---
abstract: |
    We prove martingale-ergodic and ergodic-martingale theorems with continuous parameter for vector valued Bochner integrable functions. We first provide almost everywhere convergence of vector valued martingales with continuous parameter. The norm as well as almost everywhere convergence of martingale-ergodic and ergodic-martingale averages are given. We also obtain dominant and maximal inequalities. Finally, we show that a.e. martingale-ergodic and ergodic-martingale theorems will coincide under certain assumptions.

    0.3cm [*Mathematics Subject Classification*]{}: 28D10, 46G10, 47A35, 60G44.\
    [*Key words*]{}: Continuous parameter, Vector valued martingale , vector valued martingale-ergodic process and ergodic-martingale processes, Bochner integrable functions.
address: |
    [*[Department of Mathematics,\
    The Pennsylvania State University,\
    University Park, 16802, PA, USA]{}*]{}
author:
- 'F.A. Shahidi [^1]'
title: Vector valued unified martingale and ergodic theorems with continuous parameter
---

+6pt

\[section\] \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Definition]{} \[thm\][Remark]{} \[thm\][Problem]{} ø Ø

\

Introduction
============

An interesting connection in terms of the behavior and convergence between two fundamental mathematical objects — martingales and ergodic averages has been known since S. Kakutani [@Kak], who asked for a possible unification of martingale convergence and ergodic theorems. Several attempts have been done since then (see [@Kach2] for review and references), but none of them was comprehensive. Quite recently, A.G. Kachurovskii [@Kach2],[@Kach1] solved this problem by defining a martingale-ergodic processes as the composition of martingales and ergodic averages. For $f\in L_p, p\ge 1,$ if $f_n=E(f|F_n)$ is a regular martingale, where $E(\cdot|F)$ is a conditional expectation operator and $A_mf=\frac1m\sum\limits_{i=0}^{m-1}
T^if,$ where $T$ is an $L_1-L_{\infty}$ contraction, then he proved the following

[@Kach2],[@Kach1].

1.  -   If $f\in L_p,\ p\ge 1,$ then $E(A_mf|F_n)$ converges in $L_p$ norm as $n,m\to\infty;$

    -   If $f\in L_1$ and $sup_n |E(f|F_n)|$ is integrable, then $E(A_mf|F_n)$ converges almost everywhere as $n,m\to\infty$.

2.  -   If $f\in L_p,\ p\ge 1,$ then $A_mE(f|F_n)$ converges in $L_p$ norm as $n,m\to\infty;$

    -   If $f\in L_1$ and $sup_m |A_mf|$ is integrable, then $A_mE(f|F_n)$ converges almost everywhere as $n,m\to\infty$.

While the first part of this theorem is referred as a martingale-ergodic theorem, second part is known as ergodic-martingale theorem. In fact, this theorem puts martingale convergence and ergodic theorems into one superstructure, from which both martingale convergence and ergodic theorems can be obtained as degenerate cases.

The continuous parameter analogue of the above theorem was solved by I.V. Podvigin as follows

[@pod1]

1.  -   If $f\in L_p,\ p\ge 1$ then $E(A_tf|F_s)$ converges in $L_p$ norm as $t,s\to\infty$;

    -   If $f\in L_1$ and $sup_s |E(f|F_s)|$ is integrable, then $E(A_tf|F_s)$ converges almost everywhere as $t,s\to\infty$.

2.  -   If $f\in L_p,\ p\ge 1,$ then $A_tE(f|F_s)$ converges in $L_p$ norm as $t,s\to\infty$;

    -   If $f\in L_1$ and $sup_t |A_tf|$ is integrable, then $A_tE(f|F_s)$ converges almost everywhere as $t,s\to\infty$.

Here $F_s-$ an increasing family of $\sigma-$ subalgebras $A_tf=\frac1t\int\limits_0^tT_{\tau}fd\tau$ and $\{T_t,t\ge 0\}$ is a semigroup of linear $L_1-L_{\infty}$ contractions.

