\documentclass[notitlepage]{article}
\usepackage{../../Math556}
\usepackage{listings}
\usepackage{numprint}
\usepackage{enumerate}
\usepackage{multirow}
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\begin{document}

<<setup, cache=FALSE, echo=FALSE>>=
library(knitr)
# global chunk options
opts_chunk$set(cache=TRUE, autodep=TRUE)
options(scipen=999)
options(repos=c(CRAN="https://cloud.r-project.org/"))
inline_hook <- function (x) {
  if (is.numeric(x)) {
    # ifelse does a vectorized comparison
    # If integer, print without decimal; otherwise print two places
    res <- ifelse(x == round(x),
      sprintf("%.6f", x),
      sprintf("%.6f", x)
    )
    paste(res, collapse = ", ")
  }
}
knit_hooks$set(inline = inline_hook)
@

\begin{center}
\textsclarge{MATH 556: Mathematical Statistics I}

\vspace{0.1 in} \textsc{Convergence In Distribution: Worked Examples}
\end{center}


\noindent \textbf{EXAMPLE 1: }Continuous random variable $X_n$ with support $\mathbb{X}\equiv \left( 0,n\right]$ for $n>0$ and cdf
\begin{equation*}
F_{X_{n}}(x)=1-\left( 1-\frac{x}{n}\right) ^{n}\qquad 0<x\leq n
\end{equation*}
and standard cdf behaviour outside of this support. Then as $n\lra \infty $, $\mathbb{X}\equiv \left( 0,\infty \right) $, and for all $x>0$
\begin{equation*}
F_{X_{n}}\left( x\right) \lra 1-e^{-x}\qquad \therefore
\qquad F_{X_{n}}\left( x\right) \lra F_{X}\left( x\right)
=1-e^{-x}
\end{equation*}
and hence
\begin{equation*}
X_{n}\CiD X \qquad
\end{equation*}
with $X\sim Exponential(1)$.

<<C01,comment='+', fig.width=6, fig.height=4, fig.align='center'>>=
library(plotrix)
redblue<-colorRampPalette(c("cyan","blue"))
N<-20
rbc<-redblue(N)
nvec<-c(1:N)
xvec<-seq(0,8,by=0.01)
par(mar=c(4,4,1,1))
plot(xvec,xvec*0,type='n',ylim=range(0,1),xlab='x',ylab=expression(F[X[n]](x)))
for(i in 1:N){
    xv<-pmin(xvec,nvec[i])
    yvec<-1-(1-xv/nvec[i])^nvec[i]
    lines(xvec,yvec,col=rbc[i])
}
lines(xvec,pexp(xvec,1),col='red',lwd=2)
color.legend(6,0.0,6.5,0.8,paste('n =',1:N),rbc,gradient="y",cex=0.5)
legend(7,0.8,c('Limit'),lwd=2,lty=1,col='red',cex=0.5)
@

\pagebreak

\noindent \textbf{EXAMPLE 2: \ }Continuous random variable $X_n$ with
support $\mathbb{X}\equiv \left( 0,\infty \right) $ and cdf
\begin{equation*}
F_{X_{n}}(x)=\left( 1-\frac{1}{1+nx}\right) ^{n}\qquad 0<x<\infty
\end{equation*}
and zero otherwise. \ Then as $n\lra \infty $, for all $x>0$
\begin{equation*}
F_{X_{n}}\left( x\right) \lra e^{-1/x}\qquad \therefore
\qquad F_{X_{n}}\left( x\right) \lra F_{X}\left( x\right)
=e^{-1/x}
\end{equation*}
as
\begin{equation*}
\lim_{n\lra \infty }\left( 1-\frac{1}{1+nx}\right)
^{n}=\lim_{n\lra \infty }\left( 1-\frac{1}{nx}\right)
^{n}=\lim_{n\lra \infty }\left( 1-\frac{1/x}{n}\right) ^{n}
\end{equation*}
and for any $z$
\begin{equation*}
\lim_{n\lra \infty }\left( 1+\dfrac{z}{n}\right) ^{n}=e^{z}
\end{equation*}

<<C02,comment='+', fig.width=6, fig.height=4, fig.align='center'>>=
N<-20
nvec<-c(1:N)
xvec<-seq(0,10,by=0.01)
par(mar=c(4,4,1,1))
plot(xvec,xvec*0,type='n',ylim=range(0,1),xlab='x',ylab=expression(F[X[n]](x)))
for(i in 1:N){
    yvec<-(1-1/(1+nvec[i]*xvec))^nvec[i]
    lines(xvec,yvec,col=rbc[i])
}
fx<-exp(-1/xvec)
lines(xvec,fx,col='red',lwd=2)
color.legend(8,0.0,8.5,0.8,paste('n =',1:N),rbc,gradient="y",cex=0.5)
legend(9,0.8,c('Limit'),lwd=2,lty=1,col='red',cex=0.5)
@

