\documentclass{beamer}
\usetheme[secheader]{Boadilla}

\title{Portmamteau Theorem}
\author{Jaime Frade}
\institute[Florida State University]{
\inst{}Department of Statistics\\
Florida State University
\and
\inst{}Florentina Bunea\\
STA5447
}
\date{\today}


\begin{document}

\frame{\titlepage}

\section[Outline]{}
\frame{\tableofcontents}



\section{Show 7 $\Leftrightarrow$ 9}
\frame
{
\begin{equation*}
\textrm{Show} \hspace{.15cm}  F_n \overbrace{\rightarrow}^{d} F \Leftrightarrow E[g_n(x_n)] = \int gdF_n \rightarrow \int gdF=E[g(x)]
\end{equation*}

By \textbf{Hally-Bray Theorem}, consider some $(\Omega, \mathbb{A}, P)$.  Suppose that $g$ is bdd and is continuous a.s. $F$. $(g\in C_b)$, then

\begin{equation}
E[g_n(x_n)] = \int gdF_n \rightarrow \int gdF=E[g(x)]
\end{equation}

By converse, since $E[g_n(x_n)] \rightarrow E[g(x)]$ $\Rightarrow$ $F_n \rightarrow F$
}

\section{Show 7 $\rightarrow$ 9}
\frame
{
\begin{equation*}
\textrm{Show} \hspace{.15cm} F_n \overbrace{\rightarrow}^{d} F \Rightarrow E[g_n(x_n)] = \int gdF_n \rightarrow \int gdF=E[g(x)]
\end{equation*}

Notice that for any $\delta(B) = (C_b)^c$.  For any B with $P(x \in \delta B)=0$ by continuous mapping theorem, $E(1_B(x_n)) \rightarrow E(1_B(x))$ as $n \rightarrow \infty$
\vspace{.25cm}
\\
$F_n(x) = P(x_n \in (-\infty,x])$, with $\delta(-\infty,x]=\{x\}$ so provided $P(X=x)=0$, then $P(x_n \in (-\infty,x]) \rightarrow P(x \in (-\infty,x]).$
}
\frame
{
$X \in B \leftrightarrow 1_B$.  $X \rightarrow 1_B(x)$ is not continuous at $\delta B$.  Otherwise, $h(x) = 1_B(x)$ is continuous and bounded $ \forall  x \notin \delta B$
\vspace{.25cm}
\\
$X_n \rightarrow X \Rightarrow h(x_n) \overbrace{\rightarrow}^{a.s} h(x)$ and $E[h(x_n)] \overbrace{\rightarrow}^{DCT} E[h(x)]$. 
\vspace{.25cm}
\\
$E[h(x_n)] = E[1_B(x_n)] =P(x_n) \rightarrow E[h(x)] = P(x \in B)$
}
\section{Show 7 $\leftarrow$ 9}
\frame
{
\begin{equation*}
\textrm{Show} \hspace{.15cm}  F_n \overbrace{\rightarrow}^{d} F \Leftarrow E[g_n(x_n)] = \int gdF_n \rightarrow \int gdF=E[g(x)]
\end{equation*}

Take $B=(-\infty,x]$, then $\delta B = \{x\}$
\vspace{.25cm}
\\
$P_n(B) \rightarrow P(B)$ with $P(\delta B) = 0$
\vspace{.25cm}
\\
$F_n(x) \rightarrow F(x)$  $\forall x$, $P(\{x,y\})=0$, $\forall x$ $F$ has no jumps at $x$, all $x \in C_b$
}

\section{Definition of $g(x)$}
\frame
{
For the rest of the proof, define a trapezoidal function of height 1.  Let $g_k \in C_b$.  Note the $g_k \downarrow 1_{[a,b]}$ as $k \rightarrow \infty$.  
\vspace{.25cm}
\\
Since $g_k \leq 1$ and $g_k \downarrow 1_{[a,b]}$, then
\vspace{.25cm}
\\
$\int g_k dF \downarrow F([a,b]) = F((a,b])$, then 
\vspace{.25cm}
\\
$\limsup F_n(a,b] \leq F(a,b]$
\begin{figure}
\includegraphics[scale=0.75]{plot1.png} 
\end{figure}
}