Note that there many analogues and generalizations of martingale convergence and ergodic theorems. For example, vector valued ergodic theorem for $1-$ parameter semigroup of operators was given by Sh. Hasegawa, R. Sato and Sh. Tsurumi in [@Sato1]. The result was also extended to multiparameter case under suitable assumptions in [@Sato2]. Related problems are also considered in [@Sato4]. This motivates us to provide the above theorem in other settings. The purpose of this paper is to give the latter theorem in vector valued settings. Namely, we prove martingale-ergodic and ergodic-martingale theorems with continuous parameter for vector valued Bochner integrable functions. As is done by [@Kach1], [@pod1], we also prove dominant and maximal inequalities. We also show that the condition of integrability of supremum is not necessary under the assumption that conditional expectation operator and ergodic average commute. This is the vector valued analogue of the result given in [@pod2] for continuous parameter processes. We also note that the vector valued analogue of Theorem 1.1 has been considered in [@ShGa].

To our knowledge, we do not seem to have vector valued a.e. martingale convergence theorem with continuous parameter. Hence in the next section we prove this convergence. The main result of the paper is given in section 3. We use the notation and terminology as used in [@pod1], [@ShGa].

Preliminaries
=============

In this section we prove a vector valued martingale convergence theorem with continuous parameter.

Throughout this paper by $X$ we mean a reflexive Banach space with the norm $||\cdot||_X$ and by $(\O ,\beta, \mu)$ a finite measure space. By $L_p(X)=L_p(\O, X), \ 1\le p<\infty$ we denote the Banach space of $X$ valued measurable functions $f$ on $\O$ with the norm defined as

$$||f||_p=\left(\int_{\O}||f(\o)||_X^pd\mu\right)^{\frac 1p}.$$

We just write $L_p$ when $X=R.$

Let $\{T_t, t\ge 0\}$ be a flow of linear $L_1-L_{\infty}$ contractions acting in $L_1(\O, X)$. That is, for any $t\ge 0,$

$||T_tf||_1\le ||f||$ and $||T_tf||_{\infty}\le ||f||_{\infty}$, where $$||f||_1=\int\limits_{\O}||f(\o)||_X d\mu$$ and $$||f||_{\infty}=inf\{\lambda: ||f(\o)||_X\le\lambda \ a.e\}.$$

A flow of linear operators $\{T_t, t\ge 0\}$ in $L_1(\O,X)$ is *strongly continuous semigroup* if

-   $T_0=id$

-   $T_{t_1}T_{t_2}=T_{t_1+t_2}$ for all $t_1,t_2>0$

-   $\lim\limits_{t_1\to t_2}||T_{t_1}f-T_{t_2}f||_1=0$ for any $f\in L_1(\O, X)$ and $t_2>0.$

Henceforth, $\{T_t, t\ge 0\}$ will be a strongly continuous semigroup of linear $L_1-L_{\infty}$ contractions unless otherwise mentioned.

In [@Sato1] it is shown that if $f\in L_p(\O, X), p\ge 1,$ then $\frac1t\int\limits_o^tT_{\tau}f(\o)d\tau\in L_p(\O, X).$ In this settings, we define the ergodic average as follows $$A_tf(\o)=\frac1t\int\limits_o^tT_{\tau}f(\o)d\tau,\ \ f\in L_1(\O, X),\ \ t>0.$$

The following theorem is an a.e. convergence theorem for the above ergodic average.

[@Sato1] Let $X$ be a reflexive Banach space and $\{T_t, t\ge 0\}$ be a strongly continuous semigroup of linear $L_1-L_{\infty}$ contractions on $L_1(\O, X).$ If $1\le p< \infty$ and $f\in L_p(\O, X),$ then the limit $$\lim\limits_{t\to\infty}\frac1t\int\limits_o^tT_{\tau}f(\o)d\tau$$ exists for almost all $\o\in\O.$

It is to note that the above theorems were given for slightly general type of operators $\{T_t\},$ that is, the operators $\{T_t\}$ should be contractions with respect to $L_1$ norm and bounded with respect to $L_{\infty}$ norm.

Let $F$ be a $\sigma-$ algebra and $F_1$ be its $\sigma-$ subalgebra.

