I was messing around and found an approximation for the ratio of two modified Bessel functions:
$${I_N(x_0)\over I_0(x_0)}=e^{-N^2\over2x_0}, x_0>0$$ which seems to get better with larger $x_0$.
Here is the graph showing two sides of equation with $x_0$=60.
I cant find it anywhere in the literature. Could anyone please help to prove it or give some reference ?
Some heuristics:
Take the asymptotic expansions for the modified Bessel function (large $x$):
