# Some noncommutative subsequential weighted individual ergodic theorems

Morgan O’Brien  North Dakota State University
Department of Mathematics
1210 Albrecht Boulevard, Minard Hall
Fargo, ND 58102, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: March 30, 2021)

###### Abstract.

This article is devoted to studying individual ergodic theorems for
subsequential weighted ergodic averages on the noncommutative $L_{p}$-spaces
associated to a semifinite von Neumann algebra $\mathcal{M}$. In particular,
we establish the convergence of these averages along sequences with density
one and certain types of block sequences with positive lower density, and we
extend known results along uniform sequences in the sense of Brunel and Keane.

###### Key words and phrases:

Semifinite von Neumann algebra, noncommutative ergodic theorem, positive
density sequence, block sequence, uniform sequence, bounded Besicovitch
sequence

###### 2020 Mathematics Subject Classification:

47A35, 46L52

##  1\. Introduction

Beginning with the subsequential ergodic theorems of Blum-Hanson (for norm
convergence [1]) and Brunel-Keane (for a.e. convergence [2]), proving the norm
or a.e. convergence of subsequential or weighted averages has been an active
focus of research in ergodic theory. Some of these results have been extended
to the von Neumann algebra setting.

Let $\mathcal{M}$ be a semifinite von Neumann algebra acting on a Hilbert
space $\mathcal{H}$ (containing the identity operator, denoted by $1$) with a
normal semifinite faithful (n.s.f.) trace $\tau$, and let
$L_{1}(\mathcal{M},\tau)$ be the space of $\tau$-measurable operators
affiliated to $\mathcal{M}$ such that $\|x\|_{1}:=\tau(|x|)<\infty$.

Let $T:L_{1}(\mathcal{M},\tau)+\mathcal{M}\to
L_{1}(\mathcal{M},\tau)+\mathcal{M}$ be a positive linear map which contracts
$L_{1}(\mathcal{M},\tau)$ and $\mathcal{M}$. If $L_{p}(\mathcal{M},\tau)$ is
the associated noncommutative $L_{p}$-space to $\mathcal{M}$, then $T$ extends
to a positive linear contraction on $L_{p}(\mathcal{M},\tau)$ for every $1\leq
p\leq\infty$.

Given $1\leq p\leq\infty$ and $x\in L_{p}(\mathcal{M},\tau)$, let

$M_{n}(T)(x)=\frac{1}{n}\sum_{j=0}^{n-1}T^{j}(x)$

be the $n$-th ergodic average of $x$ associated to $T$. If
$\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}$ is a strictly increasing sequence of
nonnegative integers, let

$M_{n}^{\textbf{k}}(T)(x)=\frac{1}{n}\sum_{j=0}^{n-1}T^{k_{j}}(x)$

denote the $n$-th subsequential ergodic average of $x$ associated to $T$ and
k, or just the $n$-th subsequential average for short.

In [11] it was proven that the averages $M_{n}(T)(x)$ converge almost
uniformly (a.u.) when $T$ is an automorphism on $\mathcal{M}$, $\tau$ is a
faithful $T$-invariant state, and $x\in\mathcal{M}$. It was shown in [21] that
$\tau$ can be replaced by a n.s.f. trace (i.e. not necessarily a state),
$x\in\mathcal{M}$ by $x\in L_{1}(\mathcal{M},\tau)$, and $T$ by a positive
Dunford Schwartz operator, at the price of the convergence holding bilaterally
almost uniformly (b.a.u.). This result was then extended in [9] to hold on
$L_{p}(\mathcal{M},\tau)$ for each $1\leq p<\infty$, and to hold a.u. whenever
$2\leq p<\infty$. Alternative proofs of both of these results are given in
[12] using the notions of uniform and bilaterally uniform equicontinuity in
measure at zero. In [13] it was proven that the subsequential ergodic averages
$\\{M_{n}^{\textbf{k}}(T)_{n}(x)\\}_{n=1}^{\infty}$ associated to a uniform
sequence $\\{k_{j}\\}_{j=0}^{\infty}$ (see Section 4) converge b.a.u. for
every $x\in L_{1}(\mathcal{M},\tau)$.

If $\beta=\\{\beta_{k}\\}_{k=0}^{\infty}$ is a sequence of complex numbers,
then

$M_{n}^{\beta}(T)(x)=\frac{1}{n}\sum_{k=0}^{n-1}\beta_{k}T^{k}(x)$

will be called the $n$-th weighted ergodic average of $x$, and the $n$-th
subsequential weighted ergodic average of $x$ is defined as

$M_{n}^{\beta,\textbf{k}}(T)(x)=\frac{1}{n}\sum_{j=0}^{n-1}\beta_{k_{j}}T^{k_{j}}(x).$

In this article, we will prove that the subsequential weighted averages are
b.u.e.m. at zero on $(L_{p}(\mathcal{M},\tau),\|\cdot\|_{p})$ for each $1\leq
p<\infty$ and u.e.m. at zero for $2\leq p<\infty$ when
$\\{k_{j}\\}_{j=0}^{\infty}$ is a sequence with positive lower density and
$\\{\beta_{j}\\}_{j=0}^{\infty}$ is bounded (where we write u.e.m. or b.u.e.m.
at zero for uniform or bilaterally uniform equicontinuity in measure at zero,
which are given in detail in Definition 2.1). Utilizing this fact, we will
prove that the subsequential weighted averages along any sequence of density
one converge a.u. (b.a.u.) in $L_{p}$ for $2\leq p<\infty$ ($1\leq p<\infty$)
for weights which work for the full sequence. After that, we extend the
results of Litvinov and Mukhamedov by showing that the subsequential weighted
averages along a uniform sequence converge a.u. (b.a.u.) in $L_{p}$ for $2\leq
p<\infty$ ($1\leq p<\infty$) for certain weights. Finally, we prove that the
subsequential weighted averages along certain types of block sequences with
positive density converge a.u. (b.a.u.) in $L_{p}$ for $2\leq p<\infty$
($1\leq p<\infty)$ when weighted by a bounded Besicovich sequence.

##  2\. Preliminaries

We will begin by providing a brief review of the notation and properties of
noncommutative integration theory that will be used in the sequel. For more in
depth details and proofs of these results, see [8], [14], or [19].

Let $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on
$\mathcal{H}$, with the operator norm $\|\cdot\|_{\infty}$. Let
$x:\mathcal{D}_{x}\to\mathcal{H}$ be a closed densely defined linear operator
on $\mathcal{H}$, where $\mathcal{D}_{x}\subseteq\mathcal{H}$ is the domain of
$x$. Then $x$ is said to be affiliated with $\mathcal{M}$ if $yx\subseteq xy$
for every $y\in\mathcal{M}^{\prime}$, where $\mathcal{M}^{\prime}$ is the
commutant $\\{y\in\mathcal{B}(\mathcal{H}):zy=yz\text{ for every
}z\in\mathcal{M}\\}$ of $\mathcal{M}$. Let
$\mathcal{P}(\mathcal{M})=\\{e\in\mathcal{M}:e=e^{*}=e^{2}\\}$ denote the set
of projections in $\mathcal{M}$, and if $e\in\mathcal{P}(\mathcal{M})$, we
will write $e^{\perp}=1-e$, where $1$ is the identity of $\mathcal{M}$.

A closed densely defined operator $x$ affiliated with $\mathcal{M}$ is called
$\tau$-measurable if for every $\epsilon>0$ there exists
$e\in\mathcal{P}(\mathcal{M})$ such that $\tau(e^{\perp})\leq\epsilon$ and
$e(\mathcal{H})\subseteq\mathcal{D}_{x}$. Let $L_{0}(\mathcal{M},\tau)$ denote
the space of $\tau$-measurable operators affiliated with $\mathcal{M}$.

For every $\epsilon,\delta>0$, let

$V(\epsilon,\delta)=\\{x\in
L_{0}(\mathcal{M},\tau):\|xe\|_{\infty}\leq\epsilon\text{ for some
}e\in\mathcal{P}(\mathcal{M})\text{ with }\tau(e^{\perp})\leq\delta\\}.$

The measure topology on $L_{0}(\mathcal{M},\tau)$ is the topology generated by
the system of neighborhoods of zero
$\\{V(\epsilon,\delta):\epsilon,\delta>0\\}$. With this,
$L_{0}(\mathcal{M},\tau)$ is a complete metrizable topological $*$-algebra
containing $\mathcal{M}$ as a dense subspace [14].

If $\mathcal{S}\subseteq L_{0}(\mathcal{M},\tau)$, then we will let
$\mathcal{S}^{+}=\\{x\in\mathcal{S}:x\geq 0\\}$. If $x$ is a positive, self-
adjoint operator affiliated with $\mathcal{M}$ and $x=\int_{0}^{\infty}\lambda
de_{\lambda}$ is its spectral decomposition, then

$\tau(x)=\sup_{n\geq 1}\tau\Big{(}\int_{0}^{n}\lambda de_{\lambda}\Big{)}.$

Given $1\leq p<\infty$, we will let

$L_{p}(\mathcal{M},\tau)=\\{x\in
L_{0}(\mathcal{M},\tau):\tau(|x|^{p})<\infty\\},$

where $|x|=(x^{*}x)^{1/2}$ is the absolute value of $x$. Define
$\|x\|_{p}=\tau(|x|^{p})^{1/p}$ whenever $x\in L_{p}(\mathcal{M},\tau)$ and
$1\leq p<\infty$. We will let $L_{\infty}(\mathcal{M},\tau)=\mathcal{M}$,
equipped with the operator norm $\|\cdot\|_{\infty}$. Then $\|\cdot\|_{p}$ is
a norm on $L_{p}(\mathcal{M},\tau)$ which turns it into a Banach space
whenever $1\leq p\leq\infty$. Note that $L_{1}\cap\mathcal{M}$ and
$L_{p}\cap\mathcal{M}$ are both dense in $L_{p}(\mathcal{M},\tau)$ for $1\leq
p<\infty$ with respect to the norm $\|\cdot\|_{p}$.