\frame
{
Note the $h_k \uparrow 1_{(a,b)}$ 
\vspace{.25cm}
\\
$F_n(a,b] \geq \int h_k dF_n \rightarrow \int h_k dF$ by MCT
\vspace{.25cm}
\\
$\int h_k dF \uparrow F(a,b) = F(a,b]$  as $k \rightarrow \infty$.
\vspace{.25cm}
\\
Thus, $\liminf F_n((a,b]) \geq F(a,b]$
\begin{figure}
\includegraphics[scale=0.75]{plot2.png} 
\end{figure}
}

\section{Show 9 $\rightarrow$ 12}
\frame
{
\begin{eqnarray}
\textrm{Show} \hspace{.15cm}  E[g_n(x_n)] = \int gdF_n \rightarrow \int gdF=E[g(x)] \forall  g\in C_b \nonumber 
\\
\Rightarrow \liminf P_n(B) \geq P(B)   \forall \hspace{.25cm} \textrm{open sets} \hspace{.25cm} B \nonumber 
\end{eqnarray}

Define a function, $g$, such that satisfies first part, then will show because of the way the function is setup, the function will imply the second part.
\vspace{.25cm}
\\
Let $B$ be an open set and $F=B^c$.  Consider a sequence of functions, $g_n \in C_b$, 
\begin{equation}
g_n(x) = \min(1,nd(x,F))
\end{equation}

Show that $g_n(x) \uparrow 1(x)$
\begin{description}
\item[a.]  When $x\notin F \Rightarrow$ let $\epsilon=d(x,F)>0$, then $g_n(x) = \min(1,n\epsilon)$, where $n\epsilon \geq 1$.  For $n$ large, $\Rightarrow g_n(x) =1$
\item[b.]  When $x \in F$, then $d(x,F) =0 \Rightarrow g_n(x)=0$ 
\end{description}

Therefore, 
\begin{equation*}
g_n(x) = \left\{\begin{array}{rl} 0 & x \notin F=B^c \\ 1 & x \in F=B^c \end{array}\right.
\end{equation*}
as $n\rightarrow \infty$
}
\frame
{
By 9, 
$P_n(x) \geq \int g_k dF_n \rightarrow \int g_kdF$ as $n \rightarrow \infty$
\vspace{.25cm}
\\
and
\vspace{.25cm}
\\
$\liminf P_n(B) \geq \int g_kdF$
\vspace{.25cm}
\\
by letting $k \rightarrow \infty$ and MCT, 

\begin{eqnarray}
P_n(B) & \geq & \int g_kdF_n \overbrace{\rightarrow}^{9} \int g_k \uparrow P(B), n \rightarrow \infty \nonumber
\\
\liminf_n P_n(B) & \geq & \lim_{k\rightarrow \infty} \int g_kdF \overbrace{=}^{MCT} \int \lim_{k\rightarrow \infty} g_kdF = \int gdF = P(B) \nonumber
\\
\liminf_n P_n(B) & \geq & \int 1_BdF = P(B)
\end{eqnarray}
}

\section{Show 9 $\rightarrow$ 11}
\frame
{
\begin{eqnarray}
\textrm{Show} \hspace{.15cm}  E[g_n(x_n)] = \int gdF_n \rightarrow \int gdF=E[g(x)] \forall g\in C_b \nonumber 
\\
\Rightarrow \limsup P_n(B) \leq P(B) \forall \hspace{.25cm} \textrm{closed sets} \hspace{.25cm} B \nonumber
\end{eqnarray}

\begin{eqnarray}
P(B) &=& \int 1_BdF_n \leq \int g_kdF_n \overbrace{\rightarrow}^{9} \int g_k dF \downarrow P(B^c) \nonumber
\\
\limsup_n P_n(B) & \leq & \lim_{k\rightarrow \infty} \int g_k dF_n \overbrace{=}^{MCT} \int \lim_{k\rightarrow \infty} g_kdF = \int gDF = P(B) \nonumber
\end{eqnarray}
}

