98年 - 98 國立高雄大學_博士班招生考試試題_統計學研究所：機率論#147988

(a) If $Xn \to X$ in probability and $Yn \to Y$ in probability, then $Xn + Yn \to X + Y$ in probability.

(b) If $Xn \to X$ in distribution and $Yn \to Y$ in distribution, then $Xn + Yn \to X + Y$ in distribution.

(c) If $E|Xn - X| \to 0$ as $n \to \infty$, then $Xn \to X$ in probability.

(d) If $Xn \to X$ in probability, then $E|Xn - X| \to 0$ as $n \to \infty$.

(e) If $Xn \to X$ almost surely, then $Xn \to X$ in probability.

(f) If $Xn \to X$ in probability, then $Xn \to X$ almost surely.

2. A sequence of random variables \( \{X_n, n \ge 1\} \) is called uniformly integrable if for every \( \varepsilon > 0 \), there corresponds a \( \delta > 0 \) such that \[ \sup_{n \ge 1} \int_A |X_n| dP < \varepsilon, \] whenever \( P(A) < \delta \), and, in addition, \[ \sup_{n \ge 1} E|X_n| < \infty. \] Show that if \( \{X_n, n \ge 1\} \) is uniformly integrable, then \( \{S_n/n, n \ge 1\} \) is also uniformly integrable, where \( S_n = \sum_{i=1}^n X_i \). (20%)

(a) $\rho(X + Y) \le \rho(X) + \rho(Y)$, (5%)

(b) $\rho(\sigma X) \le \max\{|\sigma|, 1\}\rho(X)$, where $\sigma$ is a real number, (5%)

\( \lim_{n \to \infty} \rho(X_n - X) = 0 \) if and only if \( X_n \to X \) in probability. (15%)

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 \[ Z_n = \sqrt{n} \frac{X_1 X_2 + X_3 X_4 + \cdots + X_{2n-1} X_{2n}}{X_1^2 + X_2^2 + \cdots + X_{2n}^2}. \quad (25\%) \]