Abelian Theorem for the Regularly Varying Measure and Its Density in Orthant

Abstract

The paper is concerned with a $\sigma$-finite measure $U$ concentrated in the positive orthant $\mathbf{R}^n_+=[0,\infty)^n$ such that there exists the Laplace transform $\widetilde{U}(\lambda)$ for $\lambda\in\operatorname{int} \mathbf{R}^n_+$. Let functions $R(t)>0$ and $b(t)=(b_1(t),\dots,b_n(t))\in\operatorname{int} \mathbf{R}^n_+$ for $t\geq0$ be such that $R(t)\to\infty$, $b_i(t)\to\infty$ for any $i=1,\dots,n$. Under certain assumptions on these functions, the weak convergence of the measures $U(b(t)\,{\cdot}\,)/R(t)$ to $\Phi{(\,\cdot\,)}$ as $t\to\infty$ is shown to imply the convergence $\widetilde{U}(\lambda/b(t))\to\widetilde{\Phi}(\lambda)<\infty$ for any $\lambda\in\operatorname{int} \mathbf{R}^n_+$ ($t\to\infty$) (the multiplication and division of vectors are defined componentwise). A function $f\colon \mathbf{R}_+^n\to \mathbf{R}_+$ is said to be regularly varying at infinity in $\mathbf{R}_+^n$ along $b(t)$ if $f(b(t)x(t))/f(b(t))\to\varphi(x)\in(0,\infty)$ as $t\to\infty$ for all $x$, $x(t) \in \mathbf{R}_+^n\setminus\{0\}$ such that $ x(t)\to x$. Sufficient conditions are given for such functions to give $\widehat{f}(\lambda/b(t))\equiv\widetilde{U}(\lambda/b(t)) \to\widehat{\phi}(\lambda)\equiv\widetilde{\Phi}(\lambda)<\infty$ for any $\lambda\in\operatorname{int} \mathbf{R}^n_+$\enskip ($t\to\infty$) for $U(dx)=f(x)\,dx$, $\Phi(dx)=\varphi(x)\,dx$. The Abelian theorem obtained here is applied at the end of the paper to investigate the limit behavior of multiple power series distributions.