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
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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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