The semifinite von Neumann algebra “of interest” throughout this paper will
always be denoted by $\mathcal{M}$ and its n.s.f. trace by $\tau$. As such, we
will often write $L_{p}=L_{p}(\mathcal{M},\tau)$ to ease notation whenever
$1\leq p<\infty$ or $p=0$ (and when confusion will not occur).

We say that $\\{x_{n}\\}_{n=1}^{\infty}\subset L_{0}$ converges to $x\in
L_{0}$ almost uniformly (a.u.) (respectively, bilaterally almost uniformly
(b.a.u.)) if for every $\epsilon>0$ there exists
$e\in\mathcal{P}(\mathcal{M})$ such that

$\tau(e^{\perp})\leq\epsilon\text{ \ and \ }\|(x_{n}-x)e\|_{\infty}\to 0\
(\text{resp., }\|e(x_{n}-x)e\|_{\infty}\to 0)\text{ as }n\to\infty.$

Clearly a.u. convergence implies b.a.u. convergence; however, in the
commutative case a.u. and b.a.u. convergences are equivalent. As shown in
Example 3.1 of [5], this is not the case in general.

######  Definition 2.1.

Let $(X,\|\cdot\|)$ be a normed space. A sequence of maps $A_{n}:X\to L_{0}$,
$n=1,2,\dots$, is said to be bilaterally uniformly equicontinous in measure
(b.u.e.m.) (uniformly equicontinuous in measure (u.e.m.)) at zero if for every
$\epsilon>0$ and $\delta>0$ there exists $\gamma>0$ such that, given $x\in X$
with $\|x\|<\gamma$, there exists $e\in\mathcal{P}(\mathcal{M})$ such that

$\tau(e^{\perp})\leq\epsilon,\ \text{ and }\
\sup_{n}\|eA_{n}(x)e\|_{\infty}\leq\delta\ (\text{resp.,\
}\sup_{n}\|A_{n}(x)e\|_{\infty}\leq\delta).$

A linear map $T:L_{1}+\mathcal{M}\to L_{1}+\mathcal{M}$ such that

$\|T(x)\|_{1}\leq\|x\|_{1}\ \ \forall\ x\in L_{1}\ \text{ and }\
\|T(x)\|_{\infty}\leq\|x\|_{\infty}\ \ \forall\ x\in\mathcal{M}$

is called a Dunford-Schwartz operator. If $T(x)\geq 0$ whenever $x\geq 0$,
then $T$ is called a positive Dunford-Schwartz operator, and we write $T\in
DS^{+}(\mathcal{M},\tau)$, or just $T\in DS^{+}$.

Note that, as was shown in Proposition 1.1 of [4], absolute contractions
considered in [21], and later in [13], can be uniquely extended to positive
Dunford-Schwartz operators. Also, it was shown in Lemma 1.1 of [9] that any
$T\in DS^{+}$ can be extended uniquely to a positive linear contraction on
$L_{p}$ for each $1<p<\infty$. Thus, denoting these extensions by $T$, we have
$\|T(x)\|_{p}\leq\|x\|_{p}$ for all $x\in L_{p}$.

Given $T\in DS^{+}$ and $1\leq p<\infty$, the positive linear maps
$M_{n}(T):L_{p}\to L_{p}$ are continuous, and are also continuous when viewed
as maps from $L_{p}$ to $L_{0}$ when $L_{p}$ has its norm topology and $L_{0}$
has the topology of convergence in measure.

Results regarding the u.e.m. and b.u.e.m. at zero of
$\\{M_{n}(T)\\}_{n=1}^{\infty}$ above were established in [12] (as
Propositions 4.1, 4.2, 4.3 and Theorem 2.1), and Proposition 2.2 below is
Theorem 2.1 of [12]. We state these results here for ease of reference.

######  Proposition 2.1.

The sequence $\\{M_{n}(T)\\}_{n=1}^{\infty}$ is b.u.e.m. (u.e.m.) at zero on
$(L_{p},\|\cdot\|_{p})$ if $1\leq p<\infty$ (respectively $2\leq p<\infty$).

######  Proposition 2.2.

Let $(X,\|\cdot\|)$ be a Banach space, $A_{n}:X\to L_{0}$ be a sequence of
linear maps that is b.u.e.m. (u.e.m.) at zero on $X$. Then the set

$C=\\{x\in X:\\{A_{n}(x)\\}\text{ converges b.a.u. (a.u.)}\\}$

is closed in $X$.

The notions of u.e.m. and b.u.e.m. at zero were used in [12] to simplify the
ending of the proof of b.a.u. convergence in the individual ergodic theorem on
$L_{1}$ from Theorem 2 of [21] and to provide alternative proofs of the
extension of that result to noncommutative $L_{p}$-spaces (first proved in
[9]).

######  Theorem 2.1.

[21, 9]. Let $\mathcal{M}$ be a semifinite von Neumann algebra with a normal
semifinite faithful trace $\tau$, and let $T\in DS^{+}$. If $1\leq p<\infty$,
then the averages $M_{n}(T)(x)$ converge b.a.u. to some $\widehat{x}\in
L_{p}(\mathcal{M},\tau)$ for every $x\in L_{p}(\mathcal{M},\tau)$.
Furthermore, this convergence occurs a.u. when $2\leq p<\infty$.

##  3\. Convergence along Sequences of Density One

For clarity, $\mathbb{N}$ will denote the set of natural numbers,
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$, and $\mathbb{C}$ will stand for the
set of complex numbers. Let $\chi_{E}:X\to\\{0,1\\}$ be the characteristic
function of a set $E\subseteq X$.

For convenience, we will let $\textbf{1}=\\{1\\}_{j=0}^{\infty}$; in what
follows, $\beta=\textbf{1}$ will correspond to the ‘non-weighted’ case of the
weighted results. Also, we will also identify a sequence
$\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}$ of nonnegative integers with its image
$\\{k_{j}:j\in\mathbb{N}_{0}\\}$ in $\mathbb{N}_{0}$.

######  Definition 3.1.

Let $T\in DS^{+}(\mathcal{M},\tau)$ and $1\leq p<\infty$. A sequence
$\beta\subset\mathbb{C}$ will be called a.u.- (b.a.u.-) good for $T$ on
$L_{p}(\mathcal{M},\tau)$ if the weighted averages
$\\{M_{n}^{\beta}(T)(x)\\}_{n=1}^{\infty}$ converge a.u. (respectively,
b.a.u.) for every $x\in L_{p}(\mathcal{M},\tau)$. Similarly, we will say that
a strictly increasing sequence $\textbf{k}\subset\mathbb{N}_{0}$ is a.u.-
(b.a.u.-) good for $T$ and $\beta$ on $L_{p}(\mathcal{M},\tau)$ if the
subsequential weighted averages
$\\{M_{n}^{\beta,\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ converge a.u.
(respectively b.a.u.) for every $x\in L_{p}(\mathcal{M},\tau)$.

A sequence $\beta$ will be called a.u.- (b.a.u.-) good in
$L_{p}(\mathcal{M},\tau)$ if $\beta$ is a.u.- (b.a.u.-) good for $T$ in
$L_{p}(\mathcal{M},\tau)$ for every $T\in DS^{+}(\mathcal{M},\tau)$.
Similarly, define k as a.u.- (b.a.u.-) good for $\beta$ in
$L_{p}(\mathcal{M},\tau)$ if it is a.u.- (b.a.u.-) good for $T$ and $\beta$ in
$L_{p}(\mathcal{M},\tau)$ for every $T\in DS^{+}(\mathcal{M},\tau)$.

######  Example 3.1.

As stated in Theorem 2.1, for each von Neumann algebra $\mathcal{M}$ with
n.s.f. trace $\tau$, the sequence $\beta=\textbf{1}$ is a.u.- (b.a.u.-) good
in $L_{p}(\mathcal{M},\tau)$ for every $2\leq p<\infty$ (respectively, $1\leq
p<\infty$).

######  Example 3.2.

Let $\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$ denote the unit circle in
$\mathbb{C}$. A function $P:\mathbb{Z}\to\mathbb{C}$ is called a trigonometric
polynomial if there exists $\\{r_{j}\\}_{j=1}^{s}\subset\mathbb{C}$ and
$\\{\lambda_{j}\\}_{j=1}^{s}\subset\mathbb{T}$ such that
$P(k)=\sum_{j=1}^{s}r_{j}\lambda_{j}^{k}$ for every $k\in\mathbb{Z}$.

A sequence $\\{\beta_{k}\\}_{k=0}^{\infty}\subset\mathbb{C}$ is called a
bounded Besicovich sequence if:

(i) $\sup_{k}|\beta_{k}|<\infty$

(ii) For any $\epsilon>0$, there exists a trigonometric polynomial $P$ such
that

$\limsup_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}|\beta_{k}-P(k)|<\epsilon.$

Assume that $\mathcal{M}$ has a separable predual. It was shown in Theorem 4.6
of [6] that a bounded Besicovich sequence $\beta$ is b.a.u.-good in
$L_{1}(\mathcal{M},\tau)$. Furthermore, it was shown in Theorem 3.1 of [4]
that $\beta$ is b.a.u.-good in $L_{p}(\mathcal{M},\tau)$ if $1\leq p<\infty$
and a.u.-good in $L_{p}(\mathcal{M},\tau)$ whenever $2\leq p<\infty$.

