The sequence {X_n}, n ³ 1, of r.v.s is said to converge completely to 0, if for every e > 0,P(|X_n|³ e) < ¥.

(i) Show that, if {X_n}, n ³ 1, converges completely to 0, then X_n 

(ii) By means of an example, show that complete convergence is not necessary for a.s. convergence.

For Part (i), use Exercise 3 here and Exercise 4 in Chapter 2. For Part (ii), take W = (0, 1], A = B_W, P = l, the Lebesgue measure, and choose the r.v.s suitably.

The most common way of establishing that X_n X is to slow that [X_n €“ X], ³ 1, converges completely to 0.

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n=1