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

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

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.

Transcribed Image Text:

Corollary 6.2. Let {{Xnk}-1}1 be a triangular array where Xn1,..., Xnn are mutually independent random variables for each n = N. Suppose that Xnk has mean pink and variance < for all k = {1,..., n} and n = N. Let nk Hn. = n =nks k=1 and . = IM k=1 nk Suppose that for some > 2 n E(|Xnk - Pnk") = 0(n.), k=1