Question: (a) Let \(X_{1}, X_{2} \ldots\) be a sequence of independent random variables with expectations \(E\left(X_{n}\right)=\mu_{n} \stackrel{n \rightarrow \infty}{\longrightarrow} \mu\) and variances \(\operatorname{Var}\left(X_{n}\right)=\sigma_{n}^{2} \stackrel{n \rightarrow \infty}{\longrightarrow} 0 .\) Show that

(b) Let \(Y_{1}, Y_{2} \ldots\) be another sequence of independent random variables with expectations \(E\left(Y_{n}\right)=\mu_{n} \stackrel{n \rightarrow \infty}{\longrightarrow} \mu\) and variances bounded \(\operatorname{Var}\left(Y_{n}\right)=\sigma_{n}^{2} \leq \sigma^{2}\) for all $n .$

Show that the weak law of large numbers holds for this sequence, that is

(c) Consider now an estimator \(\hat{\theta}_{n}\) of some parameter \(\theta\). Show that if \(\operatorname{MSE}\left(\hat{\theta}_{n}\right) \stackrel{n \rightarrow \infty}{\longrightarrow} 0\) then \(\hat{\theta}_{n}\) is consistent.

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