Geodesic
Abelian theorem for the regularly varying measure and its density in orthant
Teoriâ veroâtnostej i ee primeneniâ, Tome 64 (2019) no. 3, pp. 481-501

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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.
Keywords: weak convergence of sequence of measures, Abelian theorem for a measure and its density, regularly varying functions and measures at infinity in an orthant, integral representation theorem, multiple power series distributions. 
A. L. Yakymiv. Abelian theorem for the regularly varying measure and its density in orthant. Teoriâ veroâtnostej i ee primeneniâ, Tome 64 (2019) no. 3, pp. 481-501. http://geodesic.mathdoc.fr/item/TVP_2019_64_3_a3/
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