[@neveu]

1.  There exists a linear operator $E(\cdot|F):L_1(\O, X)\rightarrow L_1(\O, X)$ such that $$\int\limits_BE(f|F)d\mu=\int\limits_Bfd\mu$$ for any $f\in L_1(\O, X)$ and $B\subset F_1.$

2.  For every continuous linear functional $g$ and $f\in L_1(\O, X),$ the function $g(f)$ is integrable and $$g(E(f|F))=E(g(f)|F).$$

By $E(f|F)$ we denote the conditional expectation of $f\in L_p.$ Let $F_s,\ s\in R$ be a family of monotonically increasing (decreasing) sub-$\sigma-$algebras such that $F_s\uparrow F_{\infty}$ ($F_s\downarrow F_{\infty}$) as $s\to\infty.$ Unless otherwise stated, we assume that the family of sub-$\sigma-$ algebras is increasing. We also keep in mind that the results in this section, which hold for increasing family also hold for decreasing family of sub-$\sigma-$algebras. A stochastic process $f_s$ in $L_p(\O, X), \ 1\le p<\infty$ is said to be an *ordinary (reversed) martingale* if for all $s_1, s_2\in S$ with $s_1<s_2 (s_1>s_2)$ one has $E(f_{s_2}|F_{s_1})=f_{s_1}.$ A *regular* martingale is given by $f_s=E(f|F_s),$ where $f\in L_p(\O, X), \ 1\le p<\infty.$ There is a norm convergence theorem for vector valued martingales with continuous parameter [@Vaxan]. But, we were not able to find any theorem concerning a.e. convergence for them. Below we are going to provide this convergence.

Let $\{(g_s^i, s\in R), i\in I\}$ be a countable family of real valued submartingales such that $$\sup\limits_{s\in R}\int\sup\limits_{i\in I}(g_s^i)^+d\mu< \infty .$$

Then each submartingale converge a.e. to an integrable limit $g_{\infty}^i,\ i\in I$ and $$\sup\limits_{i\in I}g_s^i=\sup\limits_{i\in I}g_{\infty}^i$$ as $s\to\infty.$

The condition of the lemma implies that $\sup\limits_{s\in R}\int(g_s^i)^+d\mu$ is finite for all $i\in I,$ therefore by Doob’s convergence theorem for submartingales (see, for example [@Oksendal], Appendix C) the limits $$g_{\infty}^i=\lim\limits_{s\to\infty}g_s^i$$ exists a.e. Since $g_s^i$ is a submartingale for all $i\in I,$ then $\sup\limits_{i\in I}g_s^i$ is also a submartingale. Due to the condition of the lemma and Doob’s convergence theorem for submartingales (see [@Oksendal])we conclude again that the limit

$$g_{\infty}=\lim\limits_{s\to\infty}\sup\limits_{i\in I}(g_s^i)$$ exists a.e. This limit clearly dominates each $g_{\infty}^i(i\in
I)$ and thus also their supremum, i.e. $g_{\infty}\ge\sup\limits_{i\in I}g_{\infty}^i.$

We will show that $\int g_{\infty}d\mu=\int\sup\limits_{i\in
I}(g_{\infty}^i)d\mu$ in order to show that the above inequality in fact an equality.

Let $(I_p), p\in N$ be a sequence of finite subsets of $I$ increasing to $I$ as $p\to\infty.$ Then the integral $\int\sup\limits_{i\in I_p}g_s^id\mu$ clearly increases as $p$ increases. Moreover, it also increases with $s (s\in R)$ since $(\sup\limits_{i\in I_p}g_s^i,\ s\in R)$ is a submartingale for every $p.$

Note that the expression

$$S=\sup\limits_{p\in N, s\in R}\int\sup\limits_{i\in I_p}g_s^id\mu=
\sup\limits_{s\in R}\int\sup\limits_{i\in I}g_s^id\mu$$ is dominated by $\sup\limits_{s\in R}\int\sup\limits_{i\in
 I}(g_s^i)^+d\mu$ and hence is finite. Therefore, for every $\e>0$ there exists at least one pair $p_{\e}\in N,\ s_{\e}\in R^+$ such that $$\int\sup\limits_{i\in I_p}g_s^id\mu\ge S-\e$$ if $p=p_{\e},\ s=s_{\e}.$ Since the above supremum increases with $p$ as well as with $s,$ then the above inequality holds for $p\ge
p_{\e},\ s\ge s_{\e}.$ Note that the function $g_{\infty}-\sup\limits_{i\in I_p}g_{\infty}^i$ is the limit of positive sequence of functions $(\sup\limits_{i\in I}g_s^i-
\sup\limits_{i\in I_p}g_s^i,\ s\in R)$ so that Fatou’s lemma implies that

$$\int(g_{\infty}-\sup\limits_{i\in I_p}g_{\infty}^i)d\mu\le\liminf\limits_{s\to\infty}\int((\sup\limits_{i\in I}g_s^i-
\sup\limits_{i\in I_p}g_s^i)d\mu\le S-(S-\e)=\e.$$

