In each case below, determine whether the random variable sequence \(\left\{Y_{n}ight\}\) converges in probability and/or in mean square, and if so, define what is being converged to.

(a) \(\quad Y_{j}=(j+5)^{-1} \sum_{i=1}^{j} X_{i} \quad\) for \(\quad j=1,2,3, \ldots\); \(X_{i}{ }^{\prime} S \sim\) iid Bernoulli \((\mathrm{p})\)

(b) \(\quad Y_{j}=j^{-1} \sum_{i=1}^{j}\left(X_{i}-\lambdaight)^{2} \quad\) for \(\quad j=1,2,3, \ldots\);

(c) \(Y_{j}=j^{-1} \sum_{i=1}^{j}\left(X_{i}+Z_{i}ight) \quad\) for \(\quad j=1,2,3, \ldots\); \(\left(X_{i}, Z_{i}ight)^{\prime} s \sim\) iid \(\operatorname{Normal}\left(\left[\begin{array}{l}0 \\ 0\end{array}ight],\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}ight]ight)\)

(d) \(Y_{j}=j^{-1} \sum_{i=1}^{j} X_{i} Z_{i} \quad\) for \(\quad j=1,2,3, \ldots \quad\); \(\left(X_{i}, Z_{i}ight)^{\prime} s \sim\) iid \(\operatorname{Normal}\left(\left[\begin{array}{l}0 \\ 0\end{array}ight],\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}ight]ight)\)