2.9 Suppose that for each 1 j k, n,j , n = 1, 2, ....

2.9 Suppose that for each 1 ≤ j ≤ k, ξn,j , n = 1, 2, . . . , is a sequence of random variables such that ξn,j

−→P 0 as n→∞. Define ξn = max1≤j≤k |ξn,j |.

(i) Show that if k is fixed, then ξn

−→P 0 as n→∞.

(ii) Give an example to show that if k increases with n (i.e., k = kn →∞as n→∞), the conclusion of (i) may not be true.