$($a$)$ For a Markov process

$u_t=u_{t-1}+v_t$ with $||<mn>1</mn>$

and random variables $v_t$ are such that $E(v_t)=0,$Var$(v_t)={}_v^2$ and Cov$(v_t,v_s)=0$ for $ts.$

Show that:

$($i$)u_t={\displaystyle \underset{r=0}{\overset{}{}}}{}^r{v}_{t-r}$

$($ii$)$ Var$(u_t)=\frac{{}_v^2}{1-p^2}$

$($iii$)$ Cov$(u_t,u_{t-s})=\frac{{}^5}{(1-{}^2)}{}_v^2$