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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2006 • Question 4

Question

Math 6010 • Linear Models 2006 • Question 4

Let \(X_1,X_2,\ldots \) be i.i.d., and \(S_n:=X_1+\cdots +X_n\). Assume also that \(\mathbb {P}\{X_1=1\}=\mathbb {P}\{X_1=-1\}=1/2\), so that \(\{S_n\}_{n=1}^{\infty }\) is a simple symmetric random walk. Define \[ T:=\inf \{k\ge 1: |S_k|\ge 2\}. \] As usual, \(\inf \emptyset :=\infty \). Prove that \(\mathbb {E}[T^2]<\infty \).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be i.i.d., and \(S_n:=X_1+\cdots+X_n\). Assume also
that \(\Prob\{X_1=1\}=\Prob\{X_1=-1\}=1/2\), so that \(\{S_n\}_{n=1}^{\infty}\) is a
simple symmetric random walk. Define
\[
  T:=\inf\{k\ge 1: |S_k|\ge 2\}.
\]
As usual, \(\inf\emptyset:=\infty\). Prove that \(\E[T^2]<\infty\).

Topic Tags

Laws of Large Numbers Random Walks and Maximal Inequalities Symmetric Matrices Uniform Integrability and Truncation Methods