Probability for Statisticians by Galen R. Shorack

Author:Galen R. Shorack
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

Exercise 2.3

Provide the steps leading to (9) referred to in the Berry–Esseen proof.

Remark 2.1

(Lindeberg’s condition) (i) If Lindeberg’s condition fails, it may still be true that

(a)

Let be iid (0, 1) rvs, so that by the CLT. Now let the rv’s be independent with equal to with probabilities . Since , the Borel–Cantelli lemma shows that for the rv sequence satisfies only finitely often. Thus follows. For set , and let . Note that . So, by Slutsky’s theorem ,

(b)

(One could also let , and have in (b).) Note that . But, even so, Lindeberg’s condition fails, since

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