Therefore, $\int(g_{\infty}-\sup\limits_{i\in
I_p}g_{\infty}^i)d\mu\le\e$ and so $g_{\infty}=\sup\limits_{i\in
I}g_{\infty}^i.$

Let $X$ be a separable Banach space which is the dual of a separable Banach space and $F_s$ be an increasing family of sub-$\sigma-$algebras. Then for any $f\in L_1(\O, X)$

$$\lim\limits_{s\to\infty}E(f|F_s)= E(f|F_{\infty})$$ a.e. on $\O.$

Note that every separable reflexive Banach space satisfies the condition put on $X.$

Firstly, note that for any continuous linear functional $g\in X',$ the sequence $g(E(f|F_s))$ is a martingale as (2) of Theorem 2.2 shows that for $s_2>s_1$ $$g(E(f|F_{s_2}))=E(g(f)|F_{s_2})=E(g(f)|F_{s_1})=g(E(f|F_{s_1})).$$ One can also see that for any $g\in X'$

$$g(E(f|F_s))=E(g(f)|F_s)\rightarrow E(g(f)|F_{\infty})=g(E(f|F_{\infty}))$$ outside a set $\O_g$(which actually depends on $g$) of zero measure as $s\to\infty$ by convergence of read valued martingale [@Oksendal].

Now assume that the separable Banach space $X$ is the dual of a (necessarily) separable space $Y$ and let us identify this space with the subspace of $X',$ the dual of $X.$ Let us denote by $D$ a dense subset of unit ball in $Y$ which we can choose countable as $Y$ is separable. Then the equality $\sup\limits_{g\in
D}g(x)=||x||_X$ holds for all $x\in X. $ Indeed, one can see that $g(x)\le ||g||||x||_X$ implies $||x||_X\ge\frac{g(x)}{||g||},$ and so $||x||_X\ge \sup\limits_{g\in D}\frac{g(x)}{||g||}.$ Since there exist $x_0\in X$ and $g_0\in X'$ such that $g_0(x_0)=||x_0||_X||g_0||,$ then $||x||_X= \sup\limits_{g\in
D}\frac{g(x)}{||g||}.$

Further, take any fixed $a\in X,$ and consider the countable family of martingales $$\{(g(E(f|F_s)-a), s\in R), g\in D\}.$$

Since $$|g(E(f|F_s))|\le ||E(f|F_s)||_X\le E(||f||_X|F_s)$$ for all $g\in D$ by contraction property of the conditional expectation, then the above family satisfies the condition of Lemma 2.3 and hence by applying it we get

$$||E(f|F_s)-a||_X\rightarrow ||E(f|F_{\infty})-a||_X$$ a.e. as $s\to\infty,$ for all $a\in X.$ From this it follows that

$$\mu\{\lim\limits_{s\to\infty}||E(f|F_s)-a||_X= ||E(f|F_{\infty})-a||_X\ \forall a\in X\}=1$$

Since $X$ is separable and we can take $a=E(f(\o)|F_s)$ at every $\o\in\O,$ we find that $E(f|F_s)\rightarrow E(f|F_{\infty})$ a.e. as $s\to\infty.$

Martingale-ergodic and ergodic-martingale theorems
==================================================

In this section we prove norm as well as a.e. convergence for vector valued martingale-ergodic and ergodic-martingale averages with continuous parameter. In this section we consider only regular martingales.

Following Kachurovskii [@Kach2], we define martingale-ergodic and ergodic-martingale averages as follows.

