Continue with the previous exercise.

(i) Give an example of a sequence an, n ≥ 0, of positive integers that is strictly increasing and satisfies (14.58) for every k ≥ 0.

(ii) Show that for every k ≥ 1 and 1 ≤ i ≤ n,

(iii) Argue that P(An i. o.) = 0 implies that, with probability 1, ηan = ζn for large n.
(iv) Using the above result, argue that (14.54) and (14.55) follow if (14.59)
holds for any b > 0.
(v) Show that the distribution of ζn is the same as that of an(Xn,an−an−1 −
Xn−k+1,an−an−1 ), where Xr,n is rth order statistic of X1, . . . , Xn.
Furthermore, show that for anyr weakly converges to the Gamma(s − r, 1) distribution as n→∞.

Transcribed Image Text:

P{X(an-k+1)=X;} = 1 an-k+1