Problem 1 (qiuzhen-qualify 2024-spring)

Let $X_1,X_2,\ldots$ be independent identically distributed random variables with density
\[
 f(x)=\begin{cases}|x|^{-3}, & |x|>1,\\0, & \text{otherwise.}\end{cases}
\]
and characteristic function $\varphi(t):=\mathbb{E}e^{itX_1}$. Let $S_n:=X_1+X_2+\cdots+X_n$ for each $n\in\mathbb{N}$.

(a) Prove that $\mathbb{E}(X_1^2)=\infty$.

(b) It is known that $\lim_{t\to 0} \dfrac{1-\varphi(t)}{t^2\ln|t|}=-1$; you do not need to prove this. Using this fact, find a sequence $(a_n)_{n\in\mathbb{N}}$ and prove that $S_n/a_n$ converges in distribution to the standard normal distribution (i.e., mean 0 and variance 1).

Problem 2 (qiuzhen-qualify 2024-spring)

Let $X_1,X_2,\ldots$ be independent identically distributed random variables with exponential distribution: $\mathbb{P}(X_k>x)=e^{-x}$ for $x\ge 0$. Let $M_n:=\max_{1\le k\le n}X_k$. Show that

(a) $\limsup_{n\to\infty} \dfrac{X_n}{\ln n}=1$ almost surely.

(b) $\lim_{n\to\infty} \dfrac{M_n}{\ln n}=1$ almost surely.

Problem 3 (qiuzhen-qualify 2024-spring)

(a) Prove that: if $X_n\xrightarrow{\,P\,}X$ (i.e., $X_n$ converges to $X$ in probability) and $Y_n\xrightarrow{\,P\,}Y$ then $X_nY_n\xrightarrow{\,P\,}XY$.

(b) Is the following true or false: if $X_n\xrightarrow{\,L^1\,}X$ (i.e., $X_n$ converges to $X$ in $L^1$) and $Y_n\xrightarrow{\,L^1\,}Y$ then $X_nY_n\xrightarrow{\,L^1\,}XY$? Either prove it or give a counter example.

Problem 4 (qiuzhen-qualify 2024-spring)

Let $p\in(0,1)$ and $k\in\mathbb{N}$. Let $X_1,X_2,\ldots$ be independent random variables with $\mathbb{P}(X_n=1)=p=1-\mathbb{P}(X_n=-1)$ for each $n\in\mathbb{N}$. Let $X_0:=0$ and $S_n:=X_0+X_1+\cdots+X_n$ for each $n\in\mathbb{N}\cup\{0\}$. Let $T_k:=\inf\{n\in\mathbb{N}:S_n=k\}$ with $\inf\emptyset=\infty$. For each $t\in\mathbb{R}$, let $M(t):=\mathbb{E}e^{tX_1}$.

(a) If $p\in[1/2,1)$, prove that $e^{tk}\,\mathbb{E}\big(M(t)^{-T_k}\big)=1$ for all $t>0$.

(b) If $p\in(0,1/2)$, compute $\mathbb{P}(T_k<\infty)$.

(c) If $p\in(0,1/2)$, find the distribution of $Y:=1+\sup_{n\ge 0} S_n$.

Problem 5 (qiuzhen-qualify 2024-spring)

Let $S$ be a countable state space and $(Z_n,n\in\mathbb{N})$ be a sequence of independent identically distributed random variables taking values in a measurable space $(E,\mathcal{E})$.

(a) Suppose that $f:S\times E\to S$ is a measurable mapping and $X_0$ is independent of all $Z_n$. Let
\[
X_n:=f(X_{n-1},Z_n),\quad n\in\mathbb{N}. \tag{1}
\]
Prove that $(X_n,n\in\mathbb{N}\cup\{0\})$ is a Markov chain, and determine its transition probability.

(b) Prove that any (discrete) time-homogeneous Markov chain on $S$ can be written in the form (1).

Problem 6 (qiuzhen-qualify 2024-spring)

Let $(B_t,t\ge 0)$ be a one-dimensional Brownian motion starting from the origin (i.e., $B_0=0$). Let $\mathcal{F}_t:=\sigma(B_s:s\le t)$ be the filtration generated by $B_t$. For $a>0$, define $T_a:=\inf\{t>0:B_t\ge a\}$.

(a) Prove that $T_a$ is a $\mathcal{F}_t$ stopping time.

(b) Compute $\mathbb{E}e^{-\lambda T_a}$ for $\lambda>0$, and prove that $\mathbb{P}(T_a<\infty)=1$. Hint: one may consider $M_t:=e^{uB_t-u^2 t/2}.$

Problem 7 (qiuzhen-qualify 2024-spring)

Let $X_i$ be independent $N(\mu_i,1)$, $i=1,\ldots,n$. Let $\mu=(\mu_1,\ldots,\mu_n)^T$, $\mathbf{X}=(X_1,\ldots,X_n)^T$, $\mathbf{a}=(a_1,\ldots,a_n)^T$. Consider the following loss function
\[
\ell(\mu,\mathbf{a})=\sum_{i=1}^n (a_i-\mu_i)^2,
\]
prove that $\delta(\mathbf{X})=\mathbf{X}$ is minimax.

Problem 8 (qiuzhen-qualify 2024-spring)

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d. random variables from $Uniform(\theta,\theta+10)$ distribution with $\theta>0$.

(a) Prove that $\hat{\theta}=\min(X_1,\ldots,X_n)$ is a consistent estimator of $\theta$.

(b) Find the asymptotic distribution of
\[
\sqrt{12n}\left(\frac{\overline{X}-5-\hat{\theta}}{\hat{\theta}}\right),
\]
where $\overline{X}$ is the sample mean and $\hat{\theta}$ is the consistent estimator for $\theta$ in part (a).

Problem 9 (qiuzhen-qualify 2024-spring)

Let $(X_1,\ldots,X_n)$ be a random sample and assume that $\log X_i$ is distributed as $N(\theta,\theta)$ with an unknown parameter $\theta>0$.

(a) Derive the MLE of $\theta$, $\hat{\theta}$.

(b) What is the asymptotic distribution of $\hat{\theta}$?

Problem 10 (qiuzhen-qualify 2024-spring)

Let $X_1,\ldots,X_n$ be the incomes of $n$ persons chosen at random from a certain population. Assume that $X_i$ has the following Pareto density
\[
 f(x\mid\theta)=100^\theta\,\theta\,x^{-(1+\theta)},\quad x>100,
\]
where $\theta>1$. Let $\mu=\mathbb{E}(X)$ be the mean of the income, write down the rejection region of the uniformly most powerful (UMP) level $\alpha$ ($0<\alpha<1$) test of testing $H_0:\ \mu=\mu_0$ versus $H_1:\ \mu>\mu_0$.

Problem 11 (qiuzhen-qualify 2024-spring)

Let $\theta_1,\theta_2,$ and $\theta_3$ be nonnegative parameters with the constraint $\theta_1+\theta_2+\theta_3=1$. We observe $X_{i1}=\theta_1+\varepsilon_{i1},\ X_{i2}=\theta_2+\varepsilon_{i2},\ X_{i3}=\theta_3+\varepsilon_{i3}$ for $i=1,2,\ldots,n$, where $\varepsilon_{ik}\sim N(0,1)$ are independent normal random variables ($k=1,2,3$). Derive the uniformly minimum-variance unbiased estimator (UMVUE) for $\theta_1$.