A *martingale-ergodic* average is an average of the form $\{E(A_tf|F_s)\}_{t>0, s\ge 0},$ where $E(\cdot|F_s)$ is the conditional expectation operator and $A_tf$ is the ergodic average while an *ergodic-martingale* average is an average $A_tE(f|F_s).$

Let us introduce the following notations $$f_{\infty}(\o)=\lim\limits_{t\to\infty}A_tf(\o),$$ $$f^*(\o)=\lim\limits_{s\to\infty}E(f_{\infty}|F_s), \ \ f_*(\o)=\lim\limits_{t\to\infty}A_tE(f|F_{\infty})(\o).$$ The existence of above limit will be discussed below.

For $f\in L_p(\O, X),\ p\ge 1$ the following assertions hold.

1.  $$E(A_tf|F_s)\rightarrow f^*$$ in norm as $t,s\rightarrow \infty;$

2.  $$A_tE(f|F_s)\rightarrow f_*$$ in norm as $t,s\rightarrow \infty.$

The idea is the same with real valued cases [@Kach2], [@pod1].

Note that $$||E(A_tf|F_s)-f^*||_p\le ||E(A_tf|F_s)-E(f_{\infty}|F_s)||_p+||E(f_{\infty}|F_s)-f^*||_p$$

The expression $||E(f_{\infty}|F_s)-f^*||_p$ converges due to vector valued norm convergence theorem for continuous parameter martingales [@Vaxan].

Note that

$$||E(A_tf|F_s)-E(f_{\infty}|F_s)||_p=||E(A_tf-f_{\infty})|F_s)||_p\le ||A_tf-f_{\infty}||_p.$$

Since $||A_tf-f_{\infty}||_p$ convergent according to vector valued ergodic theorem 2.1.5 [@kren], then we get the assertion (1).

Now, we prove the second part. According to Riesz convexity theorem [@kren], [@Phil] an $L_1-L_{\infty}$ contraction is a contraction in $L_p$ norm. Therefore, we have the following estimate

$$||A_tE(f|F_s)-f_*||_p\le ||A_tE(f|F_s)-A_tE(f|F_{\infty})||_p+||A_tE(f|F_{\infty})-f_*||_p\le$$ $$\le ||E(f|F_s)-E(f|F_{\infty})||_p+||A_tE(f|F_{\infty})-f_*||_p.$$

The norm $||E(f|F_s)-E(f|F_{\infty})||_p$ converges due to vector valued norm convergence theorem for continuous parameter martingales [@Vaxan], and the norm $||A_tE(f|F_{\infty})-f_*||_p$ from theorem 2.1.5 of [@kren].

We say that a linear operator $T$ in $L_1(\O, X)$ is *positively dominated* if there exists a positive linear contraction $T'$ in $L_1,$ called a *positive dominant* of $T,$ such that $$||Tf||_X\le T'(||f||_X).$$

Let us now provide some useful examples that we will use (see [@SuchF]).

1\. If $X=R$, then it is positively dominated by some positive linear contraction on $L_1.$ For the vector valued $T$, a positive dominant may not exist in general.

2\. Let $\tau$ be a measure preserving transformation on $(\O
,\beta, \mu).$ Then the linear operator $T:L_1(\O, X)\rightarrow
L_1(\O, X)$ given by $Tf=f\circ\tau$ is said to be generated by $\tau.$ $T$ is positively dominated by $T'$ with $T'(||f||_X)=||f||_X\circ\tau.$

3\. Assume that the Banach space $X$ has the Radon-Nikodym property(the Banach space is said to have the Radon-Nykodim property with respect to $(\O, \beta, \mu)$ if any vector measure $\phi:\beta\to X$ with finite variation, which is absolutely continuous with respect to $\mu$ is just the integral of countable valued function $f:\O\to X$ ). If $X$ is reflexive, then it has the Radon-Nykodim property [@Vaxan]. Consider the conditional expectation $E(f|F)$ with respect to $\sigma-$ subalgebra $F$ of $\beta.$ For $f\in L_1(\O, X),$ the conditional expectation $E(f|F)$ is Radon-Nikodym density with respect to the finite measure $\mu$ on $F.$ Since $||E(f|F)||_X\le E'(||f||_X|F)$ a.e. for all $f\in L_1(\O, X),$ where $E'(\cdot|F)$ is a conditional expectation on $L_1,$ then the operator $E(\cdot|F)$ is positively dominated by $E'(\cdot|F).$

We say that the flow $\{T_t, t\ge 0\}$ in $L_1(\O, X)$ is positively dominated by the flow $\{P_t, t\ge 0\}$ in $L_1$ if for any $f\in L_1(\O, X)$ and $t\ge0$ one has $||T_tf||_X\le
P_t(||f||_X)$ a.e. Now, we provide a.e. convergence theorem.

