# Question

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 → ∞

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

(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 → ∞

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

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