Probability Prelim Exam 1.1 January 2019

Let \(X_n\) be a sequence of i.i.d. random variables. Assume that \(\mathbb {P}(X_n\to \infty )>0\). Prove that \(\mathbb {P}(X_0=\infty )=1\). Hint: first prove that \(\mathbb {P}(X_n\to \infty )=1\).

Let \(X_n:\Omega \to \mathbb {R}\) be a sequence of integrable random variables with the same distribution. Prove that \(X_n/n\to 0\) almost surely. Hint: Borel-Cantelli.

Let \(X\) and \(Y\) be two random variables such that \(\mathbb {P}(X\leq Y)=1\). Assume also that \(\mathbb {P}(X\leq x)\leq \mathbb {P}(Y\leq x)\) for all \(x\in \mathbb {R}\). Prove that \(\mathbb {P}(X=Y)=1\).

(Pólya’s Urn) An urn initially contains \(r\) red and \(b\) blue balls. A ball is chosen uniformly at random (i.e. with probability \(1/(r+b)\) each). If If it comes up red (resp. blue), then it is returned and another red (resp. blue) ball is added to the urn. The process is repeated indefinitely. Let \(R_n\) be the number of red balls in the urn after \(n\) draws.

a) Prove that the fraction of red balls \(M_n=R_n/(r+b+n)\) has an almost sure limit \(M_\infty \) as \(n\to \infty \).

b) Suppose now \(r=b=1\). Prove that \(\mathbb {P}(R_n=k)=1/(n+1)\) for \(1\leq k\leq n+1\). Hint: Write down the probability of choosing \(k\) red and \(n-k\) blue balls in some fixed order.

c) Still assuming \(r=b=1\), what is the distribution of \(M_\infty \)? Hint: use part b) to compute \(\mathbb {P}(M_\infty \leq x)\).

Let \(X_n\) be a sequence of independent random variables with \(\mathbb {P}(X_n=1)=1/n=1-\mathbb {P}(X_n=0)\). Prove that \[ \left (\sum _{i=1}^{n}X_i-\log n\right )/\sqrt {\log n} \] converges weakly to a standard normal.

Fix \(\alpha \in (0,1)\). Let \(X_n\) be a sequence of independent random variables with \(\mathbb {P}(X_n=n)=\mathbb {P}(X_n=-n)=n^{-\alpha }/2\) and \(\mathbb {P}(X_n=0)=1-n^{-\alpha }\). Prove that \((X_1+\cdots +X_n)/n^{(3-\alpha )/2}\) converges weakly to a centered normal.

Let \(M_n\) be a martingale with \(M_0=0\) and increments \(X_n=M_n-M_{n-1}\) that satisfy \(\mathbb {E}[X_n^2]<\infty \) for all \(n\). Assume that \((b_n)_{n=1}^{\infty }\) is an increasing sequence of positive real numbers such that \(b_n\to \infty \) and \[ \sum _{n=1}^{\infty }\frac {\mathbb {E}[X_n^2]}{b_n^2}<\infty . \] Show that \(M_n/b_n\to 0\) almost surely as \(n\to \infty \). Hint: Since the sum of the \(X_k\) is a martingale so too should be a weighted sum; prove this idea and use it to produce a new martingale out of the \(X_k\) and \(b_k\). Derive convergence properties of this new martingale, and then use Kronecker’s Lemma: if \(a_n\) is a sequence of numbers such that \(\sum _{n=1}^{\infty }a_n\) exists and is finite and \(b_n\) is a sequence of the type above, then \[ \frac {1}{b_n}\sum _{k=1}^{n}a_kb_k\to 0 \qquad \text {as } n\to \infty . \]

Let \(X_1,X_2,\ldots \) be iid random variables with common pdf \[ f(x)= \begin {cases} 0, & |x|\leq 2,\\ \dfrac {c}{x^2\log |x|}, & |x|>2. \end {cases} \]

1. Prove that there is a finite constant \(c\) which makes \(f\) a pdf.
2. Let \(S_n=X_1+X_2+\cdots +X_n\). Prove that \(S_n/n\to 0\) in probability as \(n\to \infty \). Hint: Prove that \(S_n/n\) converges to 0 in distribution, then explain why you can upgrade to convergence in probability for free.
3. Does \(S_n/n\to 0\) almost surely as \(n\to \infty \)?

Let \(X\) and \(Y\) be two integer-valued random variables defined on the same probability space. The total variation distance between \(X\) and \(Y\) is defined as \[ d_{TV}(X,Y)=2\sup _{A\subseteq \mathbb {Z}} |\mathbb {P}(X\in A)-\mathbb {P}(Y\in A)|. \] Prove that \(\mathbb {P}(X=Y)\leq 1-\frac {1}{2}d_{TV}(X,Y)\).

Let \(g:\mathbb {R}\to \mathbb {R}\) be bounded and continuous and fix \(\lambda >0\). Use properties of the Poisson distribution to show that \[ \sum _{k=0}^{\infty }g(k/n)\frac {(n\lambda )^k}{k!}e^{-n\lambda } \to g(\lambda ) \qquad \text {as } n\to \infty . \]