4. Let $X1, X2, \dots$ be independent identically distributed random variables with mean 0, variance 1, and $EX_1^4 < \infty$. Find the limiting distribution of

$$Zn = \sqrt{n} \frac{X1 X2 + X3 X4 + \dots + X{2n-1} X{2n}}{X1^2 + X2^2 + \dots + X{2n}^2}. \quad (25\%)$$

阿摩線上測驗 · 2026-11-13T14:14:16+08:00

4. Let X1, X2, be independent identically distributed random variables with mean 0, variance 1, and EX_1^4 < ∞. Find the limiting distribution ofZn = √n X1 X2 + X3 X4 + + X2n-1 X2nX1^2 + X2^2 + + X2n^2. (25%)

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4. Let $X1, X2, \dots$ be independent identically distributed random variables with mean 0, variance 1, and $EX_1^4 < \infty$. Find the limiting distribution of$$Zn = \sqrt{n} \frac{X1 X2 + X3 X4 + \dots + X{2n-1} X{2n}}{X1^2 + X2^2 + \dots + X{2n}^2}. \quad (25\%)$$

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4. Let $X1, X2, \dots$ be independent identically distributed random variables with mean 0, variance 1, and $EX_1^4 < \infty$. Find the limiting distribution of

$$Zn = \sqrt{n} \frac{X1 X2 + X3 X4 + \dots + X{2n-1} X{2n}}{X1^2 + X2^2 + \dots + X{2n}^2}. \quad (25\%)$$