######  Example 3.3.

Let k be a uniform sequence in the sense of Brunel and Keane (details given in
Section 4). If $\mathcal{M}$ has a separable predual, then it was shown in
Theorem 4 of [13] that k is b.a.u.-good for $\beta=\textbf{1}$ in
$L_{1}(\mathcal{M},\tau)$. This can be rephrased as saying that
$c=\\{\chi_{\textbf{k}}(j)\\}_{j=0}^{\infty}$ is b.a.u.-good in
$L_{1}(\mathcal{M},\tau)$; see Proposition 3.2 below.

######  Example 3.4.

A sequence $\alpha=\\{\alpha_{k}\\}_{k=0}^{\infty}\subseteq\mathbb{C}$ will be
said to have correlation if

$\gamma_{\alpha}(m)=\lim_{n\to\infty}\frac{1}{n+1}\sum_{k=0}^{n}\overline{\alpha_{k}}\alpha_{k+m}$

exists for every $m\in\mathbb{N}_{0}$. Also, set
$\gamma_{\alpha}(-m)=\overline{\gamma(m)}$ for $m\in\mathbb{N}$. Suppose that
$\alpha$ has correlation, is bounded, and satisfies the following conditions:

(i) the measure $\sigma_{\alpha}$ on $\mathbb{T}$ associated to
$\gamma_{\alpha}$ given by the Bochner-Herglotz Spectral Theorem is discrete,
where

$\gamma_{\alpha}(m)=\int_{\mathbb{T}}\lambda^{m}\,d\sigma_{\alpha}(\lambda),\,\,m\in\mathbb{Z};$

(ii) the limit
$\lim_{n\to\infty}\frac{1}{n+1}\sum_{k=0}^{n}\alpha_{k}\lambda^{k}$ exists for
all $\lambda\in\mathbb{T}$.

Suppose that $\tau(1)<\infty$ and that $T$ is a normal $\tau$-preserving
$*$-homomorphism on $\mathcal{M}$. Then, by Theorem 1.3 of [10], $\alpha$ is
a.u.- (b.a.u.-) good for $T$ on $L_{p}(\mathcal{M},\tau)$ for $2\leq p<\infty$
(respectively, for $1\leq p<\infty$).

In fact, [10] proves a stronger result. Namely, for every $\epsilon>0$ there
exists $e\in\mathcal{P}(\mathcal{M})$ such that $\tau(e^{\perp})\leq\epsilon$
and the sequence $\\{M_{n}^{\alpha}(T)(x)e\\}_{n=1}^{\infty}$ (
$\\{eM_{n}^{\alpha}(T)(x)e\\}_{n=1}^{\infty}$) converges in $\mathcal{M}$ for
every sequence $\alpha$ as above when $x\in L_{2}$ (respectively, $x\in
L_{1}$). Furthermore, it was shown that this type of sequence strictly
contains all bounded Besicovich sequences.

Recall that $E\subset\mathbb{N}_{0}$ is said to have density (lower density)
$d$ if

$\lim_{n\to\infty}\frac{\operatorname{card}(\\{0,...,n\\}\cap E)}{n+1}=d\text{
\ \ (respectively,
}\liminf_{n\to\infty}\frac{\operatorname{card}(\\{0,...,n\\}\cap E)}{n+1}=d).$

Accordingly, a sequence $\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}$ of nonnegative
integers has density (lower density) $d$ if the set
$\textbf{k}=\\{k_{j}:j\in\mathbb{N}_{0}\\}$ has density (respectively, lower
density) $d$.

######  Remark 3.1.

Note that, if k has density $d>0$, then
$\lim_{n\to\infty}\frac{k_{n}}{n}=\frac{1}{d}$. More generally, by Lemma 40 of
[16], k having lower density $d>0$ is equivalent to the condition $\sup_{n\geq
1}\frac{k_{n}}{n}<\infty$.

In what follows, $T\in DS^{+}$ and
$\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}\subset\mathbb{N}_{0}$ is a strictly
increasing sequence.

######  Proposition 3.1.

Let $\beta=\\{\beta_{j}\\}_{j=0}^{\infty}\in\ell_{\infty}(\mathbb{C})$. If
$\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}$ has lower density $d>0$, then the
sequences $\\{M_{n}^{\beta}(T)\\}_{n=1}^{\infty}$ and
$\\{M_{n}^{\beta,\textbf{k}}(T)\\}_{n=1}^{\infty}$ are b.u.e.m. at zero on
$(L_{p},\|\cdot\|_{p})$ for each $1\leq p<\infty$ and u.e.m. at zero on
$(L_{p},\|\cdot\|_{p})$ for each $2\leq p<\infty$.

###### Proof.

We will prove that the sequences are b.u.e.m. at zero for $1\leq p<\infty$.
The u.e.m. at zero claim follows similarly by using Theorem 2.3 of [4] below
instead of Theorem 2.1 of [4].

We will first prove the assertion for $\\{M_{n}^{\beta}(T)\\}_{n=1}^{\infty}$.
To that end, fix $\epsilon,\delta>0$. Since
$\beta\in\ell_{\infty}(\mathbb{C})$, there exists $C>0$ such that
$|\beta_{j}|\leq C$ for each $j\geq 0$.

Fix $1\leq p<\infty$, and assume $x\in L_{p}$. By the maximal ergodic theorem
given in Theorem 2.1 of [4], there exists $e\in\mathcal{P}(\mathcal{M})$ with

$\tau(e^{\perp})\leq 4\left(\frac{48C\|x\|_{p}}{\delta}\right)^{p}\ \text{ and
}\ \sup_{n\geq 1}\|eM_{n}^{\beta}(x)e\|_{\infty}\leq\delta.$

Let $\gamma=\dfrac{\epsilon^{1/p}\delta}{4^{1/p}\cdot 48C}$ and assume further
that $\|x\|_{p}<\gamma$. Then we still have $\sup_{n\geq
1}\|eM_{n}^{\beta}(x)e\|_{\infty}\leq\delta$, while

$\tau(e^{\perp})\leq\left(\frac{4^{1/p}\cdot
48C}{\delta}\|x\|_{p}\right)^{p}\leq\epsilon,$

implying that $\\{M_{n}^{\beta}(T)\\}_{n=1}^{\infty}$ is b.u.e.m. at zero on
$(L_{p},\|\cdot\|_{p})$.

For each $j\geq 0$, let $c_{j}=\chi_{\textbf{k}}(j)$. Then $|c_{j}|\leq 1$ for
every $j\geq 0$, so $c\beta=\\{c_{j}\beta_{j}\\}_{j=0}^{\infty}$, is a bounded
sequence. If $x\in L_{p}(\mathcal{M},\tau)$, we have

$\begin{split}M_{n}^{\beta,\textbf{k}}(T)(x)&=\frac{1}{n}\sum_{j=0}^{n-1}\beta_{k_{j}}T^{k_{j}}(x)=\frac{k_{n-1}+1}{n}\frac{1}{k_{n-1}+1}\sum_{j=0}^{k_{n-1}}c_{j}\beta_{j}T^{j}(x)\\\
&=\frac{k_{n-1}+1}{n}M_{k_{n-1}+1}^{c\beta}(T)(x).\end{split}$

Replacing $\beta$ in the first part of the proof with $c\beta$, we see that
$\\{M_{n}^{c\beta}(T)\\}_{n=1}^{\infty}$ is b.u.e.m. at zero on
$(L_{p},\|\cdot\|_{p})$.

In view of Remark 3.1, let $K=\sup_{n\geq 1}\frac{k_{n}}{n}$. Assume
$\epsilon,\delta>0$, and let $\gamma>0$ be the associated value for
$\\{M_{n}^{c\beta}(T)\\}_{n=1}^{\infty}$ in the definition of b.u.e.m. at zero
with respect to $\epsilon$ and $\frac{\delta}{K}$. Let $x\in
L_{p}(\mathcal{M},\tau)$ be such that $\|x\|_{p}<\gamma$. Then there exists
$e\in\mathcal{P}(\mathcal{M})$ such that $\tau(e^{\perp})\leq\epsilon$ and
$\sup_{n\geq 1}\|eM_{n}^{c\beta}(T)(x)e\|_{\infty}\leq\frac{\delta}{K}$. With
this projection $e$, we also find that

$\displaystyle\sup_{n\geq 1}\|eM_{n}^{\beta,\textbf{k}}(T)(x)e\|_{\infty}$
$\displaystyle=\sup_{n\geq
1}\left\\{\frac{k_{n-1}+1}{n}\,\|eM_{k_{n-1}+1}^{c\beta}(T)(x)e\|_{\infty}\right\\}$
$\displaystyle\leq K\sup_{n\geq 1}\|eM_{n}^{c\beta}(T)(x)e\|_{\infty}\leq
K\frac{\delta}{K}=\delta,$

hence $\\{M_{n}^{\beta,\textbf{k}}(T)\\}_{n=1}^{\infty}$ is b.u.e.m. at zero
on $(L_{p},\|\cdot\|_{p})$. ∎

The proof of the following lemma can be found in Lemma 3 of [13], where, the
result is proven for the b.a.u. convergence case, but a close examination of
the proof shows that the same proof will also work for a.u. convergence with
appropriate modifications.

