台灣大學數學系

九十二學年度第二學期博士班資格考試題

機率論

May 8, 2004

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(1)

(20 points) (1A) Let $(X_n)$ be a sequence of random variables. Prove that $X_n\to X$ in probability if and only if for every subsequence $X_{n(m)}$ there is a further subsequence $X_{n(m_k)}$ that converges to $X$ almost surely (a.s.). (1B) Prove that if $f$ is continuous and $X_n\to X$ in probability then $f(X_n)\to f(X)$ in probability.

(2)

(12 points) Let $(X_n)$ be a sequence of i.i.d. random variables with $X_n>0$. Let $S_n=X_1+\cdots +X_n$ and $N(t)=\sup\{n\,:\, S_n\le t\}$. Prove that $${\frac{N(t)}t\to\frac 1{\mu}}$$ a.s. as $t\to\infty$, where $\mu=E[X_1]\in (0,\infty]$ and $1/{\infty}=0$.

(3)

(36 points) (3A) Show that if random variables $X_n\to X$ in probability then $X_n$ converges weakly to $X$ (denoted as $X_n\Rightarrow X$). Conversely, show that if $X_n\Rightarrow c$, where $c$ is a constant, then $X_n\rightarrow c$ in probability. (3B) Show that if $X_n\Rightarrow X$ and $0\le Y_n\Rightarrow c$, where $c$ is a constant, then $X_nY_n\Rightarrow cX$. (3C) Let $(X_n)$ be a sequence of i.i.d. random variables with $E[X_1]=0$ and $E[X_1^2]\in (0,\infty)$. Prove that $${\sum_{k=1}^n X_k \Big/ \bigg(\sum_{k=1}^n X_k^2\bigg)^{1/2} }$$ converges weakly to a standard normal distribution.

(4)

(20 points) (4A) Let $\mathcal{F}_n, n\ge 1$, be a filtration and $(Z_n)$ be a sequence of integrable random variables adapted to $\mathcal{F}_n$. Prove that $Z_n$ can be written in a unique way as $Z_n=M_n+A_n$ such that $(M_n, \mathcal{F}_n)$ is a martingale and $A_n$ is a predictable sequence with $A_1\equiv 0$. Moreover, $(A_n)$ is increasing if and only if $(Z_n)$ is a sub-martingale. (4B) Let $(X_n)$ be a sequence of independent random variables with $E[X_n]=0$ and $E[X_n^2]<\infty$ for all $n\ge 1$. Let $\mathcal{F}_n=\sigma(X_1,...,X_n)$. Show that $Z_n=(X_1+\cdots +X_n)^2$ is a sub-martingale and find $M_n, A_n$ of the decomposition $Z_n=M_n+A_n$ described in (4A).

(5)

(12 points) Given a probability space $(\Omega, \mathcal{F}, P)$ and an integrable random variable $X$. Show that the family $\{E[X\,\vert\,\mathcal{G}] \,:\, \mathcal{G}$    is a $&sigma#sigma;$-field$\subset \mathcal{F}\}$ is uniformly integrable.

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