lecture 12 define g(h)
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@ -38,7 +38,8 @@ First, we need to prove some properties of characteristic functions.
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&=& |\bE[e^{\i t X} (e^{\i h X} - 1)]|\\
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&=& |\bE[e^{\i t X} (e^{\i h X} - 1)]|\\
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&\overset{\text{Jensen}}{\le}&
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&\overset{\text{Jensen}}{\le}&
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\bE[|e^{\i t X}| \cdot |e^{\i h X} -1|]\\
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\bE[|e^{\i t X}| \cdot |e^{\i h X} -1|]\\
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&=& \bE[|e^{\i h X} - 1|]\\
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&=& \bE[|e^{\i h X} - 1|]
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\text{\reflectbox{$\coloneqq$}} g(h)\\
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\end{IEEEeqnarray*}
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\end{IEEEeqnarray*}
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Hence $\sup_{t \in \R} |\phi_X(t + h) - \phi_X(t) | \le g(h)$.
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Hence $\sup_{t \in \R} |\phi_X(t + h) - \phi_X(t) | \le g(h)$.
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