Problem 1 (yau-contest 2019)
Section: Probability and Statistics

Suppose $(X_n)_{n\ge 1}$ is a sequence of i.i.d. random variables and the common law is exponential with parameter one. Show that $$P\left[\limsup_{n\to\infty} \frac{X_n}{\log n} = 1\right] = 1.$$

Problem 2 (yau-contest 2019)
Section: Probability and Statistics

Let $(X_n)_{n\ge 1}$ be i.i.d. real random variables and set $S_n = \sum_{i=1}^n X_i$ for $n \ge 1$. Suppose that for some constant $c \in \mathbb{R}$ we have $S_n/n \to c$ as $n \to \infty$ almost surely. Show that $X_1$ has a finite first moment and $E[X_1] = c.$

Problem 3 (yau-contest 2019)
Section: Probability and Statistics

Consider uniform permutation of $\{1,2,\ldots,n\}$ and denote by $X_n$ the number of cycles in the permutation. Find a sequence of reals $(a_n)_{n\ge 1}$ such that $$\lim_{n\to\infty} \frac{E[X_n]}{a_n} = 1,$$ and justify your answer.

Problem 4 (yau-contest 2019)
Section: Probability and Statistics

The Erdős–Rényi random graph $G(n,p)$ with parameters $n \ge 1$ and $p \in [0,1]$ is the random graph whose vertex set is $V = \{1,2,\ldots,n\}$ and where for each pair $i \ne j \in V$ the edge $i \leftrightarrow j$ is present with probability $p$ independently of all the other pairs.

(a) For $\epsilon > 0$, if $p_n \ge (1+\epsilon) \frac{\log n}{n}$, then $$P[ G(n,p_n) \text{ has an isolated vertex} ] \to 0, \quad \text{as } n \to \infty.$$

(b) For $\epsilon > 0$, if $p_n \le (1-\epsilon) \frac{\log n}{n}$, then $$P[ G(n,p_n) \text{ has an isolated vertex} ] \to 1, \quad \text{as } n \to \infty.$$