updated definition of convolution
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@ -86,17 +86,23 @@ characteristic functions of Fourier transforms.
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\subsection{Convolutions${}^\dagger$}
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\subsection{Convolutions${}^\dagger$}
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\begin{definition}+[Convolution]
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\begin{definition}+[Convolution]
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Let $\mu$ and $\nu$ be probability measures on $\R^d$
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Let $\mu$ and $\nu$ be probability measures on $\R^d$.
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with Lebesgue densities $f_\mu$ and $f_\nu$.
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Then the \vocab{convolution} of $\mu$ and $\nu$,
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Then the \vocab{convolution} of $\mu$ and $\nu$,
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$\mu \ast \nu$,
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$\mu \ast \nu$,
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is the probability measure on $\R^d$
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is the probability measure on $\R^d$
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with Lebesgue density
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defined by
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\[
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(\mu \ast \nu)(A) = \int_{\R^d} \int_{\R^d} \One_A(x + y) \mu(\dif x) \nu(\dif y).
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\]
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\end{definition}
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\begin{fact}
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If $\mu$ and $\nu$ have Lebesgue densities $f_\mu$ and $f_\nu$,
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then the convolution has Lebesgue density
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\[
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\[
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f_{\mu \ast \nu}(x) \coloneqq
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f_{\mu \ast \nu}(x) \coloneqq
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\int_{\R^d} f_\mu(x - t) f_\nu(t) \lambda^d(\dif t).
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\int_{\R^d} f_\mu(x - t) f_\nu(t) \lambda^d(\dif t).
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\]
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\]
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\end{definition}
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\end{fact}
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\begin{fact}+[Exercise 6.1]
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\begin{fact}+[Exercise 6.1]
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If $X_1,X_2,\ldots$ are independent with
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If $X_1,X_2,\ldots$ are independent with
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