0

I am trying to make intuitive sense of the ability of irreducible, infinite markov chains possibly being null recurrent

  • The way I see it, there is a possibility the process just keeps getting larger and larger towards a very distant state, so if we look at state 0, there is a non-0 probability of a path that leads forever away from 0
  • This means that E(T00) will be infinite due to the inclusion of that path multiplied by a non-0 probability

However, from the property of recurrent classes, it says that fi = P(Tii < infinity) = 1, i.e. T must be finite.

Do these two statements clash?

Thank you so much for the help!

jojorabbit
  • 23
  • Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – José Carlos Santos Feb 07 '21 at 07:12
  • It might help to study an example like a standard 1D random walk. Starting at 0 you'll eventually return to $0$ with probability $1$ (this you can prove formally!); however, the expected amount of time it takes to return to $0$ evaluates to $\infty$. – Jeroen van der Meer Feb 07 '21 at 09:52
  • What is the intuition there? Perhaps it's enlightening to run some simulations. Compute the amount of time it takes to return to 0 over and over again, and consider the average. As the computer goes on, you'll find that this average will slowly but surely increase as more and more outliers appear. Indeed the average will statistically never stabilise. – Jeroen van der Meer Feb 07 '21 at 09:54
  • $T$ is finite with probability $1$ just means $\mathbb{P}(T=\infty)=0$. It doesn't say $\mathbb{P}(T\leq n)=1$ for some $n\in\mathbb{N}$. The easiest example is to walk on the graph $(\amalg C_{2^n})/\sim$ where $\sim$ identifies a distinguished vertex from each $2^n$-cycle, and the walk is deterministic on each $2^n$-cycle and the common vertex is stochastic with probability $2^{-n}$ of going on the $2^n$-cycle, $n=1,2,\dots$. – user10354138 Feb 07 '21 at 14:36
  • thank you so much @JeroenvanderMeer! I appreciate the help very much! – jojorabbit Feb 08 '21 at 08:24
  • thank you as well @user10354138!! :) – jojorabbit Feb 08 '21 at 08:24

0 Answers0