}}I(\parallel X_{\mathbf{n}}\parallel >a_{\mathbf{n}}),\hfill \\ U_{\mathbf{n}}=\underset{\mathbf{k}\in {\mathrm{\Delta }}^{\mathbf{n}}}{sup}\parallel \sum _{\mathrm{\Gamma }(\mathbf{n})⪯\mathbf{i}⪯\mathbf{k}}{Y}_{\mathbf{i}}-E(Y_{\mathbf{i}}|{\mathcal{F}}_{i-\alpha (i)}^{\ast })\parallel ,V_{\mathbf{n}}=\underset{\mathbf{k}\in {\mathrm{\Delta }}^{\mathbf{n}}}{sup}\parallel \sum _{\mathrm{\Gamma }(\mathbf{n})⪯\mathbf{i}⪯\mathbf{k}}{Z}_{\mathbf{i}}-E(Z_{\mathbf{i}}|{\mathcal{F}}_{\mathcal{i}-\alpha (\mathcal{i})}^{\ast })\parallel .\hfill \end{array}$$

Since $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a blockwise-α-strong^∗ adapted field with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{m}},\mathbf{m}⪰\mathbf{1}\}$, it is clear that $\{Y_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ and $\{Z_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ are blockwise α-strong adapted fields with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{m}},\mathbf{m}⪰\mathbf{1}\}$. Moreover, for $\mathbf{n}\in {\mathbb{N}}^d$,

$$X_{\mathbf{n}}+E(X_{\mathbf{n}}|{\mathcal{F}}_{\mathbf{n}-\alpha (\mathbf{n})})=(Y_{\mathbf{n}}+E(Y_{\mathbf{n}}|{\mathcal{F}}_{\mathbf{n}-\alpha (\mathbf{n})}))+(Z_{\mathbf{n}}+E(Z_{\mathbf{n}}|{\mathcal{F}}_{\mathbf{n}-\alpha (\mathbf{n})})).$$

Then $T_{\mathbf{n}}\le U_{\mathbf{n}}+V_{\mathbf{n}}$ for all $\mathbf{n}⪰\mathbf{1}$. By the Markov inequality, we have

$$\begin{array}{c}\sum _{\mathbf{n}⪰\mathbf{1}}P(T_{\mathbf{n}}\ge ϵa_{\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})}{\phi }_2^{(p-1)/p}(\mathrm{\Gamma }(\mathbf{n})))\hfill \\ \le \frac{1}{2{ϵ}^p}\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{a_{\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})}^p{\phi }_2^{p-1}(\mathrm{\Gamma }(\mathbf{n}))}\mathbb{E}{U}_{\mathbf{n}}^p+\frac{1}{2ϵ}\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{a_{\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})}}\mathbb{E}{V}_{\mathbf{n}}.\hfill \end{array}$$

(3.7)

By Theorem 3.2, we have

$$\begin{array}{c}\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{a_{\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})}^p{\phi }_2^{p-1}(\mathrm{\Gamma }(\mathbf{n}))}\mathbb{E}{U}_{\mathbf{n}}^p\hfill \\ \le C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{E{\parallel U_{\mathbf{n}}\parallel }^p}{a_{\mathbf{n}}^p}\le C\sum _{\mathbf{n}⪰\mathbf{1}}E{\parallel \frac{Y_{\mathbf{n}}}{a_{\mathbf{n}}}\parallel }^p\hfill \\ \le C\sum _{\mathbf{n}⪰\mathbf{1}}E{\parallel \frac{Y_{\mathbf{n}}}{a_{\mathbf{n}}}\parallel }^{{\mu }_{\mathbf{n}}}\le C\sum _{\mathbf{n}⪰\mathbf{1}}{D}_{\mathbf{n}}\frac{E{\psi }_{\mathbf{n}}(\parallel Y_{\mathbf{n}}\parallel )}{{\psi }_{\mathbf{n}}(a_{\mathbf{n}})}\le C\sum _{\mathbf{n}⪰\mathbf{1}}{A}_{\mathbf{n}}\frac{E{\psi }_{\mathbf{n}}(\parallel X_{\mathbf{n}}\parallel )}{{\psi }_{\mathbf{n}}(a_{\mathbf{n}})}.\hfill \end{array}$$

(3.8)

