Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent random variables where \(X_{n}\) has a \(\operatorname{GAmma}\left(\theta_{n}, 2ight)\) distribution where \(\left\{\theta_{n}ight\}_{n=1}^{\infty}\) is a sequence of positive real numbers.

a. Find a non-trivial sequence \(\left\{\theta_{n}ight\}_{n=1}^{\infty}\) such that the assumptions of Theorem 6.1 (Lindeberg, Lévy, and Feller) hold, and describe the resulting conclusion for the weak convergence of \(\bar{X}_{n}\).

b. Find a non-trivial sequence \(\left\{\theta_{n}ight\}_{n=1}^{\infty}\) such that the assumptions of Theorem 6.1 (Lindeberg, Lévy, and Feller) do not hold.

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Theorem 6.1 (Lindeberg, Lvy, and Feller). Let {X} be a sequence of independent random variables where E(Xn) = n and V(Xn) = 0 < for all n = N where {n} and {0} are sequences of real numbers. Let Pn n= = N n 1 Hk, k=1