Probability qualifying exam--January, 2015. You need to get at least 50 points to pass. The value of each question is 10 points.

\(X_1,X_2,\ldots ,X_n\) be independent identically distributed positive random variables. Show that \[ \frac {1}{n}\max _{1\leq k\leq n} X_k \to 0 \qquad \text {a.s.} \] if and only if \(\mathbb {E} X_1<\infty \).

Let \(X\geq 0\) and \(\mathbb {E}|X|^p<\infty \) with some \(p>0\). Show that \[ \lim _{t\to \infty } t^p\mathbb {P}\{X>t\}=0. \]

Let \(X\) and \(Y\) be independent random variables. Show that if the distribution of \(X\) is absolutely continuous with respect to the Lebesgue measure, then the distribution of \(X+Y\) is also absolutely continuous with respect to the Lebesgue measure.

Let \(X_1,X_2,\ldots \) be independent and identically distributed random variables with \(\mathbb {E}\log (1+|X_1|)<\infty \). Show that \[ \sum _{i=1}^{\infty }\prod _{j=1}^{i}X_j \] is finite with probability one if and only if \(\mathbb {E}\log |X_1|<0\).

Let \(X_1,X_2,\ldots \) be independent and identically distributed random variables with distribution function \(F(x)=\mathbb {P}\{X_1\leq x\}\). Show that \[ \sup _{-\infty <x<\infty } |F_n(x)-F(x)| \to 0 \qquad \text {a.s.} \] as \(n\to \infty \), where \[ F_n(x)=\frac {1}{n}\sum _{i=1}^{n} I\{X_i\leq x\}. \]

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed exponential(1) random variables. Let \(k\geq 1\). Show that \[ \mathbb {P}\{X_n>\phi _{\varepsilon }(n)\ \text {i.o.}\} = \begin {cases} 0, & \text {if }\varepsilon >0,\\ 1, & \text {if }\varepsilon \leq 0, \end {cases} \] where \[ \phi _{\varepsilon }(n)=\log n+\log _{(2)}(n)+\cdots + (1+\varepsilon )\log _{(k)}(n), \] and \(\log _{(1)}(n)=\log n\), \(\log _{(k)}(n)=\log (\log _{(k-1)}(n))\).

Let \(X_1,X_2,\ldots \) be independent identically distributed positive random variables with \(\mathbb {E} X_1=\mu \). Define \[ N(t)=\min \left \{k:\sum _{i=1}^{k} X_i>t\right \}. \] Show that \[ \lim _{t\to \infty }\frac {1}{t}N(t)=\frac {1}{\mu } \qquad \text {a.s.} \]

Let \(X\) and \(Y\) be independent standard normal random variables. Compute the density function of \(Z=X/Y\).

Let \(X_1,X_2,\ldots \) be independent and identically distributed random variables uniform on \([0,1]\). Show that for all \(\alpha >-1\) there are numerical sequences \(a_n\) and \(b_n\) such that \[ Y_n= \frac {\sum _{1\leq k\leq n} k^{\alpha }X_k-a_n}{b_n} \] converges in distribution to a standard normal random variable.

Compute \[ \lim _{n\to \infty } e^{-n} \sum _{i=0}^{n+\sqrt {n}}\frac {n^i}{i!}. \]