According to Wikipedia, at https://en.wikipedia.org/wiki/Bessel_function#Asymptotic_forms, for large real $x$,
\begin{aligned}I_{\alpha }(x)={\frac {1}{{\sqrt {2\pi x}}{\sqrt[{4}]{1+{\frac {\alpha^{2}}{x^{2}}}}}}}\exp \left(-\alpha \operatorname {arsinh} \left({\frac {\alpha}{x}}\right)+x{\sqrt {1+{\frac {\alpha^{2}}{x^{2}}}}}\right)\left(1+{\mathcal {O}}\left({\frac {1}{x{\sqrt {1+{\frac {\alpha ^{2}}{x^{2}}}}}}}\right)\right).\end{aligned}
Here, \begin{equation} \operatorname {arsinh} (y) =\ln \left(y+{\sqrt {y^{2}+1}}\right) \end{equation}
The reference cited for this result on the modified Bessel function of the first kind above is Appendix B for the paper on https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-81/issue-4/The-Kosterlitz-Thouless-transition-in-two-dimensional-abelian-spin-systems/cmp/1103920388.full However, I can not figure out which part of Appendix B of this paper leads to the above result. Assuming the statement is correct, is there any other reference or derivation for this result?
My thanks for any suggestions.