I've recently come across the following limit:
$$\lim_{n \to \infty} \left[\sum_{k=1}^{\infty} \left(\frac{a}{b}\right)^k \operatorname{I}_k(ab)\right]^n$$
where $\operatorname{I}_k$ are modified Bessel functions.
Is there a known solution for this limit?
If you are wondering about the origin of my question, it has to do with the asymptotic distribution of the minima (or maxima) of the Rice distribution, where the Marcum Q- function appears in the CDF of the Rice distribution.