Modified Bessel function asymptotics via probability
type: Item
instance of: scholarly article
title: Modified Bessel function asymptotics via probability
zbMATH Open document ID: 0625.60020
DOI: 10.1016/0167-7152(87)90004-6
author: K. B. Athreya
published in: Statistics \& Probability Letters
publication date: 1987-01-01T00:00:00Z
review text: Let the modified Bessel function be defined by \[ I_{\rho}(x)=\sum^{\infty}_{k=0}\frac{1}{k!}\frac{1}{\Gamma (k+\rho +1)}(x/2)^{2k+\rho}\quad for\quad \rho \geq -1\quad and\quad -\infty <x<\infty. \] The author considers the asymptotic behaviour of \(I_{\rho}(x)\) as x and \(\rho\) both go to infinity. For instance, he shows that for a(x)/\(\sqrt{x}\to \alpha\), \(0\leq \alpha <\infty\), \[ I_{a(x)}(x)\sqrt{x}e^{-x}\quad \to \quad \sqrt{2\pi}^{-1} \exp (- \alpha^ 2/2),\quad x\to \infty. \] For the proofs the author uses local and global versions of the central limit theorem.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 4015800
zbMATH Keywords: modified Bessel function
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: central limit theorem
MaRDI profile type: Publication
full work available at URL: 0167-7152(87)90004-6
OpenAlex ID: W1981070497
cites work: Q5558293
cites work: Q5512461
cites work: Ergodic Properties of Continuous-Time Markov Processes and Their Discrete Skeletons
description: scientific article; zbMATH DE number 4015800