Prove Corollary 6.2. That is, let (left{left{X_{n k}ight}_{k=1}^{n}ight}_{n=1}^{infty}) be a triangular array where (X_{11}, ldots, X_{n 1})

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Prove Corollary 6.2. That is, let \(\left\{\left\{X_{n k}ight\}_{k=1}^{n}ight\}_{n=1}^{\infty}\) be a triangular array where \(X_{11}, \ldots, X_{n 1}\) are mutually independent random variables for each \(n \in \mathbb{N}\). Suppose that \(X_{n k}\) has mean \(\mu_{n k}\) and variance \(\sigma_{n k}^{2}\) for all \(k \in\{1, \ldots, n\}\) and \(n \in \mathbb{N}\). Let

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\[\mu_{n \cdot}=\sum_{k=1}^{n} \mu_{n k}\]

and

\[\sigma_{n .}^{2}=\sum_{k=1}^{n} \sigma_{n k}^{2}\]

Suppose that for some \(\eta>2\)

\[\sum_{k=1}^{n} E\left(\left|X_{n k}-\mu_{n k}ight|^{\eta}ight)=o\left(\sigma_{n .}^{\eta}ight)\]

as \(n ightarrow \infty\), then

\[Z_{n}=\sigma_{n \cdot}^{-1}\left(\sum_{k=1}^{n} X_{n k}-\mu_{n}ight) \xrightarrow{d} Z,\]

as \(n ightarrow \infty\), where \(Z\) is a \(\mathrm{N}(0,1)\) random variable.

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