The inequalities of partial sums have been studied by a lot of authors. The moments of random variables play an important role in limit theory of random variable sequence. By this the Marcinkiewicz-Zygmund-Burkholder type inequality, Kolmogorov type inequality, Rosenthal type inequality and Bernstein inequality are discussed.

In 1970, Rosenthal [1] proved that for real valued independent random variables, the following inequality is true

$\begin{equation} E|\sum\limits_{k=1}^n X_k|^r \leq B_r \max\{ \sum\limits_{k=1}^n E|X_k|^r,\quad (\sum\limits_{k=1}^nE|X_k|^2)^{r/2}\}, \end{equation}$

(1.1)

where $\{X_k; 1\leq k\leq n\}$ are independent random variables with zero mean, $r>2$ and $B_r$ is a positive constant only depending on $r.$

Martingale difference can be regarded as the generalization of independent random variables and some classical inequalities can also be generalized such as Rosenthal type inequality [2].

$\begin{eqnarray}&& c_1\{ E\biggl[\bigl(\sum\limits_{i=1}^n E(X_i^2|\Sigma_{i-1})\bigl)^{r/2}\biggl]+\sum\limits_{i=1}^nE|X_i|^r\}\nonumber\\ &\leq & E |f_n|^r \leq c_2\{ E\biggl[\bigl(\sum\limits_{i=1}^n E(X_i^2|\Sigma_{i-1})\bigl)^{r/2}\biggl]+\sum\limits_{i=1}^nE|X_i|^r\}, \end{eqnarray}$

(1.2)

where $(f_i, \Sigma_i, 1\leq i\leq n)$ is a real valued martingale, $X_1=f_1 ,X_i=f_i-f_{i-1}, i=2,3,\cdots,n$, $ 2\leq r<\infty $, $ c_1$ and $c_2$ are all positive constants only depending on $r$.

In 1981, De Acosta [3] proved the following inequality for Banach valued independent random variables which can be regarded as the generalization of Rosenthal type inequality:

$\begin{equation}E\biggl|\|f_n\|-E\|f_n\|\biggl |^r \leq c_r \{ (\sum\limits_{i=1}^n E\|X_i\|^2)^{r/2}+\sum\limits_{i=1}^n E\|X_i\|^r\}, \end{equation}$

(1.3)

where $r>2,$ $ (X_i; 1\leq i\leq n)$ are all independent random variables in Banach space and $f_n=\sum\limits_{i=1}^n X_i, n\geq 1$.

Let $(\Omega, \Sigma, P)$ be a probability space, $(X, \|\cdot\|)$ be a Banach space and $(\Sigma_n, n\geq -1 )$ be a increasing $\sigma$-sub-algebra sequence in $\Sigma$, where $\Sigma=\cup_{n\geq -1}\Sigma_n,\ \Sigma_{-1}=\{\Omega, \emptyset\}.$

Definition 1 [4] Let $1\leq \alpha<\infty.$ $f=(f_n,\Sigma_n, n\geq 0)$ is called a $\alpha$ quasi-martingale if

$\sum\limits_{n=1}^\infty \|E(f_{n+1}|\Sigma_n)-f_n\|_\alpha<\infty.$

When $\alpha=1,$ $(f_n,\Sigma_n, n\geq 0)$ is called a quasi-martingale.

Definition 2 [4] Banach space $X$ is called $p$ smoothable, if there exists a constant $c>0$ such that $ \rho_X(\tau)\leq c\tau^p, \tau>0,$ where

$\rho_X(\tau)=\sup\{\frac{\|x+y\|+\|x-y\|}{2}-1: \|x\|=1, \|y\|=\tau\}.$

Definition 3 [4] Banach space $X$ is called $q$ convexiable, if there exists a constant $c>0$ such that $ \delta_X(\varepsilon)\geq c\varepsilon ^q,\quad 0\leq \varepsilon\leq 2,$ where

$\delta_X(\varepsilon)=\inf \{1-\frac{\|x+y\|}{2}: \|x\|=\|y\|=1, \|x-y\|=\varepsilon\}.$

In this paper the following notations will be used.