Let $X$ be a separable Banach space. Assume that $\{T_t, t\ge 0\}$ is positively dominated by some semigroup $\{P_t, t\ge 0\}$ of strongly continuous linear $L_1-L_{\infty}$ contractions. Then for the function $f\in L_1(\O, X)$ the following assertions hold true.

1.  If $\sup\limits_{t>0}||A_tf||_X\in L_1 $ (this holds for example, if $f\in L(\O, X)logL(\O, X)$ ) then for any $t>0, s\ge 0,$ $E(A_tf|F_s)\rightarrow f^*$ a.e. as $t,s\rightarrow \infty.$

2.  If $\sup\limits_{s\ge 0}||E(f|F_s)||_X\in L_1,$ then $A_tE(f|F_s)\rightarrow
        f_*$ a.e. as $t,s\rightarrow \infty.$

We prove the first assertion. Note that

$$||E(A_tf|F_s)-f^*||_X\le ||E(A_tf|F_s)-E(f_{\infty}|F_s)||_X+||E(f_{\infty}|F_s)-f^*||_X.$$

According to martingale convergence Theorem 2.4, the norm $||E(f_{\infty}|F_s)-f^*||_X$ converges to $0$ a.e. as $s\to\infty.$

(Let $0<t_1\le t.$) Further, since conditional expectation operator is positively dominated, then

$$||E(A_tf|F_s)-E(f_{\infty}|F_s)||_X=||E(A_tf-f_{\infty}|F_s)||_X\le E'(||A_tf-f_{\infty}||_X|F_s)\le E'(h_{t_1}|F_s),$$ where $h_{t_1}(\o)=\sup\limits_{t\ge
t_1}||A_tf(\o)-f_{\infty}(\o)||_X$ and $E'$ is a positive dominant of $E.$ Due to the condition of the theorem, we have $h_{t_1}\in
L_1$ and $h_{t_1}\to 0$ a.e. from Theorem 2.1. Now applying first part of Theorem 1.2, $E'(h_{t_1}|F_s)\rightarrow 0,$ a.e. as $t_1\to\infty.$ Therefore, we have $||E(A_tf|F_s)-E(f_{\infty}|F_s)||_X\rightarrow 0,$ a.e. Hence, $||E(A_tf|F_s)-f^*||_X\rightarrow 0$ a.e. as $t,s\to\infty.$

Now we prove the second part. We have

$$||A_tE(f|F_s)-f_*||_X\le ||A_tE(f|F_s)-A_tE(f|F_{\infty})||_X+||A_tE(f|F_{\infty})-f_*||_X$$

The norm $||A_tE(f|F_{\infty})-f_*||_X$ is a.e. convergent due to Theorem 2.1.

We have the following $$||A_tE(f|F_s)-A_tE(f|F_{\infty})||_X=||\frac1t\int\limits_0^tT_{\tau}(E(f|F_s)-E(f|F_{\infty}))d\tau||_X\le$$ $$\le \frac1t\int\limits_0^t||T_{\tau}(E(f|F_s)-E(f|F_{\infty}))||_Xd\tau\le$$$$\le \frac1t\int\limits_0^tP_{\tau}\big(||(E(f|F_s)-E(f|F_{\infty}))||_X\big)d\tau=$$ $$=A'_t\big(||(E(f|F_s)-E(f|F_{\infty}))||_X\big)$$

where $P_{t}$ is a positive dominant of $T_t$ for each $t$ and $A'_tf=\frac1t\int\limits_0^tP_{\tau}fd\tau.$ According to our assumption, the flow $\{P_t, t\ge 0\}$ is strongly continuous semigroup.