Next, by Theorem 3.2,

$$\begin{array}{c}\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{a_{\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})}}\mathbb{E}{V}_{\mathbf{n}}\hfill \\ \le C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{E\parallel V_{\mathbf{n}}\parallel }{a_{\mathbf{n}}}\le C\sum _{\mathbf{n}⪰\mathbf{1}}E\parallel \frac{Z_{\mathbf{n}}}{a_{\mathbf{n}}}\parallel \hfill \\ \le C\sum _{\mathbf{n}⪰\mathbf{1}}E{\parallel \frac{Z_{\mathbf{n}}}{a_{\mathbf{n}}}\parallel }^{{\lambda }_{\mathbf{n}}}\le C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{C_{\mathbf{n}}}\frac{E{\psi }_{\mathbf{n}}(\parallel Z_{\mathbf{n}}\parallel )}{{\psi }_{\mathbf{n}}(a_{\mathbf{n}})}\le C\sum _{\mathbf{n}⪰\mathbf{1}}{A}_{\mathbf{n}}\frac{E{\psi }_{\mathbf{n}}(\parallel X_{\mathbf{n}}\parallel )}{{\psi }_{\mathbf{n}}(a_{\mathbf{n}})}.\hfill \end{array}$$

(3.9)

Then (3.7), (3.8), and (3.9) yield (3.6). □

Remark 3.5 The field of functions $\{{\psi }_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ with ${\psi }_{\mathbf{n}}(x)={|x|}^p$ satisfies the property (3.5).

Recall that the field of $\mathbb{E}$-valued random variables $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{N}}^d\}$ is said to be stochastically dominated by an $\mathbb{E}$-valued random variable X if, for some $0<C<\mathrm{\infty }$,

$$P\{\parallel X_{\mathbf{n}}\parallel \ge x\}\le CP\{\parallel X\parallel \ge x\}$$

for all $\mathbf{n}\in {\mathbb{N}}^d$ and $x>0$.

Theorem 3.6 Let $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}};\mathbf{n}\in {\mathbb{N}}^d\}$ be a blockwise-α-strong^∗ adapted field with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{k}},\mathbf{k}⪰\mathbf{1}\}$ in a real separable p-uniformly smooth Banach space $\mathbb{E}$ with $1<p\le 2$. Let ${\alpha }_1,\dots ,{\alpha }_d$ be positive constants satisfying $min\{{\alpha }_1,\dots ,{\alpha }_d\}=1$, let q be the number of integers s such that ${\alpha }_s=1=min\{{\alpha }_1,\dots ,{\alpha }_d\}$. If $\{X_{\mathbf{n}};\mathbf{n}\in {\mathbb{N}}^d\}$ is stochastically dominated by an $\mathbb{E}$-valued random variable X. Then

$$\sum _{\mathbf{n}⪰\mathbf{1}}P(T_n\ge ϵ|{\mathrm{\Gamma }}^{\alpha }(\mathbf{n}+\mathbf{1})\left|{\phi }_2^{(p-1)/p}(\mathrm{\Gamma }(\mathbf{n}))\right)<C(E\parallel X\parallel {log}_{+}^q\parallel X\parallel +1).$$

Proof For each $\mathbf{n}⪰\mathbf{1}$, set

$$\begin{array}{c}{Y}_{\mathbf{n}}=X_{\mathbf{n}}I(\parallel X_{\mathbf{n}}\parallel \le |{\mathbf{n}}^{\alpha }\left|\right),Z_{\mathbf{n}}=X_{\mathbf{n}}I(\parallel X_{\mathbf{n}}\parallel >|{\mathbf{n}}^{\alpha }\left|\right),\hfill \\ U_{\mathbf{n}}=\underset{\mathbf{k}\in {\mathrm{\Delta }}^{\mathbf{n}}}{sup}\parallel \sum _{\mathrm{\Gamma }(\mathbf{n})⪯\mathbf{i}⪯\mathbf{k}}{Y}_{\mathbf{i}}-E(Y_{\mathbf{i}}|{\mathcal{F}}_{i-\alpha (i)}^{\ast })\parallel ,\hfill \\ V_{\mathbf{n}}=\underset{\mathbf{k}\in {\mathrm{\Delta }}^{\mathbf{n}}}{sup}\parallel \sum _{\mathrm{\Gamma }(\mathbf{n})⪯\mathbf{i}⪯\mathbf{k}}{Z}_{\mathbf{i}}-E(Z_{\mathbf{i}}|{\mathcal{F}}_{i-\alpha (i)}^{\ast })\parallel .\hfill \end{array}$$