\pagebreak

\noindent \textbf{EXAMPLE 3: }Continuous random variable $X_n$ with support $%
\mathbb{X}\equiv \left[ 0,1\right] $ and cdf
\begin{equation*}
F_{X_{n}}(x)=x-\sin \left( 2n\pi x\right)/(2n\pi)\qquad 0\leq x\leq
1
\end{equation*}
and standard cdf behaviour outside of this support. \ Then as $n\lra \infty $, and for all
$0\leq x\leq 1$
\begin{equation*}
F_{X_{n}}\left( x\right) \lra x\qquad \therefore \qquad
F_{X_{n}}\left( x\right) \lra F_{X}\left( x\right) =x
\end{equation*}
and hence
\begin{equation*}
X_{n}\CiD X\qquad \text{where } X\sim Uniform(0,1)
\end{equation*}

<<C03,comment='+', fig.width=5, fig.height=5, fig.align='center'>>=
N<-20
nvec<-c(1:N)
xvec<-seq(0,1,by=0.001)
par(mar=c(4,4,1,1))
plot(xvec,xvec*0,type='n',ylim=range(0,1),xlab='x',ylab=expression(F[X[n]](x)))
for(i in 1:N){
    yvec<-xvec-sin(2*nvec[i]*pi*xvec)/(2*nvec[i]*pi)
    lines(xvec,yvec,col=rbc[i])
}
lines(xvec,xvec,col='red',lwd=2)
color.legend(0.8,0.0,0.85,0.6,paste('n =',1:N),rbc,gradient="y",cex=0.5)
legend(0.9,0.6,c('Limit'),lwd=2,lty=1,col='red',cex=0.5)
@
\noindent \textbf{NOTE}: \ for the pdf
\begin{equation*}
f_{X_{n}}(x)=1-\cos \left( 2n\pi x\right) \qquad 0\leq x\leq 1
\end{equation*}
and for all $x$ there \textbf{is no limiting value }$n\lra \infty $. \

\pagebreak

\noindent \textbf{EXAMPLE 4: }Continuous random variable $X_n$ with
support $\mathbb{X}\equiv \left[ 0,1\right] $ and cdf
\begin{equation*}
F_{X_{n}}(x)=1-\left( 1-x\right) ^{n}\qquad 0\leq x\leq 1
\end{equation*}
and standard cdf behaviour outside of this support. \ Then as $n\lra \infty $, and for $x\in
\mathbb{R}$
\begin{equation*}
F_{X_{n}}\left( x\right) \lra \left\{
\begin{array}{ll}
0 & x\leq 0\medskip  \\
1 & x>0
\end{array}
\right. .
\end{equation*}
This limiting form is \textbf{not} continuous at $x=0$, as $x=0$ is
not a point of continuity, and the \textbf{ordinary definition of
convergence in}
\textbf{distribution cannot be applied}. \ However, it is clear that for $%
\epsilon >0$ ,
\begin{equation*}
P_{X_n}\left[ \left| X_n\right| <\epsilon \right] =1-\left( 1-\epsilon
\right) ^{n}\lra 1\text{ as }n\lra \infty
\end{equation*}
so it is still correct to say
\begin{equation*}
X_{n}\CiD X\qquad \text{where }P_X\left[
X=0\right] =1
\end{equation*}
so the limiting distribution is \textbf{degenerate at }$x=0$. \

<<C04,comment='+', fig.width=5, fig.height=5, fig.align='center'>>=
N<-20;nvec<-5*c(1:N)
xvec<-seq(0,1,by=0.001)
par(mar=c(4,4,1,1))
plot(xvec,xvec*0,type='n',ylim=range(0,1),xlab='x',ylab=expression(F[X[n]](x)))
for(i in 1:N){
    yvec<-1-(1-xvec)^nvec[i]
    lines(xvec,yvec,col=rbc[i])
}
lines(c(0,1),c(1,1),col='red',lwd=2);points(0,1,pch=19,col='red')
color.legend(0.8,0.0,0.85,0.6,paste('n =',5*c(1:N)),rbc,gradient="y",cex=0.5)
legend(0.9,0.6,c('Limit'),lwd=2,lty=1,col='red',cex=0.5)
@

\pagebreak \noindent \textbf{EXAMPLE 5: }Continuous random variable $X_n$ with support $\mathbb{X}\equiv \left( 0,\infty \right) $ and cdf
\begin{equation*}
F_{X_{n}}(x)=\left( \frac{x}{1+x}\right) ^{n}\qquad x>0
\end{equation*}
and zero otherwise. \ Then as $n\lra \infty $, and for $x>0$
\begin{equation*}
F_{X_{n}}\left( x\right) \lra 0
\end{equation*}
Thus there is \textbf{no limiting distribution}.

