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

科目：研究所、轉學考(插大)◆機率 | 年份：98年 | 選擇題數：0 | 申論題數：11

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所屬科目：研究所、轉學考(插大)◆機率

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申論題 (11)

1. Let $\{Xn, n \ge 1\}$ and $\{Yn, n \ge 1\}$ be sequences of random variables and $X$ and $Y$ be random variables. Please indicate whether the following statements are true or false. If a statement is false, explain why. (30%)

(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

$$\sup{n \ge 1} \intA |X_n| \, dP < \varepsilon,$$

whenever $P(A) < \delta$, and, in addition,

$$\sup{n \ge 1} E|Xn| < \infty.$$

Show that if $\{Xn, n \ge 1\}$ is uniformly integrable, then $\{Sn/n, n \ge 1\}$ is also uniformly integrable, where $Sn = \sum{i=1}^n X_i$. (20%)
3. Let $X$ and $Y$ be random variables and $\{X_n, n \ge 1\}$ be a sequence of random variables. Define

$$\rho(X) = E\left(\frac{|X|}{1+|X|}\right).$$

Show that

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

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