course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Markov Chains |
IB |
46 |
|
Paper 1, Section II, H |
2016 |
Let $\left(X_{n}\right){n \geqslant 0}$ be a simple symmetric random walk on the integers, starting at $X{0}=0$.
(a) What does it mean to say that a Markov chain is irreducible? What does it mean to say that an irreducible Markov chain is recurrent? Show that
[Hint: You may find it helpful to use the limit
You may also use without proof standard necessary and sufficient conditions for recurrence.]
(b) What does it mean to say that an irreducible Markov chain is positive recurrent? Determine, with proof, whether
(c) Let
be the first time the chain returns to the origin. Compute