Let $0<p<\infty$ and $ f=(f_n,n\geq 0)$ be an adapted process,

$\begin{eqnarray*}f_n^*&=&\sup\limits_{n\geq k\geq 0}\|f_k\|, \quad f^*=\sup\limits_{n\geq 0}\|f_n\|;\\ df_n&=&f_{n}-f_{n-1},\quad n\geq 0,\quad f_{-1}=0;\\ S^{(p)}(f)&=&(\sum\limits_{i=0}^\infty\|df_i\|^p)^{1/p},\quad S_n^{(p)}(f)=(\sum\limits_{i=0}^n\|df_i\|^p)^{1/p};\\ \sigma^{(p)}(f)&=&\bigl(\sum\limits_{i=0}^\infty E(\|df_i\|^p\bigl|\Sigma_{i-1})\bigl)^{1/p},\quad \sigma_n^{(p)}(f)=\bigl(\sum\limits_{i=0}^n E(\|df_i\|^p\bigl|\Sigma_{i-1})\bigl)^{1/p};\\ R_n(f)&=&\sum\limits_{i=0}^n \|E(df_i|\Sigma_{i-1})\|, \quad R(f)=\sup\limits_{n\geq 0}R_n(f). \end{eqnarray*}$

In this paper, the constants $c$ may denote different constants in different contexts.

Lemma 1 [2] Let $r>1, \beta >1, \delta >0, $ $\xi$ and $\eta$ be two non-negative random variables. If for all $\delta>0$ small enough, there exists constant $\varepsilon_\delta$ satisfying $\lim\limits_{\delta\rightarrow 0}\varepsilon_\delta=0$, such that

$P(\xi>\beta \lambda, \eta\leq \delta \lambda)\leq \varepsilon_\delta P(\xi>\lambda), \quad \forall \lambda>0,$

then there exists a constant $c$ such that $E(\xi^r)\leq cE(\eta^r).$

Lemma 2 [5] Let $X$ be a Banach space. Then the following statements are equivalent:

(1) $ X$ is $p$ smoothable;

(2) There exists a constant $c>0$ such that for every $X$-valued quasi-martingale $f=(f_n,\Sigma_n,n\geq 0)$, $\|f^*\|_p\leq c\bigl(\|\sigma^{(p)}(f)\|_p+\|R(f)\|_p\bigl).$

Lemma 3 [5] Let $X$ be a Banach space. Then the following statements are equivalent:

(1) $X$ is $q$ convexifiable;

(2) There exists a constant $c>0$ such that for every $X$-valued quasi-martingale $f=(f_n,\Sigma_n,n\geq 0)$, $\|S^{(q)}(f)\|_q\leq c\bigl(\|f^*\|_q\leq +\|R(f)\|_q\bigl).$

Remark If $\|\cdot\|_q$ is replaced by any $\|\cdot\|_r, \ r\geq q$ in Lemma 3, the conclusion is also true.

Theorem 1 Let $X$ be a Banach space, $1<p\leq 2,\ p\leq r<\infty$. Then the following statements are equivalent:

(1) $X$ is $p$ smoothable;

(2) There exists a constant $c>0$ only depending on $p$ and $r$ such that for every $X$-valued quasi-martingale $f=(f_n,\Sigma_n, n\geq 0)$,

$\|f_n^*\|^r_r\leq c\bigl(\|\sigma_n^{(p)}(f)\|_r^r+\|R_n(f)\|_r^r+\sum\limits_{i=1}^nE(\|df_i\|^r)\bigl), \quad \forall n\geq 1.$

Proof $(1)\Longrightarrow (2)$ Suppose $X$ is $p$ smoothable. Let $\xi=\max_{i\leq n}\|f_i\|,$

$\eta=\max\{ \sigma^{(p)}_n(f), R_n(f), \max_{i\leq n }\|df_i\|\}$

and $\varepsilon=\delta^p/(\beta-\delta-1)^p,$ where $\beta>1, 0<\delta<\beta-1.$

