1.1. Central Limit Theorem for \(m\)-Dependent Random Variables

Definition (\(m\)-Dependent Random Variables)

Let the sequence of random variables \(X_1,X_2,X_3,\ldots\) be \(m\)-dependent if for some \(i\), \(\{X_{i},X_{i+1},\ldots,X_{i+r}\}\perp \{X_{i+s},X_{i+s+1},\ldots,\}\) whenever \(s-r>m\).

• 즉 minimum distance of indices between the two sequences가 \(m\)보다 크다면 우리는 \(m\)-dependent하다고 한다. 즉, 최소거리가 \(m+1\)보다 크다면 independent하다.

Example

Consider that the sequence \(\{X_1,X_2,X_3,X_4,X_5,X_6\}\) is \(1\)-dependent. Then, \(X_1, X_3\) are independent, \(X_4,X_5\) are dependent, \(X_5, X_6\) are dependent.

Consider that the sequence \(\{X_1,X_2,X_3,X_4,X_5,X_6\}\) is \(2\)-dependent. Then, \(X_1, X_3\) are dependent, \(X_3,X_6\) are independent.

Let \(Y_1,Y_2,Y_3,\ldots\) be iid random variables. Let \[\begin{align*} X_1 &= \frac{1}{m+1}\sum_{i=1}^{m+1} Y_i\\ X_2 &= \frac{1}{m+1}\sum_{i=2}^{m+2} Y_i\\ X_3 &= \frac{1}{m+1}\sum_{i=3}^{m+3} Y_i\\ &\mbox{ }\mbox{ }\mbox{ }\mbox{ }\vdots\\ X_{m+2} &= \frac{1}{m+1}\sum_{i=m+2}^{2m+2} Y_i. \end{align*}\] Then, the sequence \(\{X_1,X_2,X_3,\ldots \}\) are \(m\)-dependent, because \(X_1\perp X_{m+2}\).

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