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\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
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\begin{document}

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\title{Article template}

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\author{
Joshua Lee Padgett$^{1,2}$
\bigskip
\\
\small{$^1$ Department of Mathematical Sciences, University of Arkansas,}
\vspace{-0.1cm}\\
\small{Arkansas, USA, e-mail: \texttt{padgett@uark.edu}}
\smallskip
\\
\small{$^2$ Center for Astrophysics, Space Physics, and Engineering Research,}
\vspace{-0.1cm}\\
\small{Baylor University, Texas, USA, e-mail: \texttt{padgett@uark.edu}}
}

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\date{\today}

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\maketitle

\begin{abstract}
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Abstract goes here\dots
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\end{abstract}

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\tableofcontents

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\section{Introduction}
\label{sec:intro}

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Add an appropriate introduction\dots
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\section{Monte Carlo approximations}
\label{sec:monte_carlo}

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\begin{athm}{lemma}{Lp_monte_carlo}
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Let 
$p \in (2,\infty)$, 
$n \in \N$, 
let 
$ ( \Omega, \cF, \P ) $ 
be a probability space, 
and let 
$ X_i  \colon \Omega \to \R $, $ i \in \{1, 2, \dots, n\} $, 
be i.i.d.\ random variables with 
$\E[ \abs{ X_1 } ] < \infty$.
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Then it holds that
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\begin{equation}
\pr[\Big]{ \E \br[\pig]{ \abs[\big]{ \E[X_1] - \tfrac{1}{n} \pr[\big]{ \smallsum_{ i = 1 }^{ n } X_{ i } } }^p } }^{\!\!\nicefrac{1}{p}}
\le 
\br[\big]{ \tfrac{p-1}{n} }^{\nicefrac{1}{2}} 
\pr[\Big]{ \E \br[\pig]{ \abs[\big]{ X_1 - \E[ X_1 ] }^p } }^{\!\!\nicefrac{1}{p}}
\dpp
\end{equation}
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\end{athm}

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\begin{aproof}
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First, \nobs that  
\Enum{
the hypothesis that for all $i\in\{1,2,\allowbreak\dots,\allowbreak n\}$ it holds that 
$  X_i\colon \Omega\to \R $
are i.i.d.\ random variables 
}
that
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\begin{equation}
\begin{split}
\E\br[\pig]{ \abs[\big]{ \E[X_1] - \tfrac{1}{n} \pr{ \smallsum_{ i = 1 }^{ n } X_{ i } } }^p }
&
= 
\E\br[\pig]{ \abs[\big]{ \tfrac{1}{n} \pr{ \smallsum_{ i = 1 }^{ n } (\E[X_1] - X_{ i }) } }^p }
\\
&
= 
n^{-p} \, \E\br[\pig]{ \abs[\big]{ \smallsum_{ i = 1 }^{ n } (\E[X_i] - X_{ i }) }^p }
\dpp
\end{split}
\end{equation}
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\Enum{
This
;the fact that for all $i\in\{1,2,\dots,n\}$ it holds that 
$  X_i\colon \Omega\to \R $
are i.i.d.\ random variables
;
\eg Rio~\cite[Theorem 2.1]{rio2009moment} (applied with $p \with p$, $(S_i)_{i\in\{0,1,\dots,n\}} \with (\sum_{k=1}^i (\E[X_k] - X_k))_{i\in\{0,1,\dots,n\}}$, $(X_i)_{i \in\{1,2,\dots,n\}} \with (\E[X_i] - X_i)_{i\in\{1,2,\dots,n\}}$ in the notation of Rio~\cite[Theorem 2.1]{rio2009moment})
}
that
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\begin{equation}
\begin{split}
\pr[\Big]{ \E \br[\pig]{ \abs[\big]{ \E[X_1] - \tfrac{1}{n} \pr[\big]{ \smallsum_{ i = 1 }^{ n } X_{ i } } }^p } }^{\!\!\nicefrac{2}{p}}
&
= 
\tfrac{1}{n^2} \pr[\Big]{ \E\br[\pig]{ \abs[\big]{ \smallsum_{ i = 1 }^{ n } (\E[X_i] - X_{ i }) }^p } }^{\!\!\nicefrac{2}{p}} 
\\
& 
\le 
\tfrac{(p-1)}{n^2} \br[\Big]{ \smallsum_{ i = 1 }^{ n } \pr[\Big]{ \E\br[\pig]{ \abs[\big]{ \E[X_i] - X_{ i } }^p } }^{\!\!\nicefrac{2}{p}} }
\\
&
= 
\tfrac{(p-1)}{n^2} \br[\Big]{ n \pr[\Big]{ \E\br[\pig]{ \abs[\big]{ \E[X_1] - X_{ 1 } }^p } }^{\!\!\nicefrac{2}{p}} } 
\\
& 
= 
\tfrac{(p-1)}{n} \pr[\Big]{ \E\br[\pig]{ \abs[\big]{ \E[X_1] - X_{ 1 } }^p } }^{\!\!\nicefrac{2}{p}}
\dpp
\end{split}
\end{equation}
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\end{aproof}

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\begin{athm}{corollary}{prop:mlp}
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Let $p \in [2,\infty)$, 
$n \in \N$, 
let 
$ ( \Omega, \cF, \P ) $ 
be a probability space, 
and 
let 
$ X_i  \colon \Omega \to \R $,
$ i \in \{1, 2, \dots, n\} $,
be i.i.d.\ random variables with 
$\E\br[]{ \abs{ X_1 } } < \infty$. 
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Then it holds that
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\begin{equation}\label{eq:2_9}
\pr[\Big]{\E\br[\pig]{ \abs[\big]{ \E[X_1] - \tfrac{1}{n} \pr[]{ {\smallsum_{ i = 1 }^{ n }} X_{ i } } }^p } }^{\!\!\nicefrac{1}{p}}
\le 
\br[\big]{ \tfrac{p-1}{n} }^{\nicefrac{1}{2}} 
\pr[\Big]{ \E\br[\pig]{ \abs[\big]{ X_1 - \E[ X_1 ] }^p } }^{\!\!\nicefrac{1}{p}}
\dpp
\end{equation}
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\end{athm}

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\begin{aproof}
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\Nobs that
\Enum{
\eg Grohs et al.\ \cite[Lemma 2.3]{grohs2018proof}
;
\cref{Lp_monte_carlo}
}
\cref{eq:2_9}. 
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\end{aproof}

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\begin{athm}{definition}{rand_const}
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Let $p \in [2,\infty)$. 
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Then we denote by 
$\firstConstant{p} \in \R$ 
the real number given by
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\begin{equation}\label{eq:rand_const}
\firstConstant{p} 
= 
\inf\left\{ c \in \R \colon \left[ 
\begin{aligned}
& \text{It holds for every probability space $(\Omega,\cF,\P)$ and every} \\
& \text{random variable $X \colon \Omega \to \R$ with $\E[\abs{X}] < \infty$ that } \\
& \pr[\big]{ \E\br[\big]{ \abs{ X - \E[X] }^p } }^{\!\nicefrac{1}{p}} \le c \pr[\big]{ \E\br[\big]{ \abs{X}^p } }^{\!\nicefrac{1}{p}}
\end{aligned}
\right] \right\} 
\dpp
\end{equation}
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\end{athm}

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\begin{athm}{corollary}{cor:exp_bd}
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Let 
$p\in[2,\infty)$, 
$n\in\N$, 
let 
$(\Omega,\cF,\P)$ 
be a probability space, 
and let 
$X_i \colon \Omega \to \R$, $i\in\{1,2,\dots,n\}$, 
be i.i.d.\ random variables with 
$\E[\abs{X_1}] < \infty$. 
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Then
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\begin{equation}\label{prob_bd1}
\pr[\Big]{\E\br[\pig]{ \abs[\big]{ \E[X_1] - \tfrac{1}{n} \pr[]{ {\smallsum_{ i = 1 }^{ n }} X_{ i } } }^p } }^{\!\!\nicefrac{1}{p}}
\le 
\tfrac{\secondConstant{p}}{n^{\nicefrac{1}{2}}} \pr[\Big]{ \E\br[\pig]{ \abs[\big]{ X_1 }^p } }^{\!\!\nicefrac{1}{p}}
\end{equation}
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(cf.\ \cref{rand_const}).
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\end{athm}

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\begin{aproof}
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\Nobs 
that 
\Enum{
\cref{rand_const}
;
\cref{prop:mlp}
}
that \cref{prob_bd1} holds. 
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\end{aproof}

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