Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 7

Question

Math 6010 • Linear Models Spring 2020 • Question 7

Assume the same setting as in the previous problem. You can assume that the claim in that problem is true and solve this problem independently.

a) For an integer \(a\geq 0\) let \(T_a=\inf \{n\geq 0:X_n=a\}\). Fix \(a\geq 2\) and prove that \[ \mathbb {P}(T_0<\infty \text { or }T_a<\infty )=1. \] (Hint: As long as the process is not at 0 it has a positive probability to go up \(a\) times in a row. How do you turn this observation into a proof?)

b) Calculate \(\mathbb {P}(T_a<T_0\mid X_0=1)\). (Carefully explain why the conditions of the theorems you use are satisfied.)

Assume the same setting as in the previous problem. You can assume
that the claim in that problem is true and solve this problem independently.

a) For an integer \(a\geq0\) let
\(T_a=\inf\{n\geq0:X_n=a\}\). Fix \(a\geq2\) and prove that
\[
  \Prob(T_0<\infty\text{ or }T_a<\infty)=1.
\]
(\textbf{Hint:} As long as the process is not at 0 it has a positive
probability to go up \(a\) times in a row. How do you turn this observation into
a proof?)

b) Calculate \(\Prob(T_a<T_0\mid X_0=1)\). (Carefully explain why the conditions
of the theorems you use are satisfied.)

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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