Since $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a blockwise-α-strong^∗ adapted field with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{m}},\mathbf{m}⪰\mathbf{1}\}$ then it is clear that $\{Y_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ and $\{Z_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ are blockwise-α-strong adapted fields with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{m}},\mathbf{m}⪰\mathbf{1}\}$. Moreover, for $\mathbf{n}\in {\mathbb{N}}^d$,

$$X_{\mathbf{n}}+E(X_{\mathbf{n}}|{\mathcal{F}}_{\mathbf{n}-\alpha (\mathbf{n})})=(Y_{\mathbf{n}}+E(Y_{\mathbf{n}}|{\mathcal{F}}_{\mathbf{n}-\alpha (\mathbf{n})}))+(Z_{\mathbf{n}}+E(Z_{\mathbf{n}}|{\mathcal{F}}_{\mathbf{n}-\alpha (\mathbf{n})})).$$

Then $T_{\mathbf{n}}\le U_{\mathbf{n}}+V_{\mathbf{n}}$ for all $\mathbf{n}⪰\mathbf{1}$. By the Markov inequality, Theorem 3.2, and Lemma 2.9, we have

$$\begin{array}{c}\sum _{\mathbf{n}⪰\mathbf{1}}P(T_{\mathbf{n}}\ge ϵ|\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})\left|{\phi }_2^{(p-1)/p}(\mathrm{\Gamma }(\mathbf{n}))\right)\hfill \\ \le \frac{2^p}{{ϵ}^p}\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{{|\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})|}^p{\phi }_2^{p-1}(\mathrm{\Gamma }(\mathbf{n}))}\mathbb{E}{U}_{\mathbf{n}}^p+\frac{2}{ϵ}\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{|\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})|}\mathbb{E}{V}_{\mathbf{n}}\hfill \\ \le C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{{|{\mathbf{n}}^{\alpha }|}^p}E{\parallel Y_{\mathbf{n}}\parallel }^p+C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{|{\mathbf{n}}^{\alpha }|}E\parallel Z_{{\mathbf{n}}^{\alpha }}\parallel \hfill \\ \le C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{{|{\mathbf{n}}^{\alpha }|}^p}E{\parallel X\parallel }^p{I}_{\{\parallel X\parallel >|{\mathbf{n}}^{\alpha }|\}}+C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{|{\mathbf{n}}^{\alpha }|}E\parallel X\parallel I_{\{\parallel X\parallel \le |{\mathbf{n}}^{\alpha }|\}}\hfill \\ \le C\sum _{\mathbf{n}⪰\mathbf{1}}P\{\parallel X\parallel \ge |{\mathbf{n}}^{\alpha }\left|\right\}+C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{|{\mathbf{n}}^{\alpha }|}{\int }_{|{\mathbf{n}}^{\alpha }|}^{\mathrm{\infty }}P\{\parallel X\parallel \ge t\}dt\hfill \\ +C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{{|{\mathbf{n}}^{\alpha }|}^p}{\int }_0^{{|{\mathbf{n}}^{\alpha }|}^p}P\{{\parallel X\parallel }^p\ge t\}dt\hfill \\ \le C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{|{\mathbf{n}}^{\alpha }|}{\int }_{|{\mathbf{n}}^{\alpha }|}^{\mathrm{\infty }}P\{\parallel X\parallel \ge t\}dt+C\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{{|{\mathbf{n}}^{\alpha }|}^p}{\int }_0^{{|{\mathbf{n}}^{\alpha }|}^p}P\{{\parallel X\parallel }^p\ge t\}dt\hfill \\ \le C(E\parallel X\parallel {log}_{+}^q\parallel X\parallel +1).\hfill \end{array}$$

 □

4 Application to the strong law of large numbers

By applying theorems in Section 3 we establish some results of strong laws of large numbers for fields of blockwise-α-martingale differences with values in a p-uniformly smooth Banach space.

