Probability Prelim 1.3 January 3, 2008

Let \(X\) and \(Y\) have joint density \[ f(x,y)=g\left (\sqrt {x^2+y^2}\right ),\qquad (x,y)\in \mathbb {R}^2, \] for some function \(g\). Show that \(Z=X/Y\) has a Cauchy density.

(a) State the central limit theorem as well as Kolmogorov’s 0-1 law.

(b) For arbitrary events \(A_n\), \(n\geq 1\), show that \(\inf _{n\geq 1}\mathbb {P}(A_n)>0\) implies \(\mathbb {P}(A_n\ \text {i.o.})>0\).

(c) Let \(X_1,X_2,\ldots \) be i.i.d. mean 0, variance 1 random variables, and put \(S_n=X_1+\cdots +X_n\) for each \(n\geq 1\). Show that \(\limsup _{n\to \infty }S_n=\infty \) a.s.

Construct three random variables \(X,Y,Z\) such that \[ \mathbb {E}[\mathbb {E}[X\mid Y]\mid Z]\neq \mathbb {E}[\mathbb {E}[X\mid Z]\mid Y] \] with positive probability.

Let \(X_1,X_2,\ldots \) be a sequence of integrable random variables defined on the same probability space. Show that \(\{X_n\}_{n\geq 1}\) is a submartingale if and only if \(\mathbb {E}[X_T]\geq \mathbb {E}[X_1]\) for all bounded stopping times \(T\). (Take the filtration to be the natural one, \(\mathcal {F}_n=\sigma (X_1,\ldots ,X_n)\).)

Let \(X_1,X_2,\ldots ,X_n\) be i.i.d. positive random variables. Show that \[ \mathbb {E}\left [ \frac {\sum _{i=1}^m X_i}{\sum _{i=1}^n X_i} \right ] = \frac {m}{n},\qquad 1\leq m\leq n. \]

Suppose \(X_{1,n},X_{2,n},\ldots \) are i.i.d. with \(\mathbb {P}\{X_{1,n}=j\}=1/n\) for \(j=1,\ldots ,n\). Prove that if \[ T_n:=\inf \{k\geq 1:X_{1,n}+\cdots +X_{k,n}>n\}, \] then \(\lim _{n\to \infty }\mathbb {P}\{T_n=k\}\) exists for all \(k\geq 1\). Compute that limit.

Let \(X_1,X_2,\ldots \) be i.i.d. mean 0, variance 1 random variables and \(S_n=X_1+\cdots +X_n\) for each \(n\geq 1\). Show directly that the characteristic function of \(S_n/\sqrt {n}\) converges pointwise to the characteristic function of the standard normal distribution. Notice that this is a key step in the proof of the central limit theorem, so you are not allowed to use the central limit theorem in your derivation.

Let \(X_1,X_2,\ldots \) be i.i.d. with \(\mathbb {P}(X_1>x)=e^{-x}\), \(x>0\). Prove that \[ \limsup _{n\to \infty } \frac {X_n}{\ln n}=1 \] a.s.

Let \(X\) and \(Y\) be independent, each having the standard normal distribution, and let \((R,\Theta )\) be the polar coordinates for \((X,Y)\).

(a) Show that \(X+Y\) and \(X-Y\) are independent, and that \(R^2=[(X+Y)^2+(X-Y)^2]/2\), and conclude that the conditional distribution of \(R^2\) given \(X-Y\) is the chi-squared with one degree of freedom translated by \((X-Y)^2/2\).

(b) Show that the conditional distribution of \(R^2\) given \(\Theta \) is chi-squared with two degrees of freedom.

(c) If \(X-Y=0\), the conditional distribution of \(R^2\) is chi-squared with one degree of freedom. If \(\Theta =\pi /4\) or \(\Theta =5\pi /4\), the conditional distribution of \(R^2\) is chi-squared with two degrees of freedom. But the events \(\{X-Y=0\}\) and \(\{\Theta =\pi /4\}\cup \{\Theta =5\pi /4\}\) are the same. Resolve the apparent contradiction.

Show that the number of fixed points in a random permutation of length \(n\) is asymptotically Poisson with mean 1. Hint: Use characteristic functions. Write \(e^{it1_A}\) as \(1+(e^{it}-1)1_A\).