4. Let \( X_1, X_2, \cdots \) be independent identically distributed random variables with mean 0, variance 1, and \( EX_i^4 < \infty \). Find the limiting distribution of \[ Z_n = \sqrt{n} \frac{X_1 X_2 + X_3 X_4 + \cdots + X_{2n-1} X_{2n}}{X_1^2 + X_2^2 + \cdots + X_{2n}^2}. \quad (25\%) \]

阿摩線上測驗 · 2026-11-13T14:14:16+08:00

4. Let X_1, X_2, ·s be independent identically distributed random variables with mean 0, variance 1, and EX_i^4 < ∞. Find the limiting distribution of Z_n = √n X_1 X_2 + X_3 X_4 + ·s + X_2n-1 X_2nX_1^2 + X_2^2 + ·s + X_2n^2. (25%)

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4. Let \( X_1, X_2, \cdots \) be independent identically distributed random variables with mean 0, variance 1, and \( EX_i^4 < \infty \). Find the limiting distribution of \[ Z_n = \sqrt{n} \frac{X_1 X_2 + X_3 X_4 + \cdots + X_{2n-1} X_{2n}}{X_1^2 + X_2^2 + \cdots + X_{2n}^2}. \quad (25\%) \]

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