Problem 1 (yau-contest 2022)
Section: Probability and Statistics

Let $\{X_n\}$ be a sequence of Gaussian random variables. Suppose that $X$ is a random variable such that $X_n$ converges to $X$ in distribution as $n \to \infty$. Show that $X$ is also a (possibly degenerate, i.e., variance zero) Gaussian random variable.

Problem 2 (yau-contest 2022)
Section: Probability and Statistics

For two probability measures $\mu$ and $\nu$ on the real line $\mathbb{R}$, the total variation distance $\|\mu-\nu\|_{TV}$ is defined as

$$\|\mu-\nu\|_{TV} = \sup\{\mu(C)-\nu(C):\; C \in \mathcal{B}(\mathbb{R})\},$$

where $\mathcal{B}(\mathbb{R})$ is the $\sigma$-algebra of Borel sets on $\mathbb{R}$. Let $e(\mu,\nu)$ be the space of couplings of the probability measures $\mu$ and $\nu$, i.e., the space of $\mathbb{R}^2$ valued random variables $(X,Y)$ defined on some (not necessarily same) probability space $(\Omega, \mathcal{F}, P)$ such that the marginal distributions of $X$ and $Y$ are $\mu$ and $\nu$, respectively. Show that

$$\|\mu-\nu\|_{TV} = \inf\{ P(X \neq Y) : (X,Y) \in e(\mu,\nu) \}.$$

For simplicity you may assume that $\mu$ and $\nu$ are absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.

Problem 3 (yau-contest 2022)
Section: Probability and Statistics

We throw a fair die repeatedly and independently. Let $\tau_{11}$ be the first time the pattern $11$ (two consecutive 1's) appears and $\tau_{12}$ the first time the pattern $12$ (1 followed by 2) appears.

(a) Calculate the expected value $\mathbb{E}\,\tau_{11}$.

(b) Which is larger, $\mathbb{E}\,\tau_{11}$ or $\mathbb{E}\,\tau_{12}$? It is sufficient to give an intuitive argument to justify your answer. You can also calculate $\mathbb{E}\,\tau_{12}$ if you wish.

Problem 4 (yau-contest 2022)
Section: Probability and Statistics

Let $\{X_n\}$ be a Markov chain on a discrete state space $S$ with transition function $p(x,y)$, $x,y \in S$. Suppose that there is a state $y_0 \in S$ and a positive number $\theta$ such that $p(x,y_0) \ge \theta$ for all $x \in S$.

(a) Show that a positive constant $\lambda < 1$ such that for any two initial distribution $\mu$ and $\nu$,

$$\sum_{y \in S} \big| P_{\mu}\{X_1=y\} - P_{\nu}\{X_1=y\} \big| \le \lambda \sum_{y \in S} |\mu(y) - \nu(y)|.$$

(b) Show that the Markov chain has a unique stationary distribution $\pi$ and

$$\sum_{y \in S} \big| P_{\mu}\{X_n=y\} - \pi(y) \big| \le 2\lambda^n.$$

Problem 5 (yau-contest 2022)
Section: Probability and Statistics

Consider a linear regression model with $p$ predictors and $n$ observations:

$$Y = X\beta + e,$$

where $X_{n\times p}$ is the design matrix, $\beta$ is the unknown coefficient vector, and the random error vector $e$ has a multivariate normal distribution with mean zero and $\operatorname{Var}(e)=\sigma^2 I_n$ ($\sigma^2>0$ unknown and $I_n$ is the identity matrix).

Here $\operatorname{rank}(X)=k \le p$, $p$ may or may not be greater than $n$, but we assume $n-k>1$. Let $x_1=(x_{1,1},\ldots,x_{1,p})$ be the first row of $X$ and define

$$\gamma = \frac{x_1\beta}{\sigma}.$$

Find the uniformly minimum variance unbiased estimator (UMVUE) of $\gamma$ or prove it does not exist.

Problem 6 (yau-contest 2022)
Section: Probability and Statistics

Let $X_1,\ldots,X_{2022}$ be independent random variables with $X_i \sim N(\theta_i,i^2)$, $1 \le i \le 2022$. For estimating the unknown mean vector $\theta \in \mathbb{R}^{2022}$, consider the loss function

$$L(\theta,d) = \sum_{i=1}^{2022} \frac{(d_i-\theta_i)^2}{i^2}.$$

Prove that $X=(X_1,\ldots,X_{2022})$ is a minimax estimator of $\theta$.

Recall: If $Y\mid\mu \sim N(\mu,\sigma^2)$ and $\mu \sim N(\mu_0,\sigma_0^2)$ then $\mu\mid Y=y \sim N\!\left( \frac{\mu_0/\sigma_0^2 + y/\sigma^2}{1/\sigma_0^2 + 1/\sigma^2},\; \frac{1}{1/\sigma_0^2 + 1/\sigma^2} \right).$