Regular variation in 𝑅^{𝑘}

Abstract

Researchers investigating certain limit theorems in probability have discovered a multivariable analogue to Karamata’s theory of regularly varying functions. The method uses elements of real analysis and Lie groups to analyze the asymptotic behavior of functions and measures on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R Superscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {R}}^k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We present an account here which is independent of probabilistic considerations.