Question: Let Xk, k = 1, 2, 3¦ be a sequence of IID random variables with finite mean, , and let Sn be the sequence of
-1.png)
(a) Show that the characteristic function of Sn can be written as
-2.png)
(b) Use Taylor€™s theorem to write the characteristic function of the Xk as
-3.png){kind=small}
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
-4.png)
Show that as n †’ ˆž
-5.png){kind=small}
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.
71 Sn = .lk, n = 1, 2, 3, .,..
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