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Let $$ f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }}, x \in \mathbb{R}, 0<\alpha<2, $$ where $\phi_1(\alpha)=\frac{\alpha}{2} \left\{{\{\Gamma(3/\alpha)\}^{1/...

Consider a continuous random variable $X$ with the compact support $[0,1]$. For given $N\in\mathbb{N}$, we define the weighted sum as $$ S_N=\sum_{i=1}^N a_iU_i, $$ where $U_i$ are i.i.d. random ...

I am interested in the scaling of $$F(x_1,x_4)=\int_{\mathbb R^2} e^{-\vert x_1 -x_2 \vert -\varepsilon \vert x_2 -x_3 \vert- \vert x_3 -x_4 \vert } \ dx_2 dx_3 $$ In particular, I suspect that $$F(...

Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...