Let Xk, k = 1, 2, 3€¦ be a sequence of IID random variables with finite mean, , and let Sn be the sequence of sample means,

(a) Show that the characteristic function of Sn can be written as

(b) Use Taylor€™s theorem to write the characteristic function of the Xk as

Where the remainder term r2 (ω) is small compared to ω as ω†’0.Find the constants c0 and c1.
(c) Writing the characteristic function of the sample mean as

Show that as n †’ ˆž

In so doing, you have proved that the distribution of the sample mean is that of a constant in the limit as n †’ ˆž. Thus, the sample mean converges in distribution.

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71 Sn = Σ.lk, n = 1, 2, 3, .,..