Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 4

Question

Math 6010 • Linear Models Spring 2016 • Question 4

Let \(X_1,X_2,\ldots \) be i.i.d. with common distribution \(\mathbb {P}(X_1=1)=\mathbb {P}(X_1=-1)=1/2\), and put \(S_0:=0\) and \(S_n:=X_1+\cdots +X_n\) for each \(n\geq 1\). Consider the stopping time \(N:=\min \{n\geq 1:S_n=1\}\).

(a) Let \(g(u):=\mathbb {E}[u^{X_1}]\) for all \(u>1\), and show that the optional stopping theorem applies to the martingale \(M_n:=u^{S_n}g(u)^{-n}\), where \(u>1\), and the stopping time \(N\).

(b) Use part (a) to show that \[ \mathbb {E}[v^N]=\frac {1-\sqrt {1-v^2}}{v}, \qquad 0<v<1. \]

(c) Use part (b) to show that \[ \mathbb {P}(N=2m+1)= \frac {1}{m+1}\binom {2m}{m}\frac {1}{2^{2m+1}}, \qquad m\geq 0. \]

Let \(X_1,X_2,\ldots\) be i.i.d. with common distribution
\(\Prob(X_1=1)=\Prob(X_1=-1)=1/2\), and put \(S_0:=0\) and
\(S_n:=X_1+\cdots+X_n\) for each \(n\geq1\). Consider the stopping time
\(N:=\min\{n\geq1:S_n=1\}\).

\textbf{(a)} Let \(g(u):=\E[u^{X_1}]\) for all \(u>1\), and show that the optional
stopping theorem applies to the martingale \(M_n:=u^{S_n}g(u)^{-n}\), where
\(u>1\), and the stopping time \(N\).

\textbf{(b)} Use part (a) to show that
\[
  \E[v^N]=\frac{1-\sqrt{1-v^2}}{v}, \qquad 0<v<1.
\]

\textbf{(c)} Use part (b) to show that
\[
  \Prob(N=2m+1)=
  \frac{1}{m+1}\binom{2m}{m}\frac{1}{2^{2m+1}},
  \qquad m\geq0.
\]

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Topic Tags

Estimator Distributions Filtrations and Stopping Times Martingales and Supermartingales Uniform Integrability and Truncation Methods