<<C05,comment='+', fig.width=6, fig.height=4.5, fig.align='center'>>=
N<-20
nvec<-c(1:N)
xvec<-seq(0,10,by=0.01)
par(mar=c(4,4,1,1))
plot(xvec,xvec*0,type='n',ylim=range(0,1),xlab='x',ylab=expression(F[X[n]](x)))
for(i in 1:N){
    yvec<-(xvec/(1+xvec))^nvec[i]
    lines(xvec,yvec,col=rbc[i])
}
lines(c(0,10),c(0,0),col='red',lwd=2)
@
\noindent Now let $V_{n}=X_{n}/n$. \ Then $\mathbb{V}\equiv \left(
0,\infty \right) $ and the cdf of $V_{n}$ is
\begin{equation*}
F_{V_{n}}(v)=P_{V_n}\left[ V_{n}\leq v\right] =P_{X_n} \left[ X_{n}/n\leq
v\right] =P_{X_n}\left[ X_{n}\leq nv\right] =F_{X_{n}}(nv)=\left(\frac{nv}{1+nv}\right) ^{n}\qquad v>0
\end{equation*}
and as $n\lra \infty $, for all $v>0$
\begin{equation*}
F_{V_{n}}\left( v\right) \lra e^{-1/v}\qquad \therefore
\qquad F_{V_{n}}\left( v\right) \lra F_{V}\left( v\right)
=e^{-1/v}
\end{equation*}
and the limiting distribution of $V_{n}$ does exist.

<<C06,comment='+', fig.width=6, fig.height=4.5, fig.align='center'>>=
N<-20
nvec<-c(1:N)
xvec<-seq(0,10,by=0.01)
par(mar=c(4,4,1,1))
plot(xvec,xvec*0,type='n',ylim=range(0,1),xlab='x',ylab=expression(F[X[n]](x)))
for(i in 1:N){
    yvec<-(nvec[i]*xvec/(1+nvec[i]*xvec))^nvec[i]
    lines(xvec,yvec,col=rbc[i])
}
fx<-exp(-1/xvec)
lines(xvec,fx,col='red',lwd=2)
color.legend(8,0.0,8.5,0.8,paste('n =',1:N),rbc,gradient="y",cex=0.5)
legend(9,0.8,c('Limit'),lwd=2,lty=1,col='red',cex=0.5)
@

\pagebreak

 \noindent \textbf{EXAMPLE 6: }Continuous random variable
$X_n$ with support $\mathbb{X}\equiv \left( 0,\infty \right) $ and cdf
\begin{equation*}
F_{X_{n}}(x)=\frac{\exp (nx)}{1+\exp (nx)}\qquad x\in \mathbb{R}
\end{equation*}
and zero otherwise. \ Then as $n\lra \infty $
\begin{equation*}
F_{X_{n}}(x)\lra \left\{
\begin{array}{ll}
0 & x<0\medskip  \\
\dfrac{1}{2} & x=0\medskip  \\
1 & x>0
\end{array}
\right. \qquad x\in \mathbb{R}
\end{equation*}
This limiting form is \textbf{not} a cdf, as it is not right
continuous at $x=0$. \ However, as $x=0$ is not a point of
continuity, and the ordinary definition of convergence in
distribution does not apply. \ However, it is clear that for
$\epsilon >0$,
\begin{equation*}
P_{X_n}\left[ \left| X_n \right| <\epsilon \right] =\frac{\exp (n\epsilon
)}{1+\exp (n\epsilon )}-\frac{\exp (-n\epsilon )}{1+\exp (-n\epsilon
)}\lra 1 \text{ as }n\lra \infty
\end{equation*}
so it is still correct to say $X_{n}\CiD X$ where $P_X \left[ X=0 \right] = 1$, and the limiting distribution is \textbf{degenerate at }$x=0$.

<<C07,comment='+', fig.width=6, fig.height=4.5, fig.align='center'>>=
N<-20;nvec<-c(1:N);xvec<-seq(-10,10,by=0.01)
par(mar=c(4,4,1,1))
plot(xvec,xvec*0,type='n',ylim=range(0,1),xlab='x',ylab=expression(F[X[n]](x)))
for(i in 1:N){
    yvec<-(exp(nvec[i]*xvec)/(1+exp(nvec[i]*xvec)))
    lines(xvec,yvec,col=rbc[i])
}
lines(c(-10,0),c(0,0),col='red',lwd=2)
lines(c(0,10),c(1,1),col='red',lwd=2);points(0,1,pch=19,col='red')
color.legend(7,0.0,8,0.6,paste('n =',c(1:N)),rbc,gradient="y",cex=0.5)
legend(8.5,0.6,c('Limit'),lwd=2,lty=1,col='red',cex=0.5)
@
\end{document}