= Scale-free smoothed density variance
{title2=$\sigma_R\propto R^{-(n+3)/2}$}
For an unsmoothed density spectrum $P(k)=Ak^n$ and a scale-dependent smoothing window $\widetilde W(kR)$, substituting $q=kR$ in the variance integral gives $\sigma_R^2\propto R^{-(n+3)}$, provided $\int_0^\infty q^{n+2}|\widetilde W(q)|^2dq$ converges. For a window tending to one at zero, infrared convergence requires $n>-3$; the ultraviolet condition depends on the window. A real-space spherical top-hat also requires $n<1$.
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