######  Lemma 3.1.

If a sequence $\\{x_{n}\\}_{n=1}^{\infty}$ in $\mathcal{M}$ is such that for
every $\epsilon>0$ there are an a.u. (b.a.u.) convergent sequence
$\\{y_{n}\\}_{n=1}^{\infty}\subset\mathcal{M}$ and a positive integer $N$
satisfying $\|x_{n}-y_{n}\|_{\infty}<\epsilon$ for all $n\geq N$, then
$\\{x_{n}\\}_{n=1}^{\infty}$ converges a.u. (respectively, b.a.u.).

We will also need the following simple fact.

######  Lemma 3.2.

If $\mathbb{C}\ni\lambda_{n}\to\lambda$ and $L_{0}\ni x_{n}\to\widehat{x}\in
L_{0}$ a.u. (b.a.u.), then $\lambda_{n}x_{n}\to\lambda\widehat{x}$ a.u.
(respectively, b.a.u.)

###### Proof.

Assume $\epsilon>0$. Let $e\in\mathcal{P}(\mathcal{M})$ be such that
$\tau(e^{\perp})\leq\epsilon$ and $\|(x_{n}-\widehat{x})e\|_{\infty}\to 0$. It
then follows that the sequence $\\{\|x_{n}e\|_{\infty}\\}_{n}$ is bounded, and
we obtain

$\begin{split}\|(\lambda_{n}x_{n}-\lambda\widehat{x})e\|_{\infty}&\leq\|(\lambda_{n}-\lambda)x_{n}e\|_{\infty}+\|\lambda(x_{n}-\widehat{x}e)\|_{\infty}\\\
&=|\lambda_{n}-\lambda|\|x_{n}e\|_{\infty}+|\lambda|\|(x_{n}-\widehat{x})e\|_{\infty}\to
0\end{split}$

as $n\to\infty$. ∎

The following is a noncommutative extension of Proposition 1.7 in [3].

######  Proposition 3.2.

Assume $\\{k_{j}\\}_{j=0}^{\infty}$ has density $d>0$, and let
$\beta=\\{\beta_{j}\\}_{j=0}^{\infty}\subset\mathbb{C}$. Define $\gamma_{i}=1$
if $i=k_{j}$ for some $j$ and $c_{i}=0$ otherwise, and denote
$\gamma\beta=\\{\gamma_{j}\beta_{j}\\}_{j=0}^{\infty}$. Let $1\leq p<\infty$,
$T\in DS^{+}$, and $x\in L_{p}$. Then the averages
$M_{n}^{\beta,\textbf{k}}(T)(x)$ converge a.u. (b.a.u.) to $\widehat{x}\in
L_{0}$ if and only if the averages $M_{n}^{\gamma\beta}(T)(x)$ converge a.u.
(respectively b.a.u.) to $d\widehat{x}$.

###### Proof.

Assume that the averages $M_{n}^{\beta,\textbf{k}}(T)(x)$ converge a.u.
(b.a.u.) to $\widehat{x}$. Assume, without loss of generality, that $k_{0}=0$,
and let
$c(n)=\operatorname{card}(\\{k_{j}:j\in\mathbb{N}_{0}\\}\cap\\{0,...,n\\})$.
Observe that $c(n)=\max\\{j:k_{j}\leq n\\}+1$, $n\in\mathbb{N}_{0}$. It
follows that, since

$\begin{split}M_{n}^{\gamma\beta}(T)(x)&=\frac{1}{n}\sum_{j=0}^{n-1}\gamma_{j}\beta_{j}T^{j}(x)=\frac{1}{n}\sum_{j=0}^{c(n-1)-1}\beta_{k_{j}}T^{k_{j}}(x)\\\
&=\frac{c(n-1)}{n}\frac{1}{c(n-1)}\sum_{j=0}^{c(n-1)-1}\beta_{k_{j}}T^{k_{j}}(x)=\frac{c(n-1)}{n}M_{c(n-1)}^{\beta,\textbf{k}}(T)(x),\end{split}$

$\frac{c(n-1)}{n}\to d$, and the subsequence
$\\{M_{c(n-1)}^{\beta,\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ converges a.u.
(b.a.u.) to $\widehat{x}$, Lemma 3.2 implies that the sequence
$\\{M_{n}^{\gamma\beta}(T)(x)\\}_{n=1}^{\infty}$ converges a.u. (b.a.u.) to
$d\widehat{x}$.

Similarly,

$\begin{split}M_{n}^{\beta,\textbf{k}}(T)(x)&=\frac{1}{n}\sum_{j=0}^{n-1}\beta_{k_{j}}T^{k_{j}}(x)=\frac{1}{n}\sum_{j=0}^{k_{n-1}}c_{j}\beta_{j}T^{j}(x)\\\
&=\frac{k_{n-1}+1}{n}\frac{1}{k_{n-1}+1}\sum_{j=0}^{k_{n-1}}c_{j}\beta_{j}T^{j}(x)=\frac{k_{n-1}+1}{n}M_{k_{n-1}+1}^{\gamma\beta}(T)(x),\end{split}$

implies the converse. ∎

######  Theorem 3.1.

Let $\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}$ have density $1$,
$\beta=\\{\beta_{j}\\}_{j=0}^{\infty}\in\ell_{\infty}(\mathbb{C})$, and $2\leq
p<\infty$ ($1\leq p<\infty$). If $T\in DS^{+}$ and $\beta$ is a.u.-
(respectively, b.a.u.-) good for $T$ on $L_{p}(\mathcal{M},\tau)$, then k is
a.u.- (respectively, b.a.u.-) good for $T$ and $\beta$ on
$L_{p}(\mathcal{M},\tau)$. Moreover, for every $x\in L_{p}$, the sequence
$\\{M_{n}^{\beta,\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ converges a.u.
(respectively, b.a.u.) to some $\widehat{x}\in L_{p}$.

###### Proof.

We will prove the a.u. case, as the b.a.u. case will follow the same line of
proof with the appropriate modifications. As such, assume $2\leq p<\infty$.

Assume $x\in L_{1}\cap\mathcal{M}$. Then, since $\beta$ is a.u.-good for $T$
on $L_{p}(\mathcal{M},\tau)$ and since $L_{1}\cap\mathcal{M}\subset L_{p}$, we
see that the sequence $\\{M_{n}^{\beta}(T)(x)\\}_{n=1}^{\infty}$, hence the
subsequence $\\{M_{k_{n}}^{\beta}(T)(x)\\}_{n=1}^{\infty}$, converges a.u.

Let $A_{n}(T)(x)=\frac{1}{k_{n}}\sum_{j=0}^{n-1}\beta_{k_{j}}T^{k_{j}}(x),$ so
that $M_{n}^{\beta,\textbf{k}}(T)(x)=\frac{k_{n}}{n}A_{n}(T)(x)$. Since $T$ is
a contraction on $\mathcal{M}$, we see that
$\|T^{j}(x)\|_{\infty}\leq\|x\|_{\infty}$ for every $j\geq 0$. Let
$\|\beta\|_{\ell_{\infty}}=\sup_{j\geq 0}|\beta_{j}|$. Then, for each $n\geq
1$,

$\displaystyle\left\|M_{k_{n}}^{\beta}(T)(x)-A_{n}(T)(x)\right\|_{\infty}$
$\displaystyle=\left\|\frac{1}{k_{n}}\sum_{j=0,\;j\not\in\textbf{k}}^{k_{n}-1}\beta_{j}T^{j}(x)\right\|_{\infty}$
$\displaystyle\leq\frac{1}{k_{n}}\sum_{j=0,\;j\not\in\textbf{k}}^{k_{n}-1}|\beta_{j}|\|T^{j}(x)\|_{\infty}$
$\displaystyle\leq\frac{1}{k_{n}}\sum_{j=0,\;j\not\in\textbf{k}}^{k_{n}-1}\|\beta\|_{\ell_{\infty}}\|x\|_{\infty}=\frac{k_{n}-n}{k_{n}}\|\beta\|_{\ell_{\infty}}\|x\|_{\infty}.$

Let $\epsilon>0$. Since $\frac{k_{n}-n}{k_{n}}\to 0$ as $n\to\infty$ (see
Remark 3.1), there exists $N\in\mathbb{N}$ such that $n\geq N$ implies

$\|M_{k_{n}}^{\beta}(T)(x)-A_{n}(T)(x)\|_{\infty}<\epsilon.$

Thus, it follows by Lemma 3.1 that the sequence
$\\{A_{n}(T)(x)\\}_{n=1}^{\infty}$ converges a.u. since
$\\{M_{k_{n}}^{\beta}(T)(x)\\}_{n=1}^{\infty}$ does.

Since $L_{0}$ is complete with respect to a.u. (and b.a.u.) convergence (see
Theorem 2.3 and Remark 2.4 of [6]), there exists $\widehat{x}\in L_{0}$ such
that $A_{n}(T)(x)\to\widehat{x}$ a.u. Therefore, since $\frac{k_{n}}{n}\to 1$
(Remark 3.1) and $M_{n}^{\beta,\textbf{k}}(T)(x)=\frac{k_{n}}{n}A_{n}(T)(x)$,
we conclude, by Lemma 3.2, that
$\\{M_{n}^{\beta,\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ converges a.u. for every
$x\in L_{1}\cap\mathcal{M}$.

