update section titles

Author: kp992

lectures/kesten_processes.md

@@ -70,7 +70,7 @@ register_matplotlib_converters()
 ```
 
 Additional technical background related to this lecture can be found in the
-monograph of {cite}`buraczewski2016stochastic`.
+monograph by {cite}`buraczewski2016stochastic`.
 
 ## Kesten processes
 
@@ -238,7 +238,7 @@ next period as it is this period.
 
 Since $y$ was chosen arbitrarily, the distribution is unchanged.
 
-### Conditions for Stationarity
+### Conditions for stationarity
 
 The Kesten process $X_{t+1} = a_{t+1} X_t + \eta_{t+1}$ does not always
 have a stationary distribution.
@@ -267,7 +267,7 @@ As one application of this result, we see that the wealth process
 {eq}`wealth_dynam` will have a unique stationary distribution whenever
 labor income has finite mean and $\mathbb E \ln R_t  + \ln s < 0$.
 
-## Heavy Tails
+## Heavy tails
 
 Under certain conditions, the stationary distribution of a Kesten process has
 a Pareto tail.
@@ -276,7 +276,7 @@ a Pareto tail.
 
 This fact is significant for economics because of the prevalence of Pareto-tailed distributions.
 
-### The Kesten--Goldie Theorem
+### The Kesten--Goldie theorem
 
 To state the conditions under which the stationary distribution of a Kesten process has a Pareto tail, we first recall that a random variable is called **nonarithmetic** if its distribution is not concentrated on $\{\dots, -2t, -t, 0, t, 2t, \ldots \}$ for any $t \geq 0$.
 
@@ -358,13 +358,13 @@ ax.set(xlabel="time", ylabel="$X_t$")
 plt.show()
 ```
 
-## Application: Firm Dynamics
+## Application: firm dynamics
 
 As noted in our {doc}`intro:heavy_tails`, for common measures of firm size such as revenue or employment, the US firm size distribution exhibits a Pareto tail (see, e.g., {cite}`axtell2001zipf`, {cite}`gabaix2016power`).
 
 Let us try to explain this rather striking fact using the Kesten--Goldie Theorem.
 
-### Gibrat's Law
+### Gibrat's law
 
 It was postulated many years ago by Robert Gibrat {cite}`gibrat1931inegalites` that firm size evolves according to a simple rule whereby size next period is proportional to current size.
 
@@ -411,7 +411,7 @@ In the exercises you are asked to show that {eq}`firm_dynam` is more
 consistent with the empirical findings presented above than Gibrat's law in
 {eq}`firm_dynam_gb`.
 
-### Heavy Tails
+### Heavy tails
 
 So what has this to do with Pareto tails?