Ph.D. Qualification Examination in Probability January 2006

If \(X\) is distributed uniformly on \([0,1]\), then compute \(\mathbb {E}[X\mid \mathcal {G}]\), where \(\mathcal {G}\) is the \(\sigma \)-algebra generated by \(\{X\le 1/2\}\).

Let \(\{X_i\}_{i=1}^{\infty }\) be a sequence of i.i.d. random variables, and define \[ S(r):=\sum _{i=1}^{\infty }(1-r)^iX_i \qquad \text {for all } 0<r<1. \]

(a) Prove that for every fixed \(r\in (0,1)\), \(S(r)\) is a.s. absolutely convergent if and only if \(\mathbb {E}\{\log (1+|X_1|)\}<\infty \).

(b) Suppose that \(\mathbb {E}[X_1]=\mu \) and \(\operatorname {Var}(X_1)=\sigma ^2<\infty \). Then prove that there exists a constant \(c\) such that \(rS(r)\to c\) in probability, as \(r\to 0\). Compute \(c\).

Let \(\{X_n\}_{n=1}^{\infty }\) denote a martingale such that \(X_1\in L^2(\mathbb {P})\), and \[ \sum _{n=1}^{\infty }\mathbb {E}\{(X_{n+1}-X_n)^2\}<\infty . \] Prove that \(\lim _{n\to \infty }X_n\) exists a.s. and in \(L^2(\mathbb {P})\).

Let \(X_1,X_2,\ldots \) be i.i.d., and \(S_n:=X_1+\cdots +X_n\). Assume also that \(\mathbb {P}\{X_1=1\}=\mathbb {P}\{X_1=-1\}=1/2\), so that \(\{S_n\}_{n=1}^{\infty }\) is a simple symmetric random walk. Define \[ T:=\inf \{k\ge 1: |S_k|\ge 2\}. \] As usual, \(\inf \emptyset :=\infty \). Prove that \(\mathbb {E}[T^2]<\infty \).

Let \(\{X_n\}_{n=1}^{\infty }\) be a sequence of independent (but not identically distributed) random variables, such that for all \(n\ge 1\), \[ \mathbb {P}\{X_n=1\}=\mathbb {P}\{X_n=-1\}=\frac {1}{2}-\frac {1}{2n^2}, \qquad \mathbb {P}\{X_n=n\}=\mathbb {P}\{X_n=-n\}=\frac {1}{2n^2}. \] Prove that \(n^{-1/2}\sum _{i=1}^{n}X_i\Rightarrow N(0,\sigma ^2)\), and compute \(\sigma ^2\) explicitly.

Suppose \(X\) and \(Y\) are two independent standard normal random variables.

(a) Prove that for all twice continuously differentiable functions \(f,g:\mathbb {R}\to \mathbb {R}\), \[ \operatorname {Cov}(f(X),g(X))=\int _0^1 \E \!\left [f'(X)g'\!\left (sX+(1-s^2)^{1/2}Y\right )\right ]\,ds. \]

(Hint: Check it first for \(f(x):=\exp (itx)\) and \(g(x):=\exp (itx)\).)

(b) Conclude the following “Poincaré inequality,” due to J. Nash (1958): \[ \Var (f(X))\le \E \!\left [(f'(X))^2\right ]. \]

Let \(\{X_i\}_{i=1}^{\infty }\) denote a sequence of i.i.d. random variables with \(\mathbb {P}\{X_1=0\}=p\) and \(\mathbb {P}\{X_1=1\}=q:=1-p\). Consider the random sequence \(X_1X_2\cdots \), and compute \(\mathbb {E}[T]\), where \(T\) denotes the first time that the pattern “001” appears in the said sequence.

Construct an example of a sequence \(\{A_n\}_{n=1}^{\infty }\) such that:

(a) (i) \(\sum _n\mathbb {P}(A_n)=\infty \); and (ii) only finitely-many of the \(A_n\)’s occur with positive probability.

(b) (i) \(\sum _n\mathbb {P}(A_n)=\infty \); and (ii) infinitely-many of the \(A_n\)’s occur with positive probability.

Let \(X\) and \(Y\) be two standard-normal random variables.

(a) Prove that if \((X,Y)\) is Gaussian then \(X\) and \(Y\) are independent.

(b) Construct an example wherein \((X,Y)\) is not a (two-dimensional) Gaussian random variable.

Let \(\{X_i\}_{i=1}^{\infty }\) be i.i.d. with the “standard Cauchy distribution.” That is, the density function of \(X_1\) is \[ f(x)=\frac {1}{\pi (1+x^2)} \qquad -\infty <x<\infty . \] Prove that \(\max _{1\le i\le n}X_i/n\) converges weakly. Identify the limit.