2. In section 3.9 the following example is given:

Let Xn take the value n with probability 1/n, and otherwise take the value 0. Then E(Xn) = 1 for all n. However limn→∞ P{Xn = 0} = 1, so the limiting distribution puts all its mass at 0, and has mean 0.

(a) Does this example contradict the dominated convergence theorem? Explain your reasoning.

(b) Let Yn take the value √n with probability 1/n, and otherwise take the value 0.

Answer the same question.