Exercise $1$ Assume $\{Y_1,Y_2,$dots$\}$ is a sequence of independent $k1$ random vectors with $E[Y_i]=$

and $V(Y_i)=$ positive definite and bounded for all iinN.

$($a$)$ Prove the weak law of large numbers $($WLLN$)$ by using

$($i$)$ the Chebyshev or Markov inequalities,

$($ii$)$ convergence in mean square.

$($b$)$ Let $g(*)$ be a square integrable function mapping from the range of $Y_i$ to a measurable space.

Assume that $E[g(Y_i)]$ and $V(g(Y_i))$ exist, are finite, and the variance is positive definite. Show

that $p\underset{?}{lim}\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}g(Y_i)=E(Y_1).$

$($c$)$ Let $\{Y_1,Y_2,$dots$\}$ be independently $t-$distributed with $2$ degrees of freedom. What is the expectation

and variance of $Y_i?$ Which law of large numbers do you use to conclude that $\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{Y}_i{?}^{as}E[Y_i]?$ Is

it true that $\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{Y}_i{?}^{m.s.}E[Y_i]?$