Ph.D. Qualification Examination in Probability August 2004

A sequence \(X_1,X_2,\ldots \) of random variables is said to be uniformly integrable if \[ \lim _{c\to \infty }\sup _{n\ge 1}\mathbb {E}\{|X_n|;\ |X_n|>c\}=0. \] Prove that \(X_n\to X\) in \(L^1(\mathbb {P})\) if and only if both of the following happen: (i) \(X_n\to X\) in probability; and (ii) \(\{X_n\}_{n=1}^{\infty }\) is uniformly integrable.

Construct three random variables \(X,Y\), and \(Z\), such that any distinct pair of them are independent but \((X,Y,Z)\) are not independent.

Suppose \(\{X_n\}_{n=1}^{\infty }\) are i.i.d. standard normal random variables. Prove that \[ M_n:=(n+1)^{-1/2}\exp \left (\frac {S_n^2}{2n+2}\right ) \] defines a martingale, where \(S_n=\sum _{i=1}^{n}X_i\).

Choose and fix an integer \(n\ge 1\). For all continuous functions \(f:[0,1]\to \mathbb {R}\) define \[ B_nf(p)=\sum _{k=0}^{n}\binom {n}{k}p^k(1-p)^{n-k}f(k/n), \qquad \text {for all }p\in [0,1]. \] [ \(B_nf\) is the Bernstein polynomial of \(f\).] Prove that if \(f\) is increasing, then so is \(B_nf\).

Let \(\{X_n\}_{n=1}^{\infty }\) be i.i.d. random variables with distribution, \(\mathbb {P}\{X_1=2\}=\mathbb {P}\{X_1=-2\}=1/2\). Define \(S_n=\sum _{i=1}^{n}X_i\) and \(T=\inf \{n\ge 1:S_n\in \{-2,6\}\}\). Prove that \(T<\infty \) a.s., and then compute \(\mathbb {P}\{S_T=-2\}\).

Suppose \(\{X_n\}_{n=1}^{\infty }\) and \(\{Y_n\}_{n=1}^{\infty }\) are random variables on the same probability space \((\Omega ,\mathcal {F},\mathbb {P})\). Suppose also that \(X_n\Rightarrow X\) and \(Y_n\Rightarrow Y\). True or false: \((X_n,Y_n)\Rightarrow (X,Y)\). [If true, then prove it. If false, then construct a counter-example.]

Let \(\{X_n\}_{n=1}^{\infty }\) be a sequence of i.i.d. random variables with \(\mathbb {E}[X_1]=0\) and \(\mathbb {E}[X_1^2]=1\). Define \(S_k=X_1+\cdots +X_k\) for all \(k\ge 1\). Let \(N_m\) be another independent random variable with a mean-\(m\) Poisson distribution. Prove that there exist non-random \(\alpha _m\) and \(\beta _m\) such that \((S_{N_m}-\alpha _m)/\beta _m\) converges weakly to a standard-normal distribution as \(m\to \infty \). Describe an explicit example of \(\alpha _m\) and \(\beta _m\).

If \(X\ge 0\) and \(X\in L^p(\mathbb {P})\) for all \(p>1\), then prove that \(\lim _{p\to \infty }\|X\|_p\) exists.

Consider two random variables \(X,Y\in L^2(\mathbb {P})\). Prove that if \(\mathbb {E}[X\mid Y]=Y\) and \(\mathbb {E}[X^2\mid Y]=Y^2\), then \(X=Y\) a.s.

Define \(X_0=1\). Then iteratively define \(X_n\) so that for all Borel sets \(A\subseteq \mathbb {R}\), \[ \mathbb {P}\left (X_n\in A\mid X_0,\ldots ,X_{n-1}\right ) = \frac {\text {The lebesgue measure of }A\cap [0,X_{n-1}]}{X_{n-1}}. \] Prove that \(\lim _{n\to \infty }2^nX_n=0\) a.s. You may use–without proof–the following version of the Borel–Cantelli lemma [due to Paul Lévy]: If \(\{\mathcal {F}_n\}_{n=1}^{\infty }\) is a filtration and \(A_n\)’s are events such that \(A_n\in \mathcal {F}_n\) and \(\sum _{n=1}^{\infty }\mathbb {P}(A_n\mid \mathcal {F}_{n-1})=\infty \) a.s., then \(\mathbb {P}(A_n\text { infinitely often})=1\).