Probability Prelim 1.3 August 19, 2008

Let \(X_1,X_2,\ldots \) be random variables such that the partial sums \(S_n:=X_1+X_2+\cdots +X_n\) determine a martingale. Show that \(\mathbb {E}[X_iX_j]=0\) if \(i\neq j\).

Let \(X_2,X_3,\ldots \) be independent random variables such that \[ \mathbb {P}(X_n=n)=\mathbb {P}(X_n=-n)=\frac {1}{2n\log n}, \qquad \mathbb {P}(X_n=0)=1-\frac {1}{n\log n}. \] Show that this sequence obeys the weak law but not the strong law of large numbers, in the sense that \(n^{-1}\sum _{i=1}^n X_i\) converges to 0 in probability but not almost surely.

Let \(\{X_n,n\geq 1\}\) be independent \(N(0,1)\) random variables. Show that \[ \Prob \left (\limsup \frac {|X_n|}{\sqrt {\log n}}=\sqrt {2}\right )=1. \] Hint: \(1-\Phi (x)\) is asymptotic to \(\phi (x)/x\) as \(x\to \infty \).

Let \(X_1,X_2,\ldots \) be independent random variables such that \[ X_n= \begin {cases} 1 & \text {with probability }(2n)^{-1},\\ 0 & \text {with probability }1-n^{-1},\\ -1 & \text {with probability }(2n)^{-1}. \end {cases} \] Let \(Y_1:=X_1\) and for \(n\geq 2\) \[ Y_n:= \begin {cases} X_n & \text {if }Y_{n-1}=0,\\ nY_{n-1}|X_n| & \text {if }Y_{n-1}\neq 0. \end {cases} \] Show that \(Y_n\) is a martingale with respect to \(\mathcal {F}_n:=\sigma (Y_1,Y_2,\ldots ,Y_n)\). Show that \(Y_n\) does not converge almost surely. Does \(Y_n\) converge in any way? Why does the martingale convergence theorem not apply?

Let \(X\) and \(Y\) be independent exponential random variables with parameters \(\lambda \) and \(\mu \). (For example, \(\mathbb {P}(X>x)=e^{-\lambda x}\) for \(x>0\).)

(a) Show that \(Z:=\min (X,Y)\) is independent of the event \(\{X<Y\}\).

(b) Find \(\mathbb {P}(X=Z)\).

(c) Find the distribution of \(U:=(X-Y)^+=\max (X-Y,0)\).

(d) Find the distribution of \(V:=\max (X,Y)-\min (X,Y)\).

A die is thrown 10 times. What is the probability that the sum of the scores in 27?

Hint: Use probability generating functions.

Suppose \(W:=(X,Y)\) is a two-dimensional random variable and \(QW^T\) is \(N(0,1)\) for all \(2\times 2\) rotation matrices \[ Q:= \begin {pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end {pmatrix}. \] Prove that \(X\) and \(Y\) are independent and \(N(0,1)\).

Let \(X_1,X_2,\ldots \) and \(Y_1,Y_2,\ldots \) denote random variables on a common probability space. Suppose \(X_n\) converges to 0 in probability as \(n\to \infty \), and \(Y_n\) converges to \(Y\) weakly. Prove that \((X_n,Y_n)\) converges weakly to \((0,Y)\).

Construct two random variables \(X\) and \(Y\) such that:

(a) \(X\) is \(N(0,1)\),

(b) \(Y\) is \(N(0,1)\), and

(c) \((X,Y)\) is not a Gaussian random vector.

Suppose \(X\) and \(Y\) are independent bounded random variables. Prove in complete detail the following assertions:

(a) \(\mathbb {E}(XY)=\mathbb {E}(X)\mathbb {E}(Y)\),

(b) \(\mathbb {E}(X\mid Y)=\mathbb {E}(X)\).