updated version of doc

notebooks/tex_source/sections/deriving_theorem_two.tex

@@ -1,4 +1,4 @@
-Use the approximation $\ln(1+x) \approx 1 + x$ for $|x| \ll 1$ and $F(z) \approx
+Use the approximation $\ln(1+x) \approx x$ for $|x| \ll 1$ and $F(z) \approx
 1$ for large enough $z$ to derive.
 
 

notebooks/tex_source/sections/peaks_over_thresholds.tex

@@ -24,9 +24,10 @@
 \textbf{Q:} What should $u$ and the data set fulfill in order for the above
 approximation to be accurate?
 
-\textbf{A:} It should be small enough such that many data points are larger than it.
-Then the approximation in $P(X>u) \approx \frac{N_u}{N}$ holds (the estimator is
-not too biased). 
+\textbf{A:} 
+% It should be small enough such that many data points are larger than it.
+% Then the approximation in $P(X>u) \approx \frac{N_u}{N}$ holds (the estimator is
+% not too biased). 
 
 \hrulefill\\*
 
@@ -67,11 +68,12 @@
 \textbf{Q:} Intuitively, what does $u$ need to fulfill for both approximations to
 hold?
 
-\textbf{A:} $u$ should be small enough such that the approximation $P(X>u) \approx
-\frac{N_u}{N}$ holds and sufficiently large such that the generalized pareto
-distribution is a good estimate of the tail of the distribution for values
-larger than $u$. Intuitively, it should be at the *beginning of the tail*, where
-for values larger than $u$ only the tail behavior plays a role - i.e. no more
-local extrema or other specifics of the underlying distribution of the data.
+\textbf{A:} 
+% $u$ should be small enough such that the approximation $P(X>u) \approx
+% \frac{N_u}{N}$ holds and sufficiently large such that the generalized pareto
+% distribution is a good estimate of the tail of the distribution for values
+% larger than $u$. Intuitively, it should be at the *beginning of the tail*, where
+% for values larger than $u$ only the tail behavior plays a role - i.e. no more
+% local extrema or other specifics of the underlying distribution of the data.
 
 \hrulefill\\*

notebooks/tex_source/sections/taking_a_look_back.tex

@@ -3,7 +3,15 @@
 us about the quality of the fit?
 
 
-Well, the truth is, not too many. First notice the following exact equality:
+Well, the truth is, not too many.  
+
+Remember that the block-maximum is defined as
+
+\begin{equation}
+  M_n \coloneqq \max\{ X_1, X_2, \dots, X_n \}
+\end{equation}
+
+First notice the following exact equality:
 
 
 \begin{equation}