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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 4

Question

Math 6010 • Linear Models Fall 2017 • Question 4

Let \(X_1,X_2,\ldots \) be a collection of i.i.d. random variables, taking values \(\pm 1\) with probability \(1/2\) each. Define \(S_n:=X_1+\cdots +X_n\) for all \(n\geq 1\), \(T_0:=0\), and then iteratively define \[ T_{k+1}:=\inf \{n>T_k:S_n=0\} \qquad [\inf \emptyset :=\infty ]\quad \text {for all }k\geq 0. \] That is, \(T_k\) denotes the \(k\)th return time to zero. Prove that \(\lim _{k\to \infty }(T_k/k)=\infty \) a.s.

Raw LaTeX

Let \(X_1,X_2,\ldots\) be a collection of i.i.d. random variables,
taking values \(\pm1\) with probability \(1/2\) each. Define
\(S_n:=X_1+\cdots+X_n\) for all \(n\geq1\), \(T_0:=0\), and then iteratively define
\[
  T_{k+1}:=\inf\{n>T_k:S_n=0\}
  \qquad[\inf\emptyset:=\infty]\quad\text{for all }k\geq0.
\]
That is, \(T_k\) denotes the \(k\)th return time to zero. Prove that
\(\lim_{k\to\infty}(T_k/k)=\infty\) a.s.

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Topic Tags

Classical Probability and Inclusion-Exclusion Convergence in Probability and Almost Sure Convergence Measure-Theoretic Probability Uniform Integrability and Truncation Methods