Note that the real valued function $h_s(\o)=||(E(f(\o)|F_s)-E(f_{\infty}(\o)|F_s))||_X$ is integrable according to the conditions of theorem. Moreover, according to the martingale convergence Theorem 2.3 $h_s(\o)\rightarrow 0$ a.e. as $s\to\infty.$

Now applying second part of Theorem 1.2, we get $A'_t(h_s)\rightarrow 0$ a.e. as $t,s\to\infty.$ Therefore, $||A_tE(f|F_s)-A_tE(f|F_{\infty})||_X\rightarrow 0$ a.e. as $s,t\to\infty.$

**Remark**. When we consider real valued functions, that is when $X=R,$ then for any semigroup $\{T_t, t\ge 0\}$ of linear $L_1-L_{\infty}$ contractions there always exists a semigroup $\{P_t, t\ge 0\}$ of positive linear $L_1-L_{\infty}$ contractions such that $|T_tf|\le P_t|f|$ a.e. However, in vector valued positive dominant semigroup may not exist in general. It is also known that $\{T_t, t\ge 0\}$ is not positively dominated by its linear modulus [@Sato3]. Therefore in the above theorem, despite real valued case, we need an additional assumption that $\{T_t, t\ge 0\}$ should be positively dominated by $\{P_t, t\ge
0\}.$ Of course one can ask to provide the above theorems without this condition, but we fail to answer to this question.

The following theorem is dominant and maximal inequalities for martingale-ergodic processes.

Under the assumption of Theorem 3.2, $f\in L_p(\O, X),\ p>1,$ $\sup\limits_{t>0}||A_tf||_X\in L_1$ and $F_s\downarrow F,\ s\to\infty$ then the following assertions hold true.

1.  $$||\sup\limits_{t,s}||E(A_tf|F_s)||_X||_p\le \big(\frac p{p-1}\big)^2||f||_p,$$

2.  $$\mu\big\{\sup\limits_{t,s}||E(A_tf|F_s)||_X\ge \varepsilon\big\}\le \frac p{p-1}\frac{||f||_p}{\varepsilon}$$

We first prove the dominant inequality. Note that the conditional expectation operator is positively dominated, then

$$||\sup\limits_{t,s}||E(A_tf|F_s)||_X||_p\le||\sup\limits_{t,s}E'(||A_tf||_X|F_s)||_p\,$$

where $E'$ is a positive dominant of $E.$

Since $\{T_t, \ t\ge 0\}$ is positively dominated by $\{P_t, \
t\ge 0\}$, then

$$||A_tf||_X=||\frac1t\int\limits_0^tT_tfd\tau||_X\le \frac1t\int\limits_0^t||T_tf||_Xd\tau\le$$ $$\le \frac1t\int\limits_0^tP_t(||f||_X)d\tau=A'_t(||f||_X).$$

Since $E$ is positively dominated by $E'$ and $A_t$ by $A'_t,$ then we have the following inequality. $$\sup\limits_{t,s}E'(||A_tf||_X|F_s)||_p\le \sup\limits_{t,s}E'(A'_t(||f||_X)|F_s)||_p,$$

Since the flow $P_t$ is a strongly continuous semigroup, applying Theorem 3 of [@pod1] for the process $E'(A'_t(||f||_X)|F_s)$, we get

$$||\sup\limits_{t,s}E'(A'_t(||f||_X)|F_s)||_p\le \big(\frac p{p-1}\big)^2||f||_p.$$

The above chain of inequalities imply part (1) of the theorem.

Now we prove part (2). Since the operator $E$ is positively dominated by some $E'$ , then we have the following inequalities

$$\mu\big\{\sup\limits_{t,s}||E(A_tf|F_s)||_X\ge \varepsilon\big\}\le \mu\big\{\sup\limits_{t,s}E'(||A_tf||_X|F_s)\ge \varepsilon\big\}\le$$ $$\le \mu\big\{\sup\limits_{t,s}E'(A'_t||f||_X|F_s)\ge \varepsilon\big\}$$ where $A'_t(||f||_X)=\frac1t\int\limits_0^tP_{\tau}(||f||_X)d\tau.$ Now applying second part of Theorem 3 of [@pod1], for the process $E'(A'_t(||f||_X)|F_s)$, we get

$$\mu\big\{\sup\limits_{t,s}E'(A'_t||f||_X|F_s)\ge \varepsilon\big\}\le \frac p{p-1}\frac{||f||_p}{\varepsilon}.$$

Hence (2) is proved.