Let

$ A_k=\{\omega| \lambda<\max_{i\leq k-1}\|f_i(\omega)\|\leq \beta \lambda, \quad \max_{i\leq k-1}\|df_i(\omega)\|\leq \delta \lambda, \max\{\sigma^{(p)}_k(f)(\omega), R_k(f)(\omega)\}\leq \delta\lambda\}$

and $T_i=\sum\limits_{k=1}^i \chi_{A_k}df_k, 1\leq i\leq n.$ Then $T=(T_i, \Sigma_i,\ 1\leq i\leq n)$ is an $X$-valued quasi-martingale. Now we let

$k_0=\inf \{k; \lambda<\max_{i\leq k-1}\|f_i(\omega)\|, 1\leq k\leq n\}, n_0=\sup\{k;\max_{i\leq k-1}\|f_i(\omega)\|\leq \beta \lambda\}.$

Then $1\leq k_0<n_0\leq n$ and $\|f_{k_0-1}\|>\lambda ,\ \ \|f_{n_0}\|>\beta\lambda.$

On the set $\{\xi>\beta \lambda, \ \eta\leq\delta \lambda\}$, we have $T_n=\sum\limits_{i=k_0}^{n_0}df_i=f_{n_0}-df_{k_0}+f_{k_0-1}$ and

$\begin{equation}\|T_n\|\geq \|f_{n_0}\|-\|df_{k_0}\|-\|f_{k_0-1}\|\geq (\beta-\delta-1)\lambda. \end{equation}$

(3.1)

Moreover

$\begin{equation}P(\xi>\beta \lambda, \eta\leq \delta \lambda)\leq P(\|T_n\|\geq (\beta-\delta-1)\lambda)\leq (\beta-\delta-1)^{-p}\lambda^{-p}E(T^*)^p.\end{equation}$

(3.2)

Since $X$ is $p$ smoothable, by Lemma 2 we have

$E(T^*)^p\leq c\bigl(\|\sigma^{(p)}(T)\|_p+\|R(T)\|_p)^{p}\leq c\bigl( \|\sigma^{(p)}(T)\|^p_p+\|R(T)\|_p^{p}\bigl). $

By calculation we have

$\begin{eqnarray}\|\sigma^{(p)}(T)\|^p_p&=&E[\sum\limits_{k=1}^n\chi_{A_k}E(\|df_k\|^p|\Sigma_{k-1})]\notag\\ &\leq& E[\sum\limits_{k=1}^n\chi_{\{\omega| \lambda<\max_{i\leq k-1}\|f_i(\omega)\|,\ \sigma^{(p)}_n(f)(\omega)\leq \delta\lambda\}}E(\|df_k\|^p|\Sigma_{k-1})]\notag\\ &\leq& c\delta^p\lambda^p P(\max_{i\leq n}\|f_i\|>\lambda) \end{eqnarray}$

(3.3)

and

$\begin{eqnarray} \|R(T)\|_p^{p}&=&\bigl\|\sum\limits_{k=1}^{n}\|E(dT_k|\Sigma_{k-1})\|\bigl\|^p_p=\bigl\|\sum\limits_{k=1}^{n}\chi_{A_k}\|E(df_k|\Sigma_{k-1})\|\bigl\|_p^p\notag\\ &\leq& E[\sum\limits_{k=1}^n\chi_{\{\omega| \lambda<\max_{i\leq k-1}\|f_i(\omega)\|,\ R_k(f)(\omega)\leq \delta\lambda\}}\|E(df_k|\Sigma_{k-1})\|]^p\notag\\& \leq& c\delta^p\lambda^p P(\max_{i\leq n}\|f_i\|>\lambda). \end{eqnarray}$

(3.4)

By (3.2), (3.3) and (3.4) we have $P(\xi>\beta \lambda, \eta\leq \delta \lambda)\leq c(\beta-\delta-1)^{-p}\delta^p P(\max\limits_{i\leq n}\|f_i\|>\lambda)$, where $\lim\limits_{\delta\rightarrow 0}\varepsilon=\lim\limits_{\delta\rightarrow 0}(\beta-\delta-1)^{-p}\delta^p =0.$ By Lemma 1, we have $\forall n\geq 1,$

