Modified Bessel function asymptotics via probability
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.
scientific article; zbMATH DE number 4015800