lecture 12 define g(h)

2023-07-12 17:57:05 +02:00 · 2023-07-12 17:57:05 +02:00 · c926a076e7
commit c926a076e7
parent e566dff883
1 changed files with 2 additions and 1 deletions
--- a/inputs/lecture_12.tex
+++ b/inputs/lecture_12.tex
@ -38,7 +38,8 @@ First, we need to prove some properties of characteristic functions.
                                          &=& |\bE[e^{\i t X} (e^{\i h X} - 1)]|\\
                                          &\overset{\text{Jensen}}{\le}&
                                          \bE[|e^{\i t X}| \cdot  |e^{\i h X} -1|]\\
-                                          &=& \bE[|e^{\i h X} - 1|]\\
+                                          &=& \bE[|e^{\i h X} - 1|]
+                                          \text{\reflectbox{$\coloneqq$}} g(h)\\
            \end{IEEEeqnarray*}

            Hence $\sup_{t \in \R} |\phi_X(t + h) - \phi_X(t) | \le  g(h)$.