$\begin{eqnarray} \|f_n^*\|^r_r=E(\xi^r)&\leq& c E(\eta^r)\leq c\bigl(E(\sigma_n^{(p)}(f)^r)+E((\max_{i\leq n}\|df_i\|)^r)+E(R_n(f)^r)\bigl)\notag\\ &\leq&c\bigl(E(\sigma_n^{(p)}(f)^r)+\sum\limits_{i=1}^nE(\|df_i\|^r)+E(R_n(f)^r)\bigl)\notag\\ &=& c\bigl(\|\sigma_n^{(p)}(f)\|_r^r+\|R_n(f)\|_r^r+\sum\limits_{i=1}^nE(\|df_i\|^r)\bigl). \end{eqnarray}$

(3.5)

$(2)\Longrightarrow (1)$ Suppose (2) is true. Let $r=p$. For all $n\geq 1$ we have

$\begin{eqnarray} \|f_n^*\|_p&\leq& c\bigl(\|\sigma_n^{(p)}(f)\|_p^p+\|R_n(f)\|_p^p+\sum\limits_{i=1}^nE(\|df_i\|^p)\bigl)^{1/p}\notag\\ &\leq& c\bigl( \|\sigma_n^{(p)}(f)\|_p+\|R_n(f)\|_p+E(\sum\limits_{i=1}^n\|df_i\|^p)^{1/p}\bigl)\notag\\ &=&c\bigl( \|\sigma_n^{(p)}(f)\|_p+\|R_n(f)\|_p+E(\sum\limits_{i=1}^nE(\|df_i\|^p|\Sigma_{i-1}\bigl)^{1/p}\bigl)\notag\\ &\leq& c\bigl(\|\sigma_n^{(p)}(f)\|_p+\|R_n(f)\|_p\bigl). \end{eqnarray}$

(3.6)

By Lemma 2, $X$ is $p$-smoothable.

Theorem 2 Let $X$ be a Banach space, $2\leq q<\infty, q\leq r<\infty$. Then the following statements are equivalent:

(1) $ X$ is $q$ convexifiable;

(2) There exists a constant $c>0$ only depending on $p$ and $r$ such that for every $X$-valued quasi-martingale $f=(f_n,\Sigma_n, n\geq 0)$,

$ \|\sigma_n^{(q)}(f)\|_r^r+\sum\limits_{i=1}^nE(\|df_i\|^r) \leq c\{\|f_n^*\|^r_r +\|R_n(f)\|_r^r \},\quad \forall n\geq 1.$

Proof  $(1)\Longrightarrow (2)$ Suppose $X$ is $q$-convexifiable, by Remark there exists a constant $c>0$ such that for every X-valued quasi-martingale $f=(f_n,\Sigma_n,n\geq 0)$

$\begin{equation}\|S^{(q)}(f)\|_r\leq c(\|f^*\|_r +\|R(f)\|_r), \quad r\geq q.\end{equation}$

(3.7)

By the fact $\sum\limits_{i=1}^nE( \|df_i\|^r)=E(\sum\limits_{i=1}^n \|df_i\|^r) \leq E\bigl((\sum\limits_{i=1}^n \|df_i\|^q)^{r/q}=\|S_n^{(q)}(f)\|_r^r\bigl),\quad r\geq q$ we have

$\begin{eqnarray}\|\sigma_n^{(q)}(f)\|_r^r+\sum\limits_{i=1}^nE(\|df_i\|^r)&\leq& c\|S_n^{(q)}(f)\|_r^r+\sum\limits_{i=1}^nE(\|df_i\|^r)\notag\\ &\leq& c\|S_n^{(q)}(f)\|_r^r\leq cE(\max_{i\leq n}\|f_i\|^r)+\|R_n(f)\|_r^r. \end{eqnarray}$

(3.8)

