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Let $f, g \in L^1\cap L^\infty(\mathbb{R}^d)$ be probability distributions on $\mathbb{R}^d$, and suppose at large $|x|$, $f$ decays like $|x|^{-\alpha}$ while $g$ decays like $|x|^{-\beta}$, with $\alpha, \beta > 0$.

Can anything be said about the decay of the convolution, \begin{equation} h(x) := f*g(x) = \int_{\mathbb{R}^d}f(x-y)g(y)dy\quad? \end{equation}

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You can say that $h(x) = O(|x|^{-\gamma})$ where $\gamma=\min(\alpha,\beta)$. Indeed, if $|x|> 2R$, then for every $y$ either $|y|> R$ or $|x-y|>R$. Hence, $$|h(x)|\le \int_{|y|>R}|f(x-y)g(y)|\,dy + \int_{|x-y|>R}|f(x-y)g(y)|\,dy \le C_1R^{-\alpha}+ C_2R^{-\beta} $$

On the other hand, you can't say more than the above. If one of two functions is compactly supported, then the convolution follows the asymptotic of the other one.

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