In the rest of this paper, we denote by $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}}:\mathbf{n}⪰\mathbf{1}\}$ the blockwise-α-martingale difference field with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{k}},\mathbf{k}⪰\mathbf{1}\}$. When $\alpha (\mathbf{k})=\mathbf{M}$ for all k, it is called a strong blockwise-M-martingale difference field, and we set

$$S_{\mathbf{k}}=\sum _{\mathbf{1}⪯\mathbf{i}⪯\mathbf{k}}{X}_{\mathbf{i}}.$$

Let $\{a_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ be a nondecreasing field of positive constants such that

$$1<\underset{\mathbf{n}\prec \mathbf{m}⪯\mathbf{n}+\mathbf{1}}{lim\hspace{0.17em}inf}\frac{a_{\mathrm{\Gamma }(\mathbf{m})}}{a_{\mathrm{\Gamma }(\mathbf{n})}}\le \underset{\mathbf{n}\prec \mathbf{m}⪯\mathbf{n}+\mathbf{1}}{lim\hspace{0.17em}sup}\frac{a_{\mathrm{\Gamma }(\mathbf{m})}}{a_{\mathrm{\Gamma }(\mathbf{n})}}<\mathrm{\infty }.$$

(4.1)

Theorem 4.1 Let $1\le p\le 2$, and let $\mathbb{E}$ be a separable Banach space, then the following three statements are equivalent:

• (i) $\mathbb{E}$ is p-uniformly smooth.

• (ii) $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a blockwise-α-martingale difference field in $\mathbb{E}$ with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{k}},\mathbf{k}⪰\mathbf{1}\}$, $\{a_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a nondecreasing field of positive constants satisfying (4.1). If

  $$\sum _{\mathbf{n}⪰\mathbf{1}}\frac{1}{a_{\mathbf{n}}}E{\parallel X_{\mathbf{n}}\parallel }^p<\mathrm{\infty },$$

  (4.2)

then we have

$$\frac{1}{a_{\mathbf{n}}{\varphi }_2^{(p-1)/p}(\mathbf{n})}\underset{\mathbf{1}⪯\mathbf{k}⪯\mathbf{n}}{max}\parallel S_{\mathbf{k}}\parallel \to 0\mathit{\text{a.s. as }}|\mathbf{n}|\to 0.$$

(4.3)

• (iii) $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}}:\mathbf{n}⪰\mathbf{1}\}$ is a blockwise-M-martingale difference field in $\mathbb{E}$ with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{k}},\mathbf{k}⪰\mathbf{1}\}$, $\{a_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a nondecreasing sequence of positive constants satisfying (4.1). If (4.2) holds, then we have

  $$\frac{1}{a_{\mathbf{n}}{({\varphi }_1(\mathbf{n}))}^{(p-1)/p}}\underset{\mathbf{1}⪯\mathbf{k}⪯\mathbf{n}}{max}\parallel S_{\mathbf{k}}\parallel \to 0\mathit{\text{a.s. as }}|\mathbf{n}|\to 0.$$

Proof ((i) ⇒ (ii)) By (4.2) and Theorem 3.2, we have

$$\frac{T_{\mathbf{n}}}{a_{\mathrm{\Gamma }(\mathbf{n}+\mathbf{1})}{\varphi }_2(\mathbf{n})}\to 0\text{a.s.}$$

For $\mathbf{n}⪰\mathbf{1}$, let $\mathbf{m}⪰\mathbf{0}$ be such that $\mathbf{n}\in {\mathrm{\Delta }}^{(\mathbf{m})}$, by Lemma 2.9, we have

$$\begin{array}{rcl}0& \le & \frac{1}{a_{\mathbf{n}}{({\varphi }_2(\mathbf{n}))}^{(p-1)/p}}\underset{\mathbf{k}⪯\mathbf{n}}{sup}\parallel \sum _{\mathbf{1}⪯\mathbf{i}⪯\mathbf{k}}{X}_{\mathbf{i}}\parallel \\ \le & \frac{1}{a_{\mathrm{\Gamma }(\mathbf{m})}}\underset{\mathbf{k}⪯\mathbf{m}}{sup}\parallel \sum _{\mathbf{1}⪯\mathbf{i}⪯\mathbf{k}}{a}_{\mathrm{\Gamma }(\mathbf{i}+1)}\frac{T_{\mathbf{i}}}{a_{\mathrm{\Gamma }(\mathbf{i}+\mathbf{1})}{\varphi }_2(\mathrm{\Gamma }(\mathbf{i}))}\parallel \to 0\text{a.s.}\end{array}$$

((ii) ⇒ (iii)) When $\alpha (\mathbf{k})=\mathbf{M}$ for all $\mathbf{k}⪰\mathbf{1}$, with a note that ${\phi }_2(\mathbf{m})=|\mathbf{M}|{\phi }_1(\mathbf{m})$ for all $\mathbf{m}⪰\mathbf{1}$. We have (4.3).