$(2)\Longrightarrow (1)$ Suppose (2) is true. Let $r=q$. By the fact

$\|S_n^{(q)}(f)\|_q^q=E(\sum\limits_{i=0}^n\|df_i\|^q)=\|\sigma_n^{(q)}(f)\|_q^q,$

we have

$ \|S_n^{(q)}(f)\|_q^q=1/2(\|\sigma_n^{(q)}(f)\|_q^q+\sum\limits_{i=1}^nE(\|df_i\|^q)) \leq c\{\|f_n^*\|^q_q +\|R_n(f)\|_q^q \}.$

Thus

$\|S^{(q)}(f)\|_q\leq c\bigl(\|f^*\|_q\leq +\|R(f)\|_q\bigl).$

By Lemma 3 $X$ is $q$-convexifiable.

Corollary 1 Let $X$ be a Banach space, $2\leq r<\infty$, Then the following statements are equivalent:

(1) $X$ is a Hilbert space;

(2) There exists a constant $c$ such that

$\begin{eqnarray*}&& c^{-1} \bigl(\|\sigma_n^{(2)}(f)\|_r^r +\sum\limits_{i=1}^n E(\|df_i\|^r) \bigl )\\ &\leq& \|f_n^*\|^r_r +\|R_n(f)\|_r^r \leq c \bigl(\|\sigma_n^{(2)}(f)\|_r^r +\sum\limits_{i=1}^n E(\|df_i\|^r)+\|R_n(f)\|_r^r \bigl), \forall n\geq 1.\end{eqnarray*}$

Theorem 3 Let $X$ be a $p$-smoothable Banach space, $1<p\leq 2, p\leq r<\infty$. If $f=(f_n,\Sigma_n, n\geq 0)$ is an $X$-valued quasi-martingale satisfying $\sum\limits_{n=1}^\infty (\frac{E(\|df_n\|^r|\Sigma_{n-1})}{n^r})^{1/r}<\infty\quad {\rm a.s.},$ then $\frac{1}{n}f_n=\frac{1}{n}\sum\limits_{i=1}^n df_i\rightarrow 0 \quad {\rm a.s.}$.

Proof Since $\sum\limits_{n=1}^\infty (\frac{E(\|df_n\|^r|\Sigma_{n-1})}{n^r})^{1/r}<\infty\quad {\rm a.s.}$ for $0< \varepsilon<1$, there exists $\mathbf{N}$, such that

$\sum\limits_{n=\mathbf{N}+1}^\infty (\frac{E(\|df_n\|^r|\Sigma_{n-1})}{n^r})^{1/r}\leq \varepsilon<1 \quad {\rm a.s.}.$

Now, for any $ m>n\geq \mathbf{N}$, let $dg_{i}=\frac{df_i}{i},$ then $\sum\limits_{i=n+1}^m\frac{df_i}{i}=\sum\limits_{i=n+1}^mdg_i=g_m-g_n:=g^{(n)}_m.$ By the fact $dg^{(n)}_i=g^{(n)}_i-g^{(n)}_{i-1}=g_i-g_{i-1}=dg_i=\frac{df_i}{i}, (i>n)$ and

$\begin{eqnarray*}&&\sum\limits_{m=n+1}^\infty \|E(g^{(n)}_{m+1}|\Sigma_m)-g^{(n)}_m\|_1 =\sum\limits_{m=n+1}^\infty \|\frac{E(df_{m+1}|\Sigma_m)}{m+1}\|_1\\ &\leq&\sum\limits_{m=n+1}^\infty \|E(f_{m+1}|\Sigma_m)-f_m)\|_1<\infty,\end{eqnarray*}$

we know $g^{(n)}=(g^{(n)}_m, m\geq n)$ is a quasi-martingale.