As $\\{\frac{k_{n}}{n}\\}_{n=1}^{\infty}$ and $\beta$ are a bounded sequences,
$\\{M_{n}^{\beta,\textbf{k}}(T)\\}_{n=1}^{\infty}$ is u.e.m. at zero on
$(L_{p},\|\cdot\|_{p})$ by Proposition 3.1. This in turn implies, by
Proposition 2.2, that the set

$C=\\{x\in
L_{p}(\mathcal{M},\tau):\\{M_{n}^{\beta,\textbf{k}}(T)(x)\\}_{n=1}^{\infty}\
\text{ converges a.u.}\\}$

is closed in $L_{p}$. Since $L_{1}\cap\mathcal{M}$ is dense in $L_{p}$ it
follows that $L_{p}=C$.

Therefore, given $x\in L_{p}$, there exists $\widehat{x}\in L_{0}$ such that
$M_{n}^{\beta,\textbf{k}}(T)(x)\to\widehat{x}$ a.u., hence in measure. Since
the unit ball in $L_{p}$ is closed with respect to the measure topology (see,
for example, Theorem 1.2 in [6]) and
$\|M_{n}^{\beta,\textbf{k}}(T)(x)\|_{p}\leq\|\beta\|_{\ell_{\infty}}\|x\|_{p}$
for all $n$, we conclude that $\widehat{x}\in L_{p}$. ∎

If $\beta=\textbf{1}$, this establishes the following.

######  Corollary 3.1.

If $\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}$ has density $1$ and $x\in L_{p}$,
then the subsequential averages $M_{n}^{\textbf{k}}(T)(x)$ converge b.a.u.
(a.u.) to some $\widehat{x}\in L_{p}$ if $1\leq p<\infty$ (respectively, if
$2\leq p<\infty$).

Combining Theorem 3.1 with Theorem 3.1 of [4], we arrive at the following.

######  Corollary 3.2.

Assume $\mathcal{M}$ has a separable predual. If k has density $1$ and $\beta$
is a bounded Besicovich sequence, then, given $x\in L_{p}$, the subsequential
weighted averages $M_{n}^{\beta,\textbf{k}}(T)(x)$ converge b.a.u. (a.u.) to
some $\widehat{x}\in L_{p}$ if $1\leq p<\infty$ (respectively, if $2\leq
p<\infty$.

Suppose $\alpha,\beta\in\ell_{\infty}(\mathbb{C})$ are such that there exists
a set $E\subset\mathbb{N}_{0}$ with density $1$ such that
$\alpha_{j}\neq\beta_{j}$ implies $j\in E^{c}$. Then, similar to the proof of
Theorem 3.1, it follows that for every $x\in L_{1}\cap\mathcal{M}$ and
$n\in\mathbb{N}$,

$\displaystyle\|M_{n}^{\alpha}(T)(x)$ $\displaystyle-
M_{n}^{\beta}(T)(x)\|_{\infty}\leq\frac{1}{n}\sum_{j=0,\;j\not\in
E}^{n-1}|\alpha_{j}-\beta_{j}|\|T^{j}(x)\|_{\infty}$
$\displaystyle\leq\frac{n-\operatorname{card}(\\{0,...,n-1\\}\cap
E)}{n}(\|\alpha\|_{\ell_{\infty}}+\|\beta\|_{\ell_{\infty}})\|x\|_{\infty},$

which tends to $0$ as $n\to\infty$. Since both weighted averages are b.u.e.m.
at zero on $L_{p}$ for $1\leq p<\infty$ and u.e.m. at zero on $L_{p}$ for
$2\leq p<\infty$, an application of Lemma 3.1 and Proposition 2.2 then implies
the following.

######  Theorem 3.2.

Let $\alpha,\beta\in\ell_{\infty}(\mathbb{C})$ be such that there exists a set
$E\subseteq\mathbb{N}_{0}$ with density $1$ such that
$\alpha_{j}\neq\beta_{j}$ implies $j\in E^{c}$. Then for $2\leq p<\infty$
($1\leq p<\infty$) and $T\in DS^{+}$, $\alpha$ is a.u.- (respectively b.a.u.-)
good for $T$ on $L_{p}(\mathcal{M},\tau)$ if and only if $\beta$ is a.u.-
(respectively b.a.u.-) good for $T$ on $L_{p}(\mathcal{M},\tau)$.

######  Remark 3.2.

1. Theorem 3.2 together with Proposition 3.2 shows that the conclusion of Theorem 3.1 is actually an equivalence.

2. The proof of Theorem 3.1 also shows that the a.u. limit of $\\{M_{n}^{\beta,\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ is the same as that of $\\{M_{n}^{\beta}(T)(x)\\}_{n=1}^{\infty}$ whenever $x\in L_{1}\cap\mathcal{M}$. To see this, let $x\in L_{1}\cap\mathcal{M}$ and let $\widehat{x}\in L_{0}$ be the a.u. limit of $\\{M_{n}^{\beta}(T)(x)\\}_{n=1}^{\infty}$. Then, for any $\epsilon>0$ with corresponding projection $e$ from the definition of the a.u. convergence of $\\{M_{k_{n}}^{\beta}(T)(x)\\}_{n=1}^{\infty}$ to $\widehat{x}$, we find

$\|(A_{n}(x)-\widehat{x})e\|_{\infty}\leq\|A_{n}^{\beta}(x)-M_{k_{n}}^{\beta}(T)(x)\|_{\infty}+\|(M_{k_{n}}^{\beta}(T)(x)-\widehat{x})e\|_{\infty}\xrightarrow{n\to\infty}0,$

which shows that $\\{A_{n}(x)\\}_{n=1}^{\infty}$ shares its a.u. limit with
$\\{M_{k_{n}}^{\beta}(T)(x)\\}_{n=1}^{\infty}$. The rest of the proof is
similar to that of Theorem 3.1. The same can be said for b.a.u. convergence,
as well as the convergences in Theorem 3.2 for $x\in L_{1}\cap\mathcal{M}$.

##  4\. Convergence along Uniform Sequences

We will now extend the main result of [13] regarding the convergence of
subsequential averages along uniform sequences to the general non-commutative
$L_{p}$ setting for every $1\leq p<\infty$. Furthermore, we also show that we
can replace the b.a.u. convergence as stated there with a.u. convergence for
$2\leq p<\infty$, as well give a condition that allows one to add weights to
these averages.

We will first give a very brief review of the terminology used in that paper,
namely that relating to uniform sequences. See [2] and [18] for more details
and properties of uniform sequences.

Let $X$ be a compact Hausdorff space, and let $\phi:X\to X$ be a continuous
map such that the family $\\{\phi^{n}\\}_{n=0}^{\infty}$ is equicontinuous.

The pair $(X,\phi)$ is called strictly ergodic if there exists a unique
$\phi$-invariant measure $\mu$ on $(X,\mathcal{B}_{X})$ (so that
$\mu\circ\phi=\mu$), where $\mathcal{B}_{X}$ denotes the $\sigma$-algebra of
all Borel subsets of $X$, such that $\operatorname{supp}(\mu)=X$ and such that
for any $z\in X$ and $f\in C(X)$ we have

$\int_{X}f(t)d\mu(t)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(\phi^{k}(z)),$

where $C(X)$ is the Banach space of continuous functions $f:X\to\mathbb{C}$
with the supremum norm, and the convergence above is with respect to that
norm.

Fix a strictly ergodic system $(X,\mathcal{B}_{X},\mu,\phi)$ and a set
$Y\in\mathcal{B}_{X}$ such that $\mu(Y)>0=\mu(\partial Y)$, where $\partial Y$
denotes the boundary of $Y$. A sequence
$\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}$ of non-negative integers is said to be
uniform if there exists a point $\omega_{0}\in X$ such that

$k_{0}=\min\\{k\geq 0:\phi^{k}(\omega_{0})\in Y\\},$

$k_{n}=\min\\{k>k_{n-1}:\phi^{k}(\omega_{0})\in Y\\},n\geq 1.$

The triple $(X,\mathcal{B}_{X},\mu,\phi)$, $Y$, and $\omega_{0}$ will be
called the apparatus for k [BK].

For the rest of this section $\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}$ will be a
fixed uniform sequence with the apparatus $(X,\mathcal{B}_{X},\mu,\phi)$, $Y$,
and $\omega_{0}$. Details of the next two lemmas can be found in [18].

######  Lemma 4.1.

If $\\{k_{j}\\}_{j=0}^{\infty}$ is a uniform sequence with the apparatus
above, then

$\lim_{n\to\infty}\frac{n}{k_{n}}=\mu(Y).$

######  Lemma 4.2.

If $\\{k_{j}\\}_{j=0}^{\infty}$ is a uniform sequence with the apparatus as
given above, then for any $\epsilon>0$ there exist open subsets $Y_{1}$,
$Y_{2}$, and $W$ of $X$ such that

(i) $Y_{1}\subseteq Y\subseteq Y_{2}$, $\mu(Y_{2}\setminus Y_{1})<\epsilon$,
and $\mu(\partial Y_{1})=\mu(\partial Y_{2})=0$.

(ii) $\omega_{0}\in W$ and, for every $\omega\in W$ and all $k\geq 0$,

$\chi_{Y_{1}}(\phi^{k}(\omega))\leq\chi_{Y}(\phi^{k}(\omega_{0})\leq\chi_{Y_{2}}(\phi^{k}(\omega)).$

The following is the main result of [13].

######  Theorem 4.1.

[13] Let a semifinite von Neumann algebra $(\mathcal{M},\tau)$ have a
separable predual, and let $T\in DS^{+}(\mathcal{M},\tau)$. If k is a uniform
sequence and $x\in L_{1}(\mathcal{M},\tau)$, then the subsequential averages
$M_{n}^{\textbf{k}}(T)(x)$ converge b.a.u. to some $\widehat{x}\in
L_{1}(\mathcal{M},\tau)$.

