fixed typos

inputs/lecture_12.tex

@@ -29,14 +29,14 @@ First, we need to prove some properties of characteristic functions.
 \begin{refproof}{charfprops}
     \begin{enumerate}[(i)]
         \item $\phi_X(0) = \bE[e^{\i 0 X}] = \bE[1] = 1$.
-            For $t \in \R$, we have $|\phi_X(t)| = |\bE[e^{\i t X}]| \overset{\text{Jensen}}{\le} \bE[|e^{\i t X}|] = 1$.
+            For $t \in \R$, we have $|\phi_X(t)| = |\bE[e^{\i t X}]| \overset{\yaref{jensen}}{\le} \bE[|e^{\i t X}|] = 1$.
 
         \item Let $t, h \in \R$.
             Then
             \begin{IEEEeqnarray*}{rCl}
                 |\phi_X(t+h) - \phi_X(t)| &=& |\bE[e^{\i (t+h) X} - e^{\i t X}]|\\
                                           &=& |\bE[e^{\i t X} (e^{\i h X} - 1)]|\\
-                                          &\overset{\text{Jensen}}{\le}&
+                                          &\overset{\yaref{jensen}}{\le}&
                                           \bE[|e^{\i t X}| \cdot  |e^{\i h X} -1|]\\
                                           &=& \bE[|e^{\i h X} - 1|]
                                           \text{\reflectbox{$\coloneqq$}} g(h)\\
@@ -73,7 +73,7 @@ First, we need to prove some properties of characteristic functions.
                 \begin{IEEEeqnarray*}{rCl}
                     |e^{\i y} - 1| &=& |\int_0^y \cos(s) \dif s + \i \int_0^y \sin(s) \dif s|\\
                                    &=& |\int_0^y e^{\i s} \dif s|\\
-                                   &\overset{\text{Jensen}}{\le}& \int_0^y |e^{\i s}| ds = y.
+                                   &\overset{\yaref{jensen}}{\le}& \int_0^y |e^{\i s}| ds = y.
                 \end{IEEEeqnarray*}
                 For $y < 0$, we have $|e^{\i y} - 1| = |e^{-\i y} - 1|$
                 and we can apply the above to  $-y$.

inputs/lecture_19.tex

@@ -61,7 +61,7 @@ However, some subsets can be easily described, e.g.
 
     Since $\sup_n \bE[|X_n|^{1+\delta}] < \infty$,
     we have that $\sup_n \bE[|X_n|] < \infty$ by \yaref{jensen}.
-    Hence for $K$ large enough relevant term is less than $\epsilon$.
+    Hence for $K$ large enough the relevant term is less than $\epsilon$.
 \end{proof}
 
 \begin{fact}\label{lec19f2}

inputs/lecture_20.tex

@@ -60,7 +60,7 @@
     \begin{IEEEeqnarray*}{rCl}
          \|X_n - X_n'\|_{L^p}^p
          &=& \bE[\bE[X - X' | \cF_n]^{p}]\\
-         &\overset{\text{Jensen}}{\le}& \bE[\bE[(X - X')^p | \cF_n]]\\
+         &\overset{\yaref{cjensen}}{\le}& \bE[\bE[(X - X')^p | \cF_n]]\\
          &=& \|X - X'\|_{L^p}^p\\
          &<& \epsilon.
     \end{IEEEeqnarray*}