From 298e812476136624d60c7ec644c0bff8e768459d Mon Sep 17 00:00:00 2001
From: Longye Tian <longye.tian@outlook.com>
Date: Mon, 17 Jun 2024 20:22:36 +1000
Subject: [PATCH] add definition of accessible

Update related to #463
---
 lectures/markov_chains_II.md | 9 ++-------
 1 file changed, 2 insertions(+), 7 deletions(-)

diff --git a/lectures/markov_chains_II.md b/lectures/markov_chains_II.md
index 7928bb14..59b8e23c 100644
--- a/lectures/markov_chains_II.md
+++ b/lectures/markov_chains_II.md
@@ -58,14 +58,9 @@ import numpy as np
 
 To explain irreducibility, let's take $P$ to be a fixed stochastic matrix.
 
-Two states $x$ and $y$ are said to **communicate** with each other if
-there exist positive integers $j$ and $k$ such that
+State $x$ is called **accessible** (or **reachable**) from state $y$ if $P^t(x,y)>0$ for some integer $t\ge 0$. 
 
-$$
-P^j(x, y) > 0
-\quad \text{and} \quad
-P^k(y, x) > 0
-$$
+Two states, $x$ and $y$, are said to **communicate** if $x$ and $y$ are accessible from each other.
 
 In view of our discussion {ref}`above <finite_mc_mstp>`, this means precisely
 that