Update markov_chains_I.md

Author: Jingni0117

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
 
 $$
     \psi_0 P^t \to \psi^*
-    \quad \text{as } t \to \infty
+    \quad \text{ as } t \to \infty
 $$
 ```