fixed typos
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@ -29,14 +29,14 @@ First, we need to prove some properties of characteristic functions.
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\begin{refproof}{charfprops}
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\begin{enumerate}[(i)]
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\item $\phi_X(0) = \bE[e^{\i 0 X}] = \bE[1] = 1$.
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For $t \in \R$, we have $|\phi_X(t)| = |\bE[e^{\i t X}]| \overset{\text{Jensen}}{\le} \bE[|e^{\i t X}|] = 1$.
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For $t \in \R$, we have $|\phi_X(t)| = |\bE[e^{\i t X}]| \overset{\yaref{jensen}}{\le} \bE[|e^{\i t X}|] = 1$.
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\item Let $t, h \in \R$.
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Then
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\begin{IEEEeqnarray*}{rCl}
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|\phi_X(t+h) - \phi_X(t)| &=& |\bE[e^{\i (t+h) X} - e^{\i t X}]|\\
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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{\yaref{jensen}}{\le}&
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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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\text{\reflectbox{$\coloneqq$}} g(h)\\
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@ -73,7 +73,7 @@ First, we need to prove some properties of characteristic functions.
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\begin{IEEEeqnarray*}{rCl}
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|e^{\i y} - 1| &=& |\int_0^y \cos(s) \dif s + \i \int_0^y \sin(s) \dif s|\\
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&=& |\int_0^y e^{\i s} \dif s|\\
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&\overset{\text{Jensen}}{\le}& \int_0^y |e^{\i s}| ds = y.
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&\overset{\yaref{jensen}}{\le}& \int_0^y |e^{\i s}| ds = y.
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\end{IEEEeqnarray*}
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For $y < 0$, we have $|e^{\i y} - 1| = |e^{-\i y} - 1|$
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and we can apply the above to $-y$.
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@ -61,7 +61,7 @@ However, some subsets can be easily described, e.g.
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Since $\sup_n \bE[|X_n|^{1+\delta}] < \infty$,
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we have that $\sup_n \bE[|X_n|] < \infty$ by \yaref{jensen}.
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Hence for $K$ large enough relevant term is less than $\epsilon$.
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Hence for $K$ large enough the relevant term is less than $\epsilon$.
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\end{proof}
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\begin{fact}\label{lec19f2}
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@ -60,7 +60,7 @@
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\begin{IEEEeqnarray*}{rCl}
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\|X_n - X_n'\|_{L^p}^p
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&=& \bE[\bE[X - X' | \cF_n]^{p}]\\
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&\overset{\text{Jensen}}{\le}& \bE[\bE[(X - X')^p | \cF_n]]\\
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&\overset{\yaref{cjensen}}{\le}& \bE[\bE[(X - X')^p | \cF_n]]\\
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&=& \|X - X'\|_{L^p}^p\\
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&<& \epsilon.
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\end{IEEEeqnarray*}
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