臺灣大學數學系

八十九學年度第二學期碩博士班資格考試試題

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1

(16 points). Let $X_1,X_2,...$ be uncorrelated random variables with $X_n=m \ \ \forall n$ and $Var(X_n)=O(n^{\delta}), \ \ 0<\delta<1$. Prove that the weak law of large numbers holds for $X_1,X_2,...$.

2

(16 points). Let $X_1,X_2,...$ be independent with $P(X_n=1)=p_n$ and $P(X_n=0)=1-p_n$. Prove that (i) $X_n\rightarrow 0$ in probability if and only if $p_n\rightarrow 0$ (ii) $X_n\rightarrow 0$ a.s. if and only if $\sum p_n<\infty$.

3

(18 points). (i) Prove that $X_n\Rightarrow c$ if only if $X_n\rightarrow c$ in probability, where $c$ is a constant. (ii) Let $X_1,X_2,...$ and $Y_1,Y_2,...$ be independent. Suppose $X_n\Rightarrow X$ and $Y_n\Rightarrow Y$, prove that $X_n+Y_n\Rightarrow X+Y$.

4

(16 points). Let $X$ be of standard normal distribution. Use characteristic function to compute the moments of $X$, that is , $EX^k,\ \ k=1,2,3,4,...$.

5

(18 points). Define the ``upcrossing number" $U_n$ of $X_0,X_1,X_2,...$ over an interval [a,b], and prove the inequality

$$(b-a)EU_n\leq E(X_n-a)^+ -E(X_0-a)^+$$

for submartingale $X_0,X_1,X_2,...$.

6

(16 points). Let $X_1,X_2,...$ be a martingale. Prove that there exists an $X_\infty$ such that $X_1,X_2,.., X_\infty$ forms a martingale if and only if $X_1,X_2,...$ are uniformly integrable.

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