By Theorem 1, we have

$\|g^{(n)}_m\|^r_r \leq c\bigl(\sum\limits_{i=n+1}^mE(\|dg^{(n)}_i\|^r)+\|\sigma_m^{(p)}(g^{(n)})\|_r^r+\|R_m(g^{(n)})\|_r^r\bigl),$

i.e.,

$\begin{eqnarray}&&E\|\sum\limits_{i=n+1}^m\frac{df_i}{i}\|^r\notag\\ &\leq& c\{ \sum\limits_{i=n+1}^m\frac{E\|df_i\|^r}{i^r}+E\bigl(\sum\limits_{i=n+1}^m \frac{ E(\|df_i\|^p|\Sigma_{i-1})}{i^p}\bigl)^{r/p}+E\bigl(\sum\limits_{i=n+1}^m \frac{\|E(df_i|\Sigma_{i-1})\|} {i} \bigl)^r\}.\notag\\ &=&:I+II+III. \end{eqnarray}$

(3.9)

Since

$\begin{equation*}\lim\limits_{n\rightarrow \infty }\frac{E(\|df_i\|^r|\Sigma_{i-1})}{i^r}=0 \quad {\rm a.s.},\end{equation*}$

then

$\begin{equation}\sup\limits_{n\geq 1} \frac{E(\|df_n\|^r|\Sigma_{n-1})}{n^r}<\infty \ {\rm a.s.}. \end{equation}$

(3.10)

Thus we have

$\begin{eqnarray}I&=&\sum\limits_{i=n+1}^\infty\frac{E(\|df_i\|^r)}{i^r}\notag\\ &=&E( \sum\limits_{i=n+1}^\infty\frac{E(\|df_i\|^r|\Sigma_{i-1})}{i^r})\notag\\ &=&E( \sum\limits_{i=n+1}^\infty(\frac{E(\|df_i\|^r|\Sigma_{i-1})}{i^r})^{1/r }(\frac{E(\|df_i\|^r|\Sigma_{i-1})}{i^r})^{1-1/r })\notag\\ &\leq& E((\sup\limits_{n\geq 1} \frac{E(\|df_n\|^r|\Sigma_{n-1})}{n^r})^{1-1/r}\sum\limits_{i=n+1}^\infty(\frac{E(\|df_i\|^r|\Sigma_{i-1})}{i^r})^{1/r })\notag\\ &\rightarrow& 0. \end{eqnarray}$

(3.11)

Since $r\geq p$, by Jensen inequality

$ \frac{ E(\|df_i\|^p|\Sigma_{i-1})}{i^p}\leq (\frac{ E(\|df_i\|^r|\Sigma_{i-1})}{i^r})^{p/r}\leq (\frac{ E(\|df_i\|^r|\Sigma_{i-1})}{i^r})^{1/r}, \quad \forall i\geq 1,$

then

$\begin{equation}II=E\bigl(\sum\limits_{i=n+1}^m \frac{ E(\|df_i\|^p|\Sigma_{i-1})}{i^p}\bigl)\leq E\bigl( \sum\limits_{i=n+1}^m(\frac{ E(\|df_i\|^r|\Sigma_{i-1})}{i^r})^{1/r}\bigl)\rightarrow 0 ( n\rightarrow \infty).\end{equation}$

(3.12)

Since

$\begin{eqnarray*}\sum\limits_{i=n+1}^m \frac{\|E(df_i|\Sigma_{i-1})\|} {i}&\leq& \sum\limits_{i=n+1}^m \frac{E(\|df_i\|\bigl|\Sigma_{i-1})} {i}\\ &\leq& \sum\limits_{i=n+1}^m (\frac{E(\|df_i\|^r|\Sigma_{i-1})} {i^r})^{1/r}\rightarrow 0 \quad (n\rightarrow \infty).\end{eqnarray*}$

Then

$\begin{equation}III=E\bigl( \sum\limits_{i=n+1}^m \frac{\|E(df_i|\Sigma_{i-1})\|} {i}\bigl)^r\rightarrow 0\quad (n\rightarrow \infty).\end{equation}$

(3.13)

By (14), (15) and (16) we have $\lim\limits_{n\rightarrow \infty}E||\sum\limits_{i=n+1}^m \frac{df_i}{i}\|^r=0$ a.s. and $\sum\limits_{i=1}^m\frac{df_i}{i}$ is convergent almost everywhere. By Kronecher lemma $\frac{1}{n}f_n=\frac{1}{n}\sum\limits_{i=1}^n df_i\rightarrow 0$ a.s..