As they will be necessary for the statement and proof of the next theorem, we
will let $\widetilde{L}_{p}$ and $\widetilde{\mathcal{M}}$ denote the spaces
of Bochner $p$-integrable and bounded weakly measurable functions respectively
(where $1\leq p<\infty$) on the measure space $(X,\mathcal{B}_{X},\mu)$. If
$\mathcal{M}$ has n.s.f. trace $\tau$, then $\widetilde{\mathcal{M}}$ is a
semifinite von Neumann algebra with n.s.f. trace $\widetilde{\tau}$ given by

$\widetilde{\tau}(f)=\int_{X}\tau(f(z))d\mu(z)$

whenever $f\in\widetilde{\mathcal{M}}^{+}$. If $\mathcal{M}$ has a separable
predual then, in view of Proposition 1.22.12 and Theorem 1.22.13 of [17], we
see that predual $L_{1}(\widetilde{\mathcal{M}},\widetilde{\tau})$ of
$\widetilde{M}$ is isomorphic to $\widetilde{L_{1}}$, and
$L_{p}(\widetilde{M},\widetilde{\tau})$ is isomorphic to $\widetilde{L}_{p}$
for $1<p<\infty$.

######  Theorem 4.2.

Assume $\mathcal{M}$ has a separable predual. Let
$\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}$ be a uniform sequence and
$\beta\in\ell_{\infty}(\mathbb{C})$. If $\beta$ is a.u.- (b.a.u.-) good in
$L_{p}(\widetilde{\mathcal{M}},\widetilde{\tau})$ for $2\leq p<\infty$
(respectively, $1\leq p<\infty$), then k is a.u.- (respectively, b.a.u.-) good
for $\beta$ in $L_{p}(\mathcal{M},\tau)$, with the limit $\widehat{x}\in
L_{p}(\mathcal{M},\tau)$.

###### Proof.

Our proof follows closely the proof of Theorem 4 of [13]. We will only prove
the a.u. case. The b.a.u. case will follow similarly. Accordingly, assume that
$2\leq p<\infty$ and $T\in DS^{+}(\mathcal{M},\tau)$.

Define
$\widetilde{T}:\widetilde{L}_{1}+\widetilde{\mathcal{M}}\to\widetilde{L}_{1}+\widetilde{\mathcal{M}}$
by, for $z\in X$,

$(\widetilde{T}f)(z)=T(f(\phi(z))).$

Then $\widetilde{T}\in DS^{+}(\widetilde{M},\widetilde{\tau})$, and so we may
consider the corresponding weighted averages

$\frac{1}{n}\sum_{j=0}^{n-1}\beta_{j}\widetilde{T}^{j}(f)=\frac{1}{n}\sum_{j=0}^{n-1}\beta_{j}T^{j}(f\circ\phi^{j})$

for $f\in\widetilde{L}_{1}+\widetilde{\mathcal{M}}$. Since $\beta$ is
a.u.-good in $L_{p}(\widetilde{M},\widetilde{\tau})$, it follows that the
averages
$\\{\frac{1}{n}\sum_{j=0}^{n-1}\beta_{j}\widetilde{T}^{j}(f)\\}_{n=1}^{\infty}$
converge a.u. for every $f\in\widetilde{L}_{1}\cap\widetilde{M}$. By Lemma 4.1
in [6], it follows that

$\frac{1}{n}\sum_{j=0}^{n-1}\beta_{j}T^{j}(f(\phi^{j}(z)))$

converge a.u. for $\mu$-a.e. $z\in X$.

Assume $x\in L_{1}\cap\mathcal{M}$ and fix $\epsilon>0$. Let
$W,Y_{1},Y_{2}\subset X$ satisfy the conclusion of Lemma 4.2 with this
$\epsilon$. Define $\widetilde{x}:X\to\widetilde{L}_{1}$ by

$\widetilde{x}(z)=\chi_{Y_{1}}(z)x$

for $z\in X$. Then
$\widetilde{x}\in\widetilde{L}_{1}\cap\widetilde{\mathcal{M}}$, so that the
averages

$\frac{1}{n}\sum_{j=0}^{n-1}\beta_{j}T^{j}(x)\chi_{Y_{1}}(\phi^{j}(z))$

converge a.u. for $\mu$-a.e. $z\in X$. Since $\mu(W)>0$, we can find
$\omega\in W$ such that

$\frac{1}{n}\sum_{j=0}^{n-1}\beta_{j}T^{j}(x)\chi_{Y_{1}}(\phi^{j}(\omega))$

converges a.u.

Lemma 1 of [18] tells us that, since $\mu(\partial Y_{j})=0$, we have

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\chi_{Y_{k}}(\phi^{j}(\omega))=\mu(Y_{k})\
\text{ for }k\in\\{1,2\\}.$

The properties of $Y_{1}$, $Y_{2}$, and $W$ from Lemma 4.2 then entail that
there exists $N$ such that $n\geq N$ implies

$0\leq\frac{1}{n}\sum_{j=0}^{n-1}[\chi_{Y}(\phi^{j}(\omega_{0}))-\chi_{Y_{1}}(\phi^{j}(\omega))]<\epsilon.$

With this, letting $\|\beta\|_{\ell_{\infty}}=\sup_{j\geq 0}|\beta_{j}|$, we
find that

$\displaystyle\left\|\frac{1}{n}\sum_{j=0}^{n-1}\beta_{j}T^{j}(x)\chi_{Y}(\phi^{j}(\omega_{0}))\right.$
$\displaystyle\left.-\frac{1}{n}\sum_{j=0}^{n-1}\beta_{j}T^{j}(x)\chi_{Y_{1}}(\phi^{j}(\omega))\right\|_{\infty}$
$\displaystyle\leq\frac{1}{n}\sum_{j=0}^{n-1}\|\beta_{j}T^{j}(x)(\chi_{Y}(\phi^{j}(\omega_{0}))-\chi_{Y_{1}}(\phi^{j}(\omega))\|_{\infty}$
$\displaystyle\leq\|\beta\|_{\ell_{\infty}}\|x\|_{\infty}\frac{1}{n}\sum_{j=0}^{n-1}(\chi_{Y}(\phi^{j}(\omega_{0}))-\chi_{Y_{1}}(\phi^{j}(\omega)))$
$\displaystyle<\|\beta\|_{\ell_{\infty}}\|x\|_{\infty}\epsilon,$

which, by Lemma 3.1, implies that the averages

$\frac{1}{n}\sum_{j=0}^{n-1}\beta_{j}T^{j}(x)\chi_{Y}(\phi^{j}(\omega_{0}))$

converge a.u. as $n\to\infty$.

Next, we have

$M_{n}^{\beta,\textbf{k}}(T)(x)=\frac{1}{n}\sum_{j=0}^{n-1}\beta_{k_{j}}T^{k_{j}}(x)=\frac{k_{n-1}+1}{n}\frac{1}{k_{n-1}+1}\sum_{j=0}^{k_{n-1}}\beta_{j}T^{j}(x)\chi_{Y}(\phi^{j}(\omega_{0})).$

Since $\frac{k_{n-1}+1}{n}\to\frac{1}{\mu(Y)}$ as $n\to\infty$ by Lemma 4.1,
Lemma 3.2 implies that the averages
$\\{M_{n}^{\beta,\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ converge a.u. for every
$x\in L_{1}\cap\mathcal{M}$.

As the sequences $\\{\frac{k_{n}}{n}\\}_{n=1}^{\infty}$ and $\beta$ are
bounded, Proposition 3.1 implies that
$\\{M_{n}^{\beta,\textbf{k}}(T)\\}_{n=1}^{\infty}$ is u.e.m. at zero on
$(L_{p},\|\cdot\|_{p})$. By Proposition 2.2, this fact implies that the set

$C=\\{x\in L_{p}:\\{M_{n}^{\beta,\textbf{k}}(T)(x)\\}_{n=1}^{\infty}\text{
converges a.u.}\\}$

is closed in $L_{p}$. Since $L_{1}\cap\mathcal{M}$ is a subset of $C$ which is
dense in $L_{p}$, it follows that $L_{p}=C$. Therefore, since $L_{0}$ is
complete with respect to a.u. convergence, given $x\in L_{p}$, there is
$\widehat{x}\in L_{0}$ such that
$M_{n}^{\beta,\textbf{k}}(T)(x)\to\widehat{x}$ a.u. Finally, since
$\|M_{n}^{\beta,\textbf{k}}(T)(x)\|_{p}\leq\|\beta\|_{\ell_{\infty}}\|x\|_{\infty}$,
we conclude as in Theorem 3.1 that $\widehat{x}\in L_{p}$. ∎

Similar to what we did in the density $1$ case, by setting $\beta=\textbf{1}$,
this yields a generalization of Theorem 4.1.

######  Corollary 4.1.

Assume $\mathcal{M}$ has a separable predual and $T\in DS^{+}$. If k is a
uniform sequence, then, given $1\leq p<\infty$, the subsequential averages
$M_{n}^{\textbf{k}}(T)(x)$ converge b.a.u. to some $\widehat{x}\in
L_{p}(\mathcal{M},\tau)$ for every $x\in L_{p}(\mathcal{M},\tau)$.
Furthermore, this convergence occurs a.u. when $2\leq p<\infty$.

