The pdf and cdf of the non-central chi squared RV (under the scenario I am studying) is given as follows: \begin{align} &f(x)=\frac{1}{v} \exp\left(\frac{-(a+x)}{v}\right)I_{0}\left(\frac{\sqrt{xa}}{v/2}\right) \\ &1-F(x)=\exp\left(\frac{-(a+x)}{v}\right)\sum_{m=0}^{\infty}\left(\sqrt{\frac{a}{x}}\right)^m I_{m}\left(\frac{\sqrt{xa}}{v/2}\right) \end{align} The approximations to the tails of $f(x)$ and $1-F(x)$ (as $x\rightarrow \infty$), are given in one of the papers as, \begin{align} &f(x) \sim \frac{1}{2\sqrt{v\pi}(xa)^{1/4}}\exp\left(-\frac{\left(\sqrt{x}-\sqrt{a} \right)^2}{v} \right), \\ &1-F(x) \sim \frac{\sqrt{v}}{2\sqrt{\pi}(xa)^{1/4}}\exp\left(-\frac{\left(\sqrt{x}-\sqrt{a} \right)^2}{v} \right). \end{align} I was easily able to derive the approximation for $f(x)$ using $I_{0}(x)\sim \frac{\exp(x)}{\sqrt{2\pi x}}$ as $x\rightarrow \infty$. I am stuck at how to get the approximation for the CDF tail? Apparently they are using some convergence result for $\sum_{m=0}^{\infty}\left(\sqrt{\frac{a}{x}}\right)^m I_{m}\left(\frac{\sqrt{xa}}{v/2}\right)$ as $x \rightarrow \infty$. But I cant seem to figure it out.

Any help would be greatly appreciated.


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