Now, we provide dominant and maximal inequalities for ergodic-martingale average.

Under the assumption of Theorem 3.2, $f\in L_p(\O, X),\ p>1,$ $\sup\limits_{s\ge0}||E(f|F_s)||_X\in L_1$ and $F_s\downarrow F,\ s\to\infty.$ and the following assertions hold true.

1.  $$||\sup\limits_{t,s}||A_tE(f|F_s)||_X||_p\le \big(\frac p{p-1}\big)^2||f||_p,$$

2.  $$\mu\big\{\sup\limits_{t,s}||A_tE(f|F_s)||_X\ge \varepsilon\big\}\le \frac p{p-1}\frac{||f||_p}{\varepsilon}.$$

This theorem can easily be proven using Theorem 4 of [@pod1] and the way of proof of Theorem 3.3. So we omit details.

It is known that in $L_1,$ the condition of integrability of supremum can not be omitted in all unified theorem [@ArgRos]. The following theorem is given without this assumption, but the conditional expectation operator and ergodic average should commute.

Let $F_s\downarrow F,\ s\to\infty$ and $T_t$ be a semigroup of strongly continuous measure preserving transformation and $T_tE(f|F_s)=E(T_tf|F_s),$ for all $t,s\ge 0.$ Then for any $f\in L_1(\O, X),$ the averages $A_tE(f|F_s)$ and $E(A_tf|F_s)$ converge a.e. as $t,s\to\infty.$

The idea is almost the same as Theorem 4 of [@pod2].

Let $n=[t],$ then $n=t+\a,$ where $0\le\a<1.$ For any $t>0,\ s\ge
0$ we have

$$A_tE(f|F_s)=\frac1t\int\limits_0^tT_{\tau}E(f|F_s)d\tau=\frac1t\int\limits_0^nT_{\tau}E(f|F_s)d\tau+\frac1t\int\limits_n^{n+\a}T_{\tau}E(f|F_s)d\tau=$$ $$=\frac1t\sum\limits_{k=0}^{n-1}\int\limits_k^{k+1}T_{\tau}E(f|F_s)d\tau+\frac1t\int\limits_n^{n+\a}T_{\tau}E(f|F_s)d\tau=$$ $$=\frac1t\sum\limits_{k=0}^{n-1}\int\limits_0^1T_{\tau+k}E(f|F_s)d\tau+\frac1t\int\limits_0^{\a}T_{\tau+n}E(f|F_s)d\tau=$$ $$=\frac1t\sum\limits_{k=0}^{n-1}T_k\int\limits_0^1T_{\tau}E(f|F_s)d\tau+\frac1tT_n\int\limits_0^{\a}T_{\tau}E(f|F_s)d\tau=$$ $$=\frac1t\sum\limits_{k=0}^{n-1}(T_1)^kE(A_1f|F_s)d\tau+\frac{\a}t(T_1)^nA_{\a}E(f|F_s)=$$ $$=\frac nt[S_n(T_1)E(g_1|F_s)+\frac{\a}n(T_1)^nA_{\a}E(f|F_s)],$$ where $g_1=A_1f$ and $S_n(T)f=\frac
1n\sum\limits_{i=0}^{n-1}T_if.$

Now let us estimate the expressions $S_n(T_1)E(g_1|F_s)$ and $\frac{\a}n(T_1)^nA_{\a}E(f|F_s).$ Evidently, the former is a.e. convergent. If $P^1$ and $E'$ be positive dominants of $T_1$ and $E$ respectively, then the latter converges a.e. since $$||\frac{\a}n(T_1)^nA_{\a}E(f|F_s)||_X\le \frac 1n (P^1)^nE'(A_1||f||_X|F_s)=$$ $$=\frac{n+1}nS_{n+1}(P^1)E'(||f||_X|F_s)-S_n(P^1)E'(||f||_X|F_s)\rightarrow 0$$ a.e. as $s,n\to\infty$ from Theorem 1.2.

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[^1]: Email: farruh.shahidi@@gmail.com
