Probability Prelim Exam 1.3 August 2013

Let \(X,X_1,X_2,\ldots \) be independent random variables, all distributed according to the standard normal distribution.

(a) Prove that there exist finite constants \(c_1,c_2,x_0>0\), such that \[ \frac {c_1}{x}e^{-x^2/2}\leq \mathbb {P}\{X>x\}\leq \frac {c_2}{x}e^{-x^2/2} \qquad \text {for all }x\geq x_0. \]

(b) Use the inequalities of part (a) in order to prove that if \(X_1,X_2,\ldots \) are independent standard normal random variables then \[ \limsup _{n\to \infty }\frac {X_n}{\sqrt {2\ln n}} = -\liminf _{n\to \infty }\frac {X_n}{\sqrt {2\ln n}} =1 \qquad \text {a.s.} \]

Let \(X_1,X_2,\ldots \) be i.i.d. with \(\mathbb {P}\{X_1=1\}=1/4\) and \(\mathbb {P}\{X_1=0\}=3/4\). Let \(N\) denote the smallest integer \(k\geq 2\) such that \(X_{k-1}=0,=X_k=1\); that is, \(N\) is the first time the pattern “01” occurs in the infinite sequence \(X_1,X_2,\ldots \). Compute \(\mathbb {E}(N)\).

Suppose \(X_1,X_2,\ldots ,X_n\) are i.i.d. random variables, and \(1\leq X_1\leq 2\) a.s. Compute, for every \(1\leq i\leq j\leq n\), the quantity \[ \mathbb {E}\left (\frac {X_i+\cdots +X_j}{X_1+\cdots +X_n}\right ). \]

Suppose \(\mu \) is a probability measure on the Borel subsets of \((0,1)\) such that \[ \int |\ln (x)|\,\mu (dx)<\infty . \] Prove that \[ I(n):=\left (\int x^{1/n}\,\mu (dx)\right )^n \to \exp \left (\int \ln (x)\,\mu (dx)\right ) \qquad \text {as } n\to \infty . \]

[Hint: \(I(n)=\mathbb {E}\left (\prod _{j=1}^{n}X_j^{1/n}\right )\), where \(X_1,\ldots ,X_n\) are i.i.d.]

Let \(\mathbf {X}_n:=(X_{1,n},\ldots ,X_{d,n})\) be a random variable in \(\mathbf {R}^d\) for every \(n\geq 1\). Let \(\mathbf {Y}:=(Y_1,\ldots ,Y_d)\) be another random variable in \(\mathbf {R}^d\). Prove that \(\mathbf {X}_n\Rightarrow \mathbf {Y}\) if and only if the 1-dimensional random variable \(\sum _{i=1}^{d}\lambda _iX_{i,n}\) converges weakly to \(\sum _{i=1}^{d}\lambda _iY_i\) for every \(\lambda :=(\lambda _1,\ldots ,\lambda _d)\in \mathbf {R}^d\).

For every random vector \((X,Y)\) in \(\mathbf {R}^2\) we can define a function \(\mathcal {Q}\) of two variables as follows: \[ \mathcal {Q}(a,b):=\Prob \{X>a,\ Y>b\}-\Prob \{X>a\}\Prob \{Y>b\}. \]

(a) Suppose \(f,g:\mathbf {R}\to \mathbf {R}\) are bounded and have integrable, bounded, and continuous derivatives. Then prove that \[ \E [f(X)g(Y)]-\E [f(X)]\E [g(Y)] = \iint _{\mathbf {R}^2}\mathcal {Q}(a,b)f'(a)g'(b)\,da\,db. \]

(b) Suppose \(\mathcal {Q}(a,b)\geq 0\) for all \(a,b\in \mathbf {R}\). Prove that if \(X\) and \(Y\) have two finite moments, then \(\operatorname {Cov}(X,Y)\geq 0\) and there exists a finite constant \(\alpha \), depending only on \(f\) and \(g\), such that \[ |\E [f(X)g(Y)]-\E [f(X)]\E [g(Y)]|\leq \alpha \operatorname {Cov}(X,Y). \]

[In other words, if \(\mathcal {Q}\geq 0\) and \(X\) and \(Y\) have a small covariance, then \(X\) and \(Y\) are almost independent.]

Suppose that, for every integer \(n\geq 1\), \(X(n)\) has a Gamma distribution with parameters \(n+1\) and 1; that is, the probability density function of \(X(n)\) is \[ f_n(x):=\frac {x^ne^{-x}}{n!}\mathbf {1}_{(0,\infty )}(x). \]

(a) Compute \(\mathbb {E}(X(n))\) and \(\operatorname {Var}(X(n))\) for every integer \(n\geq 1\).

(b) Prove that, as \(n\to \infty \), \[ \frac {X(n)-n}{\sqrt {n}}\Rightarrow \mathrm {N}(0,1). \]

Let \(\{W(t)\}_{t\geq 0}\) be a standard Brownian motion, and define \[ U(t):=e^{-t/2}W(e^t)\qquad \text {for all }t\geq 0. \]

(a) Prove that \(\{U(t)\}_{t\geq 0}\) is Gaussian process; compute its mean and covariance functions.

(b) Prove that \[ \limsup _{t\to \infty }\frac {U(t)}{\sqrt {2\ln t}} = -\liminf _{t\to \infty }\frac {U(t)}{\sqrt {2\ln t}} =1 \qquad \text {a.s.} \]

Let \(X\) be an integer-valued random variable that is non-degenerate; this means that there exist at least two integers \(n,m\) such that \(\mathbb {P}\{X=n\}\) and \(\mathbb {P}\{X=m\}\) are both strictly positive. Prove that the set \[ \mathcal {V}:=\{t\in \mathbf {R}:|\mathbb {E}[e^{itX}]|=1\} \] has zero Lebesgue measure. [Hint: If \(t\in \mathcal {V}\) then \(Z:=e^{it(X-Y)}\) has mean one, where \(Y\) is an independent copy of \(X\). Proceed by proving that \(\mathbb {E}(|Z-1|^2)=0\).]

Consider i.i.d. \(X_1,X_2,\ldots \) that are strictly positive, and define \[ S_n:=\sqrt {X_1+\sqrt {X_2+\sqrt {X_3+\sqrt {\cdots +\sqrt {X_n}}}}}, \qquad \text {for all }n\geq 1. \] In other words, if \(f(a,b):=\sqrt {a+\sqrt {b}}\) for all \(a,b>0\), then \(S_1:=f(X_1,0)\) and \(S_{k+1}:=f(S_k,X_{k+1})\) for all \(k\geq 0\).

Prove that \(\lim _{n\to \infty }S_n\) exists a.s. and is finite a.s.