For this assignment do not upload R code or output.

Please summarize your findings on paper, scan and upload the pdf$,$ for the following two questions:

Use the pieces of code below to create the transition matrix for a Ladder Chain with k$=10$

states, and then run some experiments $($run at least $10$ times with new lists of random numbers$)$ to estimate how far back in time would the Propp$-$Wilson algorithm have to start for at time $0$

to be only one state occupied.

What would be your estimate if the Ladder Chain had k$=100$

states? Explain your reasoning for the answer.

Transition matrix

Copy

k$=5$

P$=$matrix$(0,$ncol$=5,$ nrow$=5)$

P$[1,1]=\mathrm{.}5$

P$[$k$,$k$]=\mathrm{.}5$

for$($i in $1$:$($k$-1))\{$P$[$i$,$i$+1]=\mathrm{.}5$

P$[$i$+1,$i$]=\mathrm{.}5\}$

Preparatory function

Copy

ppsi$<-$ function$($x$,$p$)\{$

cum$\_$prob $<-$ cumsum$($p$)$

for $($j in $1$:length$($p$))\{$

if $($x $<$ cum$\_$prob$[$j$])\{$

j

break

$\}$

$\}$

return$($j$)$

$\}$

Update function

Copy

phi$<-$function$($s$,$x$)\{$ppsi$($x$,$P$[$s$,])\}$

List of random numbers

Copy

n$=10$

Ul$=$runif$(2^$n$)$

Run the Propp$-$Wilson algorithm

Copy

for$($j in $0$:n$)\{$ts$=1$:k

Os$=1$:k

for$($i in $2^$j:$1)\{$

ts$=$lapply$($ts$,$phi,x$=$Ul$[$i$])$

Os$=$cbind$($Os$,$ts$)\}$

print$($Os$)$

if$($length$($unique$($ts$))==1)\{$

X$=$unlist$($unique$($ts$))$

print$($X$)$

break$\}$

$\}$

## Os ts

## $[1,]12$

## $[2,]23$

## $[3,]34$

## $[4,]45$

## $[5,]55$

## Os ts ts

## $[1,]112$

## $[2,]212$

## $[3,]323$

## $[4,]434$

## $[5,]545$

## Os ts ts ts ts

## $[1,]11112$

## $[2,]21112$

## $[3,]32112$

## $[4,]43212$

## $[5,]54323$

## Os ts ts ts ts ts ts ts ts

## $[1,]123211112$

## $[2,]234321112$

## $[3,]345432112$

## $[4,]455432112$

## $[5,]555432112$

## $[1]2$