6.29 We see that, in the i.i.d. case, the same condition (i.e., a finite second moment)

is necessary and sufficient for both CLT and LIL. In other words, a sequence of i.i.d. random variables obeys the CLT if and only if it obeys the LIL.

It is a different story, however, if the random variables are independent but not identically distributed. For example, Wittmann (1985) constructed the following example. Let nk be an infinite sequence of integers such that nk+1 >

2nk, k ≥ 1. Let X1,X2, . . . be independent such that for nk + 1 ≤ i ≤ 2nk, we have P(Xi = 1) = P(Xi = −1) = 1/4, P(Xi =

√

2nk) = P(Xi =

−

√

2nk) = 1/8nk, and P(Xi = 0) = 1−1/2−1/4nk; for all other i ≥ 1, we have P(Xi = 1) = P(Xi = −1) = 1/2.

(i) Show that E(Xi ) = 0 and σ2 i

= E(X2 i ) = 1, therefore an = s2 n

=



n i=1 σ2 i

= n. It follows that (6.40) is satisfied.

(ii) Show that Lindeberg’s condition (6.42) does not hold for  = 1.

(iii) Show by Theorem 6.12 that Xi , i ≥ 1, does not obey the CLT. (Hint: You may use the result of Example 1.6.)

Wittmann (1985) further showed that the sequence obeys the LIL. On the other hand, Marcinkiewicz and Zygmund (1937b) constructed a sequence of independent random variables that obeys the CLT but not the LIL.