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lecture 10 thm 5

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Josia Pietsch 2023-07-20 12:58:53 +02:00
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@ -154,7 +154,8 @@ However, Fourier analysis is not only useful for continuous probability density
Let $\phi$ be the characteristic function of $\bP \in M_1(\lambda)$. Let $\phi$ be the characteristic function of $\bP \in M_1(\lambda)$.
Then Then
\begin{enumerate}[(a)] \begin{enumerate}[(a)]
\item $\phi(0) = 1$, $|\phi(t)| \le 1$ and $\phi(\cdot )$ is continuous. \item $\phi(0) = 1$, $|\phi(t)| \le 1$, $\phi(-t) = \overline{\phi(t)}$
and $\phi(\cdot )$ is continuous.
\item $\phi$ is a \vocab{positive definite function}, \item $\phi$ is a \vocab{positive definite function},
i.e.~ i.e.~
\[\forall t_1,\ldots, t_n \in \R, (c_1,\ldots,c_n) \in \C^n ~ \sum_{j,k = 1}^n c_j \overline{c_k} \phi(t_j - t_k) \ge 0 \[\forall t_1,\ldots, t_n \in \R, (c_1,\ldots,c_n) \in \C^n ~ \sum_{j,k = 1}^n c_j \overline{c_k} \phi(t_j - t_k) \ge 0