((iii) ⇒ (i)) Assume that (iii) holds. Let $\{X_n,{\mathcal{G}}_n,n\ge 1\}$ be a martingale difference sequence in $\mathbb{E}$ such that

$$\sum _{n=1}^{\mathrm{\infty }}\frac{E{\parallel X_n\parallel }^p}{n^p}<\mathrm{\infty }.$$

For $\mathbf{n}⪰\mathbf{1}$, put $X_{\mathbf{n}}=X_{n_1}$ if $n_i=1$ ($2\le i\le d$) and $X_{\mathbf{n}}=0$ if there exists a positive integer i ($2\le i\le d$) such that $n_i\ge 2$,

$${\mathcal{F}}_{\mathbf{n}}={\mathcal{G}}_{n_1}\text{for all }\mathbf{n}=(n_1,\dots ,n_d)\in {\mathbb{N}}^d.$$

Then $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a blockwise-1-martingale difference field with respect to the blocks ${\prod }_{k=1}^d[2^k,2^{k+1})$ in $\mathbb{E}$ and ${\sum }_{\mathbf{n}⪰\mathbf{1}}^{\mathrm{\infty }}\frac{E{\parallel X_{\mathbf{n}}\parallel }^p}{{|\mathbf{n}|}^p}={\sum }_{n=1}^{\mathrm{\infty }}\frac{E{\parallel X_n\parallel }^p}{n^p}<\mathrm{\infty }$. Let $a_{\mathbf{n}}=|\mathbf{n}|$ for all $\mathbf{n}⪰\mathbf{1}$. Then (4.1) and (4.2) hold. Thus, by (iii),

$$\underset{|\mathbf{n}|\to \mathrm{\infty }}{lim}\frac{1}{|\mathbf{n}|{({\varphi }_1(\mathbf{n}))}^{(p-1)/p}}\sum _{\mathbf{1}⪯\mathbf{i}⪯\mathbf{n}}{X}_{\mathbf{i}}=0\text{a.s.}$$

Note that ${\varphi }_1(\mathbf{n})=1$, $\mathbf{n}⪰\mathbf{1}$, and we have

$$\underset{|\mathbf{n}|\to \mathrm{\infty }}{lim}\frac{1}{|\mathbf{n}|}\sum _{\mathbf{1}⪯\mathbf{i}⪯\mathbf{n}}{X}_{\mathbf{i}}=0\text{a.s.}$$

Taking $n_i=1$ for all $2\le i\le d$ and letting $n_i\to \mathrm{\infty }$, we obtain

$$\underset{n_1\to \mathrm{\infty }}{lim}\frac{1}{n_1}\sum _{i=1}^{n_1}{X}_i=0\text{a.s.}$$

Then by Theorem 2.2 of Hoffmann-Jørgensen and Pisier [11], $\mathbb{E}$ is p-uniformly smooth. □

Remark 4.2 In Theorem 4.1, when $d=2$, $\alpha (\mathbf{n})=\mathbf{1}$, ${\mathrm{\Delta }}_{\mathbf{k}}={\prod }_{k=1}^d[2^k,2^{k+1})$ we have the result in Theorem 3.2 in [3]. When $d=2$, $\alpha (\mathbf{n})=\mathbf{M}$, $\mathbb{E}=\mathbb{R}$, ${\mathrm{\Delta }}_{\mathbf{k}}={\prod }_{k=1}^d[2^k,2^{k+1})$, $\{X_{\mathbf{n}};\mathbf{n}⪰\mathbf{1}\}$ is a double of mean zero random variables and we have a part of Theorem 3.1 in [6].

$\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}};\mathbf{n}⪰\mathbf{1}\}$ is said to be a strong^∗ blockwise-α-martingale difference field if it is a blockwise-α-strong^∗ adapted field as well as a blockwise-α-martingale difference field.

Theorem 4.3 Let $1\le p\le 2$, and let $\mathbb{E}$ be a separable Banach space, then the following two statements are equivalent:

• (i) $\mathbb{E}$ is p-uniformly smooth.

• (ii) $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a strong^∗ blockwise-α-martingale difference field in $\mathbb{E}$ with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{k}},\mathbf{k}⪰\mathbf{1}\}$, $\{{\psi }_{\mathbf{n}},\mathbf{n}\ge \mathbf{1}\}$ is a field of positive Borel functions satisfying (3.5). If

  $$\sum _{\mathbf{n}⪰\mathbf{1}}{A}_{\mathbf{n}}\frac{E{\psi }_{\mathbf{n}}(\parallel X_{\mathbf{n}}\parallel )}{{\psi }_{\mathbf{n}}(a_{\mathbf{n}})}<\mathrm{\infty },$$

  (4.4)

where $A_{\mathbf{n}}=max\{\frac{1}{C_{\mathbf{n}}},D_{\mathbf{n}}\}$, then we have (4.3).

