Consider the stochastic process generated by $$ X_{t}=X_{t-1}+Z_{t}, \quad t=0, \pm 1, \pm 2, \ldots $$ Under certain conditions on $X_{0}$ and the $z_{t}$, this process is known as a simple random walk and can be used as a simple model for financial asset prices. a. (3 pts) Suppose $t>0$. Then, by recursion $$ \begin{aligned} X_{t} &=\underbrace{X_{t-1}}_{=X_{t-2}+Z_{t- 1}}+Z_{t}=\underbrace{X_{t-2}}_{=X_{t-3}+Z_{t-2}}+Z_{t- 1}+Z_{t}=\underbrace{X_{t-3}}_{=X_{t-4}+Z_{t-3}}+Z_{t-2}+Z_{t- 1}+Z_{t} &=\cdots=x_{0}+Z_{1}+\cdots+Z_{t} \end{aligned} $$ For example, for $t=3$, $$ X_{3}=X_{0}+Z_{1}+Z_{2}+Z_{3} $$ Let $\mathbf {Z}=\left(Z_{1}, \ldots, Z_{t} ight)$, and let $\mathbf{A}=(1, \ldots, 1$ be the vector of same dimension as $\mathbf {Z} $ with all components equal to one. Show that $X_{t}=X_{0}+\mathbf{A Z}^{\prime} $. b. $(4 pts) $ In addition, suppose $X_{0}=0$ and $\mathbf {Z}$ is a $t$-dimensional Gaussian random vector with parameters $\boldsymbol{\mu}=\mathbf {0}$ and $\boldsymbol{\Sigma}=\mathbf{I}_{t}$. Show that $X_{t}$ is $NCO, t $ distributed. Hint: You may use the formula for linear transformations of a Gaussian random vector. SP.AS. 10561