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lecture 12 define g(h)

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Josia Pietsch 2023-07-12 17:57:05 +02:00
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inputs/lecture_12.tex
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@ -38,7 +38,8 @@ First, we need to prove some properties of characteristic functions.
&=& |\bE[e^{\i t X} (e^{\i h X} - 1)]|\\ &=& |\bE[e^{\i t X} (e^{\i h X} - 1)]|\\
&\overset{\text{Jensen}}{\le}& &\overset{\text{Jensen}}{\le}&
\bE[|e^{\i t X}| \cdot |e^{\i h X} -1|]\\ \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*} \end{IEEEeqnarray*}
Hence $\sup_{t \in \R} |\phi_X(t + h) - \phi_X(t) | \le g(h)$. Hence $\sup_{t \in \R} |\phi_X(t + h) - \phi_X(t) | \le g(h)$.