I am studying a chemical system in which a statistic has the following form:

$$ E \ = \ \frac{I_0(2\sqrt{\lambda})^2}{I_1(2\sqrt{\lambda})^2} - \frac{1}{2\sqrt{\lambda}} \frac{I_0(2\sqrt{\lambda})}{I_1(2\sqrt{\lambda})} - 1.$$

Where $I_0(x)$ is the modified Bessel Function of the First Kind and $\lambda \geq 0$. I have shown that $0 \leq E$ and I am trying to find an asymptotic upper bound for $E$ (expression for upper bound that depends on $\lambda$) but don't know how to do it because I don't have the math background for it! Any ideas on how this can be approached? Any help is appreciated.

Edit: Looks like $E \leq \frac{1}{2\lambda}$ from simulations. Might be wrong! Still, don't know if that is provable.


First, let $2 \sqrt \lambda=x$ to make $$E \ = \ \frac{I_0(x)^2}{I_1(x)^2} - \frac{1}{x} \frac{I_0(x)}{I_1(x)} - 1$$ and use the expansions $$I_0(x)=1+\frac{x^2}{4}+\frac{x^4}{64}+\frac{x^6}{2304}+O\left(x^8\right)$$ $$I_1(x)=\frac{x}{2}+\frac{x^3}{16}+\frac{x^5}{384}+\frac{x^7}{18432}+O\left(x^8\right)$$ which give $$E=\frac{2}{x^2}-\frac{1}{4}+\frac{x^2}{32}-\frac{5 x^4}{1536}+\frac{7 x^6}{23040}+O\left(x^8\right)$$ Back to $\lambda$ $$E=\frac{1}{2 \lambda }-\frac{1}{4}+\frac{\lambda }{8}-\frac{5 \lambda ^2}{96}+O\left(\lambda ^3\right)$$

  • $\begingroup$ I am a bit confused by this answer because when I use Mathematica, I get $E \leq \frac{1}{2\lambda}$ for $\lambda > \frac{1}{2}$. But this expression for $E$ would grow fast with large $\lambda$! Also, I know $\lim_{\lambda \to \infty} E = 0$ but this won't yield that either. $\endgroup$ – Analytic Potato Jul 16 at 12:07
  • $\begingroup$ @AnalyticPotato. As far as I understood, you are looking for tha asymptotics when $\lambda \to 0$ (for large values of $\lambda$ the answer was given in your othe question. $\endgroup$ – Claude Leibovici Jul 17 at 3:13

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