######  Remark 4.1.

1. It is known that $\mathcal{M}$ having a separable predual is equivalent to being able to assume that the Hilbert space $\mathcal{H}$ it acts on which is separable (see [20]). In this case it is also known that $\widetilde{M}$ is $*$-isomorphic to $L_{\infty}(X,\mathcal{B}_{X},\mu)\overline{\otimes}\mathcal{M}$, and that this acts on $L_{2}(X,\mathcal{B}_{X},\mu)\otimes\mathcal{H}$. If $L_{1}(X,\mathcal{B}_{X},\mu)$ is separable, then $L_{2}(X,\mathcal{B}_{X},\mu)$ is too, so that the tensor product Hilbert space $L_{2}(X,\mathcal{B}_{X},\mu)\otimes\mathcal{H}$ is separable. Therefore $\widetilde{M}$ acts on a separable Hilbert space, implying it has a separable predual. So if $(X,\mathcal{B}_{X},\mu)$ is such that $L_{1}(X,\mathcal{B}_{X},\mu)$ is separable, then Theorem 4.2 will hold for every bounded Besicovich sequence.

2. Lemma 4.1 can be shown to be equivalent to saying that a uniform sequence k with apparatus $(X,\mathcal{B}_{X},\mu)$, $Y$, and $\omega_{0}$ has density $\mu(Y)$.

##  5\. Convergence along Block Sequences of Positive Lower Density

In this section, we will study the a.u. and b.a.u. convergence of the
subsequential averages along certain types of block sequences with positive
lower density. After this, we show that this can be extended by allowing
bounded Besicovich sequences as weights when $\mathcal{M}$ has a separable
predual. For more information about block sequences, see [3] or [15]. The norm
convergence of the subsequential ergodic averages of any $T\in DS^{+}$ along
block sequences was considered in [7].

Let $\mathcal{I}=\\{I_{n}=[a_{n},b_{n}]\\}_{n=0}^{\infty}$ be a sequence of
intervals in $[0,\infty)$ whose endpoints are in $\mathbb{N}_{0}$ such that
$b_{n}<a_{n+1}$ for every $n\geq 0$.

If $\textbf{k}=\\{k_{n}\\}_{n=0}^{\infty}$ is the sequence determined by the
strictly increasing enumeration of $\bigcup_{n}I_{n}\cap\mathbb{N}_{0}$, then
k is called a block sequence. Define a function
$N_{\mathcal{I}}:\mathbb{N}_{0}\to\mathbb{N}_{0}$ to be such that

$k_{n}\in[a_{N_{\mathcal{I}}(n)},b_{N_{\mathcal{I}}(n)}]$

(which exists and is well-defined by the restrictions on the intervals).
Observe that $N_{\mathcal{I}}(n)$ counts the number of intervals of
$\mathcal{I}$ one has to skip before finding the interval that $k_{n}$ lives
in.

######  Example 5.1.

For each $n\in\mathbb{N}_{0}$, let $I_{n}=[n^{2},n^{2}+n]$. Let
$\mathcal{I}=\\{I_{n}\\}_{n=0}^{\infty}$, and let k be the strictly increasing
enumeration of $\bigcup_{n=0}^{\infty}I_{n}\cap\mathbb{N}_{0}$, so that k is a
block sequence. One can show that k has lower density $\frac{1}{2}$ and that
$N_{\mathcal{I}}(n)=[\sqrt{n}]$, where $[x]$ denotes the greatest integer less
than or equal to $x$ for each $x\in\mathbb{R}$, which implies that
$0\leq\frac{N_{\mathcal{I}}(n)}{n}\leq\frac{1}{\sqrt{n}}\to 0$ as
$n\to\infty$.

######  Theorem 5.1.

Let $\textbf{k}=\\{k_{j}\\}_{j=0}^{\infty}$ be a block sequence with lower
density $d>0$ such that $\frac{N_{\mathcal{I}}(n)}{n}\to 0$ as $n\to\infty$
and $T\in DS^{+}$. Then, given $x\in L_{p}(\mathcal{M},\tau)$, the
subsequential averages $M_{n}^{\textbf{k}}(T)(x)$ converge a.u. (b.a.u.) to
some $\widehat{x}\in L_{p}(\mathcal{M},\tau)$ if $2\leq p<\infty$
(respectively, $1\leq p<\infty$).

###### Proof.

First we will prove the claim for $x\in L_{2}(\mathcal{M},\tau)$. Since $T$ is
a contraction on all of $L_{2}$, and not just its self-adjoint part, the set

$\\{x\in L_{2}:T(x)=x\\}+\\{x=y-T(y):y\in L_{2}\cap\mathcal{M}\\}$

is dense in $L_{2}$ (see Theorem 4.1 in [12]). If $x\in L_{2}$ is such that
$T(x)=x$, then $M_{n}^{\textbf{k}}(T)(x)=x$ for every $n$. Since for every
$\epsilon>0$ there exists $e\in\mathcal{P}(\mathcal{M})$ such that
$(M_{n}^{\textbf{k}}(T)(x)-x)e\in\mathcal{M}$ for every $n$ and
$\tau(e^{\perp})\leq\epsilon$, we conclude that
$\\{M_{n}^{\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ converges a.u. to $x$.

Now assume that $x=y-T(y)$ for some $y\in L_{2}\cap\mathcal{M}$. Fix $n\geq
1$. Then

$\displaystyle M_{n}^{\textbf{k}}$
$\displaystyle(T)(x)=\frac{1}{n}\sum_{j=0}^{n-1}T^{k_{j}}(y-T(y))=\frac{1}{n}\sum_{j=0}^{n-1}\left(T^{k_{j}}(y)-T^{k_{j}+1}(y)\right)$
$\displaystyle=\frac{1}{n}\left((T^{a_{N_{\mathcal{I}}(n-1)}}(y)-T^{k_{n-1}+1}(y))+\sum_{i=0}^{N_{\mathcal{I}}(n-1)-1}\sum_{j=a_{i}}^{b_{i}}\left(T^{j}(y)-T^{j+1}(y)\right)\right)$
$\displaystyle=\frac{1}{n}\left((T^{a_{N_{\mathcal{I}}(n-1)}}(y)-T^{k_{n-1}+1}(y))+\sum_{i=0}^{N_{\mathcal{I}}(n-1)-1}\left(T^{a_{i}}(y)-T^{b_{i}+1}(y)\right)\right)$

and since $T$ is a contraction on $\mathcal{M}$, we obtain

$\left\|M_{n}^{\textbf{k}}(T)(x)\right\|_{\infty}\leq
2\|y\|_{\infty}\frac{N_{\mathcal{I}}(n-1)+1}{n}.$

By assumption we know that $\lim_{n\to\infty}\frac{N_{\textbf{k}}(n)}{n}=0$,
implying $\|M_{n}^{\textbf{k}}(T)(x)\|_{\infty}\to 0$ as $n\to\infty$.
Therefore $\\{M_{n}^{\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ converges uniformly,
and so a.u., as $n\to\infty$.

Since k has lower density $d>0$, $\\{M_{n}^{\textbf{k}}(T)\\}_{n=1}^{\infty}$
is u.e.m. at zero on $(L_{2},\|\cdot\|_{2})$ by Proposition 3.1. Therefore,
since $\\{M_{n}^{\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ converges a.u. on dense
subset of $L_{2}$, it follows that
$\\{M_{n}^{\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ converges a.u. for every $x\in
L_{2}$ by Proposition 2.2. In particular, this convergence holds for every
$x\in L_{1}\cap\mathcal{M}$.

Since the averages $\\{M_{n}^{\textbf{k}}(T)\\}_{n=1}^{\infty}$ are also
u.e.m. (b.u.e.m.) at zero on $(L_{p},\|\cdot\|_{p})$ for $2\leq p<\infty$
(respectively, $1\leq p<\infty$) by Proposition 3.1, since a.u. convergence
implies b.a.u. convergence, and since $L_{1}\cap\mathcal{M}$ is dense in
$L_{p}$, it follows by Proposition 2.2 that
$\\{M_{n}^{\textbf{k}}(T)(x)\\}_{n=1}^{\infty}$ converges a.u. (respectively,
b.a.u.) for every $x\in L_{p}$ when $2\leq p<\infty$ (respectively, $1\leq
p<\infty$). Finally, we observe as in the proof of Theorem 3.1, that the limit
$\widehat{x}\in L_{p}$. ∎

As mentioned in the proof, the proof of Theorem 4.2 above very closely follows
Theorem 4 of [13]. One also sees that the proof of Theorem 4 of [13] is
similar to the proof of Lemma 4.2 in [6], which itself implies Theorem 4.6 of
[6] and Theorem 3.1 in [4]. Following the commonalities in the core parts of
the proofs of these statements, we omit the proof of the following extension
of Theorem 5.1 since it follows the same lines.

######  Theorem 5.2.

Let $\mathcal{M}$ have a separable predual and $2\leq p<\infty$ ($1\leq
p<\infty$). If $\beta$ is a bounded Besicovich sequence and k is a block
sequence with lower density $d>0$ such that $\frac{N_{\mathcal{I}}(n)}{n}\to
0$, then k is a.u.- (respectively, b.a.u.-) good for $\beta$ in
$L_{p}(\mathcal{M},\tau)$, with the limits in $L_{p}(\mathcal{M},\tau)$.

Acknowledgements: The author is grateful to Dr. Semyon Litvinov for his help
in simplifying and improving numerous aspects of this paper. The author is
also thankful to Dr. Doğan Çömez for his guidance and support with writing
this paper.

