From f0d3fd69de04ff5d5600a88a0a5afc3f57039219 Mon Sep 17 00:00:00 2001
From: Jingni <112955268+Jingni0117@users.noreply.github.com>
Date: Fri, 15 Dec 2023 02:28:29 +1100
Subject: [PATCH 1/2] Update markov_chains_I.md

---
 lectures/markov_chains_I.md | 9 ++++-----
 1 file changed, 4 insertions(+), 5 deletions(-)

diff --git a/lectures/markov_chains_I.md b/lectures/markov_chains_I.md
index 40285910..d175040e 100644
--- a/lectures/markov_chains_I.md
+++ b/lectures/markov_chains_I.md
@@ -294,7 +294,7 @@ Looking at the data, we see that democracies tend to have longer-lasting growth
 regimes compared to autocracies (as indicated by the lower probability of
 transitioning from growth to growth in autocracies).
 
-We can also find a higher probability from collapse to growth in democratic regimes
+We can also find a higher probability from collapse to growth in democratic regimes.
 
 
 ### Defining Markov chains
@@ -411,7 +411,6 @@ def mc_sample_path(P, ψ_0=None, ts_length=1_000):
     X = np.empty(ts_length, dtype=int)
 
     # Convert each row of P into a cdf
-    n = len(P)
     P_dist = np.cumsum(P, axis=1)  # Convert rows into cdfs
 
     # draw initial state, defaulting to 0
@@ -683,7 +682,7 @@ P = np.array([[0.4, 0.6],
 ψ @ P
 ```
 
-Notice that `ψ @ P` is the same as `ψ`
+Notice that `ψ @ P` is the same as `ψ`.
 
 
 
@@ -772,11 +771,11 @@ For example, we have the following result
 (strict_stationary)=
 ```{prf:theorem}
 Theorem: If there exists an integer $m$ such that all entries of $P^m$ are
-strictly positive, with unique stationary distribution $\psi^*$, and
+strictly positive, with unique stationary distribution $\psi^*$, then
 
 $$
     \psi_0 P^t \to \psi^*
-    \quad \text{as } t \to \infty
+    \quad \text{ as } t \to \infty
 $$
 ```
 

From 221d2c25f84b089a7102f1f0bfed3513ed2ce12e Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Wed, 20 Dec 2023 16:10:04 +1100
Subject: [PATCH 2/2] fix markup issues

---
 lectures/markov_chains_I.md | 6 ------
 1 file changed, 6 deletions(-)

diff --git a/lectures/markov_chains_I.md b/lectures/markov_chains_I.md
index d175040e..16aed135 100644
--- a/lectures/markov_chains_I.md
+++ b/lectures/markov_chains_I.md
@@ -107,8 +107,6 @@ From  US unemployment data, Hamilton {cite}`Hamilton2005` estimated the followin
 
 ```
 
-+++
-
 Here there are three **states**
 
 * "ng" represents normal growth
@@ -836,8 +834,6 @@ ax.scatter(ψ_star[0], ψ_star[1], ψ_star[2], c='k', s=60)
 plt.show()
 ```
 
-+++ {"user_expressions": [], "tags": []}
-
 Here
 
 * $P$ is the stochastic matrix for recession and growth {ref}`considered above <mc_eg2>`.
@@ -1082,8 +1078,6 @@ Solution 1:
 
 ```
 
-+++
-
 Since the matrix is everywhere positive, there is a unique stationary distribution.
 
 Solution 2: