Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 5

Question

Math 6010 • Linear Models Spring 2009 • Question 5

Let \(X_1,X_2,\ldots \) be an i.i.d. sequence of integer-valued random variables, and let \(\lambda _0\) be a root of the equation \(\mathbb {E}[\lambda ^{X_1}]=1\). (If \(\lambda _0\) is complex, it suffices to notice that expectations, conditional expectations, and martingales extend easily to complex-valued random variables.) Let \(S_n:=X_1+\cdots +X_n\) and \(M_n:=\lambda _0^{S_n}\) for each \(n\geq 0\), where \(S_0:=0\).

(a) Show that \(\{M_n\}_{n\geq 0}\) is a martingale with respect to \(\mathcal {F}_n:=\sigma (X_1,\ldots ,X_n)\).

(b) Consider the special case that \(\mathbb {P}\{X_1=1\}=p\) and \(\mathbb {P}\{X_1=-1\}=1-p\), where \(0<p<1\). Show that, in this case, \(\lim _{n\to \infty }M_n\) exists a.s. and is finite.

Let \(X_1,X_2,\ldots\) be an i.i.d. sequence of integer-valued random
variables, and let \(\lambda_0\) be a root of the equation
\(\E[\lambda^{X_1}]=1\). (If \(\lambda_0\) is complex, it suffices to notice that
expectations, conditional expectations, and martingales extend easily to
complex-valued random variables.) Let \(S_n:=X_1+\cdots+X_n\) and
\(M_n:=\lambda_0^{S_n}\) for each \(n\geq0\), where \(S_0:=0\).

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\textbf{(a)} Show that \(\{M_n\}_{n\geq0}\) is a martingale with respect to
\(\mathcal{F}_n:=\sigma(X_1,\ldots,X_n)\).

\textbf{(b)} Consider the special case that \(\Prob\{X_1=1\}=p\) and
\(\Prob\{X_1=-1\}=1-p\), where \(0<p<1\). Show that, in this case,
\(\lim_{n\to\infty}M_n\) exists a.s. and is finite.

\textbf{(c)} Find another example to show that \(\lim_{n\to\infty}M_n\) need not
exist a.s.

Topic Tags

Conditional Expectation and Conditional Independence Convergence in Probability and Almost Sure Convergence Filtrations and Stopping Times Martingales and Supermartingales