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immediately have that
$\sqrt{n}\beta_{n}\to_{\mathcal{P}_{0}}0,\sqrt{n}\gamma_{n}\to_{\mathcal{P}_{0}}0$
and $\sqrt{n}\delta_{n}\to_{\mathcal{P}_{0}}0$ by following the same arguments
as in the proof of Theorem 4.4.2. The fact that $\sqrt{n}\alpha_{n}$ converges
uniformly in distribution to a
$\mathcal{N}(0,\Sigma\otimes\Sigma)$-distribution over $\mathcal{H}_{0}$
follows from the fact that the distribution of $(U_{1,i},U_{2,i})_{i=1}^{n}$
is unchanged whenever $P\in\mathcal{H}_{0}$. By Lemma B.2.2 we have that

$\displaystyle\sqrt{n}\hat{\rho}_{n}=\sqrt{n}\alpha_{n}+\sqrt{n}\beta_{n}+\sqrt{n}\gamma_{n}+\sqrt{n}\delta_{n}\Rightarrow_{\mathcal{H}_{0}}\mathcal{N}(0,\Sigma\otimes\Sigma)$

which shows part (i) of the theorem. Next we turn to part (ii) of the theorem.
Analogously to the proof of Theorem 4.4.6 we have that
$\beta_{n}\to_{\mathcal{P}_{0}}0,\gamma_{n}\to_{\mathcal{P}_{0}}0$ and
$\delta_{n}\to_{\mathcal{P}_{0}}0$ under Assumption 4.5.1. Now consider
writing

$\displaystyle\alpha_{k\ell,n}=\frac{1}{n}\sum_{i=1}^{n}A_{i}\quad\text{where}\quad
A_{i}=\varphi_{k}(U_{1,i})\varphi_{\ell}(U_{2,i})$

for $k,\ell=1,\dots,q$. Then $(A_{i})_{i=1}^{n}$ are i.i.d. with
$E_{P}(A_{i})=\rho_{k\ell}$ and

$\displaystyle
V_{P}(A_{i})=E_{P}(\varphi_{k}(U_{1,i})^{2}\varphi_{\ell}(U_{2,i})^{2})-\rho^{2}\leq\|\varphi_{k}\|_{\infty}\|\varphi_{\ell}\|_{\infty}<\infty$

for all $P\in\mathcal{P}_{0}$. Therefore, for given $\varepsilon>0$, we have
by Chebyshev’s inequality that

$\displaystyle\sup_{P\in\mathcal{P}_{0}}P(|\alpha_{k\ell,n}-\rho_{k\ell}|>\varepsilon)\leq\sup_{P\in\mathcal{P}_{0}}\frac{V_{P}(\frac{1}{n}\sum_{i=1}^{n}A_{i})}{\varepsilon^{2}}=\sup_{P\in\mathcal{P}_{0}}\frac{V_{P}(A_{i})}{n\varepsilon^{2}}\leq\frac{||\varphi_{1}||^{2}_{\infty}||\varphi_{2}||^{2}_{\infty}}{n\varepsilon^{2}}\to
0$

for $n\to\infty$ which shows that $\alpha_{n}\to_{\mathcal{P}_{0}}\rho$. From
this we get $\hat{\rho}_{n}\to_{\mathcal{P}_{0}}\rho$ as wanted.

###  A.15 Proof of Corollary 4.5.3

Note that due to Theorem 4.5.2 (i) we have that
$nT_{n}\Rightarrow_{\mathcal{H}_{0}}\chi^{2}_{q^{2}}$ under Assumption 4.5.1
using the same argument as in the proof of Corollary 4.4.3. Then the result is
obtained by the same argument as in the proof of Corollary 4.4.5 by noting
that $\sup_{P\in\mathcal{H}_{0}}|F_{nT_{n}}(t)-\Phi(t)|\to 0$ as $n\to\infty$
for all $t\in\mathbb{R}$. $\square$

###  A.16 Proof of Corollary 4.5.4

Let $\lambda>0$ be fixed. By Theorem 4.5.2 (ii) we have
$\hat{\rho}_{n}\to_{\mathcal{A}_{\lambda}}\rho$ where
$|\rho_{k\ell}|>\lambda>0$ for some $k,\ell=1,\dots,q$. Therefore
$\inf_{P\in\mathcal{A}_{\lambda}}|\rho_{k\ell}|\geq\lambda>0$ and so

$\displaystyle
T_{n}=\|\Sigma^{-1/2}\hat{\rho}_{n}\Sigma^{-1/2}\|_{F}^{2}\to_{\mathcal{A}_{\lambda}}\|\Sigma^{-1/2}\rho\Sigma^{-1/2}\|_{F}^{2}>0$

since $\inf_{P\in\mathcal{A}_{\lambda}}|\rho^{P}_{k\ell}|>0$ and
$\Sigma^{-1/2}$ is positive definite. Therefore
$nT_{n}\to_{\mathcal{A}_{\lambda}}\infty$, and so we have

$\displaystyle\inf_{P\in\mathcal{A}_{\lambda}}P(nT_{n}>c)$
$\displaystyle=\inf_{P\in\mathcal{A}_{\lambda}}(1-P(nT_{n}\leq c))$
$\displaystyle=-\sup_{P\in\mathcal{A}_{\lambda}}(P(nT_{n}\leq c))-1)\to 1$

as $n\to\infty$ for all $c\in\mathbb{R}$. From this we have

$\displaystyle\liminf_{n\to\infty}\inf_{P\in\mathcal{A}_{\lambda}}E_{P}(\hat{\Psi}_{n})=\liminf_{n\to\infty}\inf_{P\in\mathcal{A}_{\lambda}}P(nT_{n}>z_{1-\alpha})=1$

for all $\alpha\in(0,1)$. $\square$

##  Appendix B Modes of Stochastic Convergence

Let $\mathcal{M}$ denote some class of distributions. We start by defining the
notions of small and big O in probability.

###  B.1 Small and big-O in probability

All sequences $(a_{n})$ and $(b_{n})$ below are assumed to be non-zero.

######  Definition B.1.1.

Let $(X_{n})$ and $(a_{n})$ be sequences of random variables in $\mathbb{R}$.
If for every $\varepsilon>0$

$\displaystyle\sup_{P\in\mathcal{M}}P(|X_{n}/a_{n}|>\varepsilon)\to 0$

for $n\to\infty$ then we say that $X_{n}$ is small O of $a_{n}$ in probability
uniformly over $\mathcal{M}$ and write $X_{n}\in o_{\mathcal{M}}(a_{n})$. If
for every $\varepsilon>0$ there is $M>0$ such that

$\displaystyle\sup_{n\in\mathbb{N}}\sup_{P\in\mathcal{M}}P(|X_{n}/a_{n}|>M)<\varepsilon$

then we say that $X_{n}$ is big O of $a_{n}$ in probability uniformly over
$\mathcal{M}$ and write $X_{n}\in\mathcal{O}_{\mathcal{M}}(a_{n})$.

When $X_{n}\in\mathcal{O}_{\mathcal{M}}(a_{n})$ we also say that $X_{n}$ is
stochastically bounded by $a_{n}$ uniformly over $\mathcal{M}$. When $X_{n}\in
o_{\mathcal{M}}(1)$ we will typically write $X_{n}\to_{\mathcal{M}}0$.

######  Lemma B.1.2.

Let $(X_{n}),(a_{n})$ and $(b_{n})$ be sequences of random variables in
$\mathbb{R}$ such that $X_{n}\in\mathcal{O}_{\mathcal{M}}(a_{n})$. Then it
holds that $b_{n}X_{n}\in\mathcal{O}_{\mathcal{M}}(a_{n}b_{n})$.

######  Lemma B.1.3.

Assume that $X_{n}\in\mathcal{O}_{\mathcal{M}}(a_{n})$ and
$Y_{n}\in\mathcal{O}_{\mathcal{M}}(b_{n})$. Then
$X_{n}Y_{n}\in\mathcal{O}_{\mathcal{M}}(a_{n}b_{n})$.

######  Lemma B.1.4.

Assume $X_{n}\in\mathcal{O}_{\mathcal{M}}(a_{n})$ and that $a_{n}\in o(1)$.
Then $X_{n}\in o_{\mathcal{M}}(1)$.

######  Lemma B.1.5.

Assume that $X_{n}\in o_{\mathcal{M}}(1)$ and that $|X_{n}|\leq C$ for all
$n\geq 1$ for a constant $C$ that does not depend on $P$. Then
$\sup_{P\in\mathcal{M}}E_{P}|X_{n}|\to 0$ for $n\to\infty$.

We now turn to uniform convergence in distribution.

###  B.2 Uniform convergence in distribution

We follow Kasy (2019) and Bengs and Holzmann (2019).

######  Definition B.2.1.

Let $X,X_{1},X_{2},\dots$ be real valued random variables with distribution
determined by $P\in\mathcal{M}$. If it holds that

$\displaystyle\sup_{P\in\mathcal{M}}|E_{P}(f(X_{n}))-E_{P}(f(X))|\to 0$

for $n\to\infty$ for all functions $f:\mathbb{R}\to\mathbb{R}$ that are
bounded and continuous, then we say that $(X_{n})$ converges uniformly in
distribution to $X$ over $\mathcal{M}$. In this case we write
$X_{n}\Rightarrow_{\mathcal{M}}X$.

######  Lemma B.2.2 (Uniform Slutsky’s Lemma).

Assume that $X_{n}\Rightarrow_{\mathcal{M}}X$ and that
$Y_{n}\to_{\mathcal{M}}0$. Then $X_{n}+Y_{n}\Rightarrow_{\mathcal{M}}X$.

###### Proof.

See Bengs and Holzmann (2019) Theorem 6.3. ∎

## Acknowledgements

This work was supported by a research grant (13358) from VILLUM FONDEN.

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