Let \(\left\{X_{n}ight\}_{n=1}^{\infty},\left\{Y_{n}ight\}_{n=1}^{\infty}\), and \(\left\{Z_{n}ight\}_{n=1}^{\infty}\) be independent sequences of random variables that converge in probability to the random variables \(X, Y\), and \(Z\), respectively. Suppose that \(X, Y\), and \(Z\) are independent and that each of these random variables has a \(\mathrm{N}(0,1)\) distribution.

a. Let \(\left\{\mathbf{W}_{n}ight\}_{n=1}^{\infty}\) be a sequence of three-dimensional random vectors defined by \(\mathbf{W}_{n}=\left(X_{n}, Y_{n}, Z_{n}ight)^{\prime}\) and let \(\mathbf{W}=(X, Y, Z)^{\prime}\). Prove that \(\mathbf{W}_{n} \xrightarrow{p} \mathbf{W}\) as \(n ightarrow \infty\). Identify the distribution of \(\mathbf{W}\). Would you be able to completely identify this distribution if the random variables \(X, Y\), and \(Z\) are not independent? What additional information would be required?

b. Prove that \(\frac{1}{3} \mathbf{1}^{\prime} \mathbf{W}_{n} \xrightarrow{p} \frac{1}{3} \mathbf{1}^{\prime} \mathbf{W}\) and identify the distribution of \(\frac{1}{3} \mathbf{1}^{\prime} \mathbf{W}\).