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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.

Henry Lee
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lbagua
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  • Are there any conditions on $a$ and $b$? Note that the limit in question is of the form $\lim_{n\to+\infty} c^n$ with $$c= \sum\limits_{k = 1}^\infty {\left( {\frac{a}{b}} \right)^k I_k (ab)} . $$ – Gary May 27 '21 at 07:00
  • I would add $a>0$ and $b>0$ but I do not have any other restrictions. Is there a known result for "your" $c$? – lbagua May 27 '21 at 17:47

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