Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 2

Question

Math 6010 • Linear Models Fall 2016 • Question 2

Choose and fix two integers \(N\geq 1\) and \(x\in \{0,1,\ldots ,N\}\). Let \(X_0=x\) and suppose that, conditionally, \(X_{n+1}\) has a \(\operatorname {Binomial}(N,X_n/N)\) distribution given \(X_1,\ldots ,X_n\). More precisely, for all \(n\geq 0\) and \(k=0,\ldots ,N\), \[ \mathbb {P}(X_{n+1}=k\mid \mathcal {F}_n) = \binom {N}{k} \left (\frac {X_n}{N}\right )^k \left (1-\frac {X_n}{N}\right )^{N-k}, \] where \(\mathcal {F}_n=\sigma (\{X_1,\ldots ,X_n\})\) for all \(n\geq 1\). Prove that:

(a) \(X_{\infty }=\lim _{n\to \infty }X_n\) exists a.s. and in \(L^p(\mathbb {P})\) for all \(1\leq p<\infty \).

(b) Prove that \(\mathbb {E}(X_{\infty })=x\) and \(\mathbb {E}(X_{\infty }^2)=Nx\).

Choose and fix two integers \(N\geq1\) and
\(x\in\{0,1,\ldots,N\}\). Let \(X_0=x\) and suppose that, conditionally,
\(X_{n+1}\) has a \(\operatorname{Binomial}(N,X_n/N)\) distribution given
\(X_1,\ldots,X_n\). More precisely, for all \(n\geq0\) and \(k=0,\ldots,N\),
\[
  \Prob(X_{n+1}=k\mid\mathcal{F}_n)
  =
  \binom{N}{k}
  \left(\frac{X_n}{N}\right)^k
  \left(1-\frac{X_n}{N}\right)^{N-k},
\]
where \(\mathcal{F}_n=\sigma(\{X_1,\ldots,X_n\})\) for all \(n\geq1\). Prove that:

\textbf{(a)} \(X_{\infty}=\lim_{n\to\infty}X_n\) exists a.s. and in
\(L^p(\Prob)\) for all \(1\leq p<\infty\).

\textbf{(b)} Prove that \(\E(X_{\infty})=x\) and \(\E(X_{\infty}^2)=Nx\).

\textbf{(c)} Compute the distribution of \(X_{\infty}\).

Topic Tags

Conditional Expectation and Conditional Independence Convergence in Probability and Almost Sure Convergence Filtrations and Stopping Times Uniform Integrability and Truncation Methods