Proof ((i) ⇒ (ii)) By Theorem 3.4 and by the same argument as in the proof of Theorem 4.1.

((ii) ⇒ (i)) Assume that (ii) holds. Let $\{X_n,{\mathcal{G}}_n,n\ge 1\}$ be a martingale difference sequence in $\mathbb{E}$ such that

$$\sum _{n=1}^{\mathrm{\infty }}\frac{E{\parallel X_n\parallel }^p}{n^p}<\mathrm{\infty }.$$

For $\mathbf{n}⪰\mathbf{1}$, put $X_{\mathbf{n}}=X_{n_1}$ if $n_i=1$ ($2\le i\le d$) and $X_{\mathbf{n}}=0$ if there exists a positive integer i ($2\le i\le d$) such that $n_i\ge 2$,

$${\mathcal{F}}_{\mathbf{n}}={\mathcal{G}}_{n_1}\text{for all }\mathbf{n}=(n_1,\dots ,n_d)\in {\mathbb{N}}^d.$$

Then $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}},\mathbf{n}⪰\mathbf{1}\}$ is a blockwise-1-martingale difference and strong^∗ adapted field with respect to the blocks ${\prod }_{k=1}^d[2^k,2^{k+1})$ in $\mathbb{E}$ and we have ${\sum }_{\mathbf{n}⪰\mathbf{1}}^{\mathrm{\infty }}\frac{E{\parallel X_{\mathbf{n}}\parallel }^p}{{|\mathbf{n}|}^p}={\sum }_{n=1}^{\mathrm{\infty }}\frac{E{\parallel X_n\parallel }^p}{n^p}<\mathrm{\infty }$. Put ${\psi }_{\mathbf{n}}(x)=x^p$, ${\lambda }_{\mathbf{n}}=1$, ${\mu }_{\mathbf{n}}=p$, $C_{\mathbf{n}}=1$, $D_{\mathbf{n}}=1$, $\mathbf{n}⪰\mathbf{1}$ and $a_{\mathbf{n}}=|\mathbf{n}|$ for all $\mathbf{n}⪰\mathbf{1}$. Thus, by (ii) and by the same argument as in the proof of Theorem 4.1, we have (i). □

Theorem 4.4 Let $\{X_{\mathbf{n}},{\mathcal{F}}_{\mathbf{n}};\mathbf{n}⪰\mathbf{1}\}$ be a blockwise-α-strong^∗ adapted field with respect to the blocks $\{{\mathrm{\Delta }}_{\mathbf{k}},\mathbf{k}⪰\mathbf{1}\}$ in a real separable p-uniformly smooth Banach space $\mathbb{E}$ with $1<p\le 2$. Let ${\alpha }_1,\dots ,{\alpha }_d$ be positive constants satisfying $min\{{\alpha }_1,\dots ,{\alpha }_d\}=1$, let q be the number of integers s such that ${\alpha }_s=1=min\{{\alpha }_1,\dots ,{\alpha }_d\}$. If $\{X_{\mathbf{n}};\mathbf{n}\in {\mathbb{N}}^d\}$ is stochastically dominated by an $\mathbb{E}$-valued random variable X such that $E(\parallel X\parallel {log}_{+}^q\parallel X\parallel )<\mathrm{\infty }$. Then

$$\frac{1}{|{\mathbf{n}}^{\alpha }|{\varphi }_2(\mathbf{n})}\underset{\mathbf{1}⪯\mathbf{k}⪯\mathbf{n}}{max}\parallel S_{\mathbf{k}}\parallel \to 0\mathit{\text{a.s. as }}|\mathbf{n}|\to 0.$$

(4.5)

Proof Using Theorem 4.3 and by the same argument as in the proof of Theorem 4.1, we have (4.5). □

Remark 4.5 In Theorem 4.3, when $d=2$, $\alpha (\mathbf{n})=1$, ${\mathrm{\Delta }}_{\mathbf{k}}={\mathrm{\Delta }}^{\mathbf{k}}={\prod }_{k=1}^d[2^k,2^{k+1})$ we have the result in Theorem 3.2(ii) in [7].