\section{Show 11 $\rightarrow$ 12}
\frame
{
\begin{eqnarray}
\textrm{Show} \hspace{.15cm}  \limsup P_n(B) & \leq & P(B) \forall \hspace{.25cm} \textrm{closed sets} \hspace{.25cm} B \nonumber 
\\
\Rightarrow \liminf P_n(B) & \geq & P(B)	\forall \hspace{.15cm} \textrm{open sets} \hspace{.15cm} B \nonumber
\end{eqnarray}

\begin{eqnarray}
\limsup P_n(B) & \leq & P(B)	\forall \hspace{.15cm} \textrm{closed sets} \hspace{.15cm} B  \nonumber
\\
1-\limsup P_n(B) & \geq & P(B^c)	\nonumber
\\
1+\liminf(-P_n(B)) & \geq & P(B^c)	\nonumber
\\
\liminf(1-P_n(B)) & \geq & P(B^c)	\nonumber
\\
\liminf(P_n(B^c)) & \geq & P(B^C)
\end{eqnarray}
}

\section{Show 12 $\rightarrow$ 11}
\frame
{
\begin{eqnarray}
\textrm{Show} \hspace{.15cm}  \liminf P_n(B) & \geq & P(B)	\forall \hspace{.15cm} \textrm{open sets} \hspace{.15cm} B  \nonumber 
\\
\Rightarrow  \limsup P_n(B)  & \leq & P(B) \forall \hspace{.25cm} \textrm{closed sets} \hspace{.25cm} B \nonumber
\end{eqnarray}

\begin{eqnarray}
\liminf P_n(B) & \geq & P(B)	\forall \hspace{.15cm} \textrm{open sets} \hspace{.15cm} B \nonumber
\\
1-\liminf P_n(B) & \leq & P(B^c)	\nonumber
\\
1+\limsup(-P_n(B)) & \leq & P(B^c)	\nonumber
\\
\limsup(1-P_n(B)) & \leq & P(B^c)	\nonumber
\\
\limsup P_n(B^c) & \leq & P(B^c)
\end{eqnarray}
}

\section{Show 11 and 12 $\rightarrow$ 13}
\frame
{
\begin{eqnarray}
\textrm{Show} \hspace{.15cm}  \liminf P_n(B) & \geq & P(B)	\forall \hspace{.15cm} \textrm{open sets} \hspace{.15cm} B  \nonumber 
\\
\textrm{and} \hspace{.25cm} \limsup P_n(B) & \leq & P(B) \forall \hspace{.25cm} \textrm{closed sets} \hspace{.25cm} B \nonumber
\\
\Rightarrow \lim P_n(B) &=& P(B) \nonumber
\end{eqnarray}

Define the following:

\begin{description}
\item[a.]  $B^0$ is the interior of $B$, where $B^0 =  \cup\{U \subset B, \hspace{.15cm} \textrm{s.t.} \hspace{.15cm} U \hspace{.15cm} \textrm{is open}\}$, (Union of all open subsets of $B$.
\item[b.]  $\bar{B}$ is the closure of $B$, where $\bar{B} =  \bigcap \{B \subset C, \hspace{.15cm} \textrm{s.t.} \hspace{.15cm} C \hspace{.15cm} \textrm{is closed}\}$, (Intersection of all closed subsets of $B$.
\end{description}

\begin{equation*}
P(B)  \leq  \liminf P_n(B) \leq \limsup P_n(B) \leq P(B) 
\end{equation*}

$\forall$ $P-$continuity sets $B$, $(P(\delta B)=0)$, then the boundary $B$, $\delta B = \bar{B}/B^0$

Since $P(\delta B)=0$, then $P(\bar{B})=P(B^0) = P(B) \rightarrow \lim P_n(B) = P(B)$
}

\section{Show 13 $\rightarrow$ 7}
\frame
{
\begin{eqnarray}
\textrm{Show} \hspace{.15cm}  \lim P_n(B) = P(B) \rightarrow F_n \overbrace{\rightarrow}^{d} F \nonumber
\end{eqnarray}

For $\lim P_n(B) = P(B)$ $\forall$ $P-$ continuity sets $B$, , $(P(\delta B)=0)$, where $B \subset \Omega=\mathbb{R}$.

Let $B=(-\infty, x] \rightarrow \delta(-\infty,x] = \{x\}$.  Then $P(-\infty,x])=0$ $\forall$ continuity points $x \in \Omega$

\begin{equation*}
F_n \overbrace{\rightarrow}^{d} F \Rightarrow F_n((-\infty,x]) \rightarrow F((-\infty,x])
\end{equation*}
}

\end{document}