diff --git a/inputs/lecture_12.tex b/inputs/lecture_12.tex index 63d95b8..1fa9a32 100644 --- 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 $\sup_{t \in \R} |\phi_X(t + h) - \phi_X(t) | \le g(h)$.