Regular variation (${\mathrm{RV}}_{\rho }$)

A positive measurable function $q$ has regular variation at infinity with index $\rho$ if $q(tu)/q(t)\to u^{\rho }$ for every $u>0$. Equivalently, $q(t)=t^{\rho }\ell (t)$ with $\ell$ a slowly varying function.

A slowly varying function has regular variation with index zero: $\ell (tu)/\ell (t)\to 1$ for every $u>0$. A power of $\log t$ is an example.