Let $x_1,x_2,$dots be independent random variables such that $P\{x_1=j\}=$

${}_1,j0.$ Say that a record occurs at time $n$ if $x_n>$max$(x_1,$dots,$x_{n-1}),$

where $x_0=-,$ and if a record does occur at time $n$ call $x_n$ the record

value. Let $R_i$ denote the $i$ th record value.

$($a$)$ Argue that $\{R_i,i1\}$ is a Markov chain and compute its transi$-$

tion probabilities.

$($b$)$ Let $T_1$ denote the time between the $i$ th and $(i+1)$ st record. Is $\{\begin{array}{c}{T}_1\end{array},$

$i1$ a Markov chain? What about $\{(R_1,T_1),i1\}?$ Compute

transition probabilities where appropriate.

$($c$)$ Let $S_n={\displaystyle \underset{i=1}{\overset{n}{}}}{T}_i,n1.$ Argue that $\{S_n,n1\}$ is a Markov chain

and find its transition probabilities.