6.17 Suppose that Sn is distributed as Poisson(n), where nas n. Use two different methods to show...

6.17 Suppose that Sn is distributed as Poisson(λn), where λn→∞as n→∞. Use two different methods to show that Sn obeys the CLT; that is, ξn = λ

−1/2 n (Sn−

λn)

−→d N(0, 1).

(i) Show that the mgf of ξn converges to the mgf of N(0, 1).

(ii) Let Yni, 1 ≤ i ≤ n, be independent and distributed as Poisson(n

−1λn), n ≥ 1. Show that



n i=1 Yni has the same distribution as Sn. Furthermore, show that Xni = λ

−1/2 n (Yni−n

−1λn) satisfy Liapounov’s condition (6.37)

with δ = 2. [Hint: You may use the fact that the fourth central moment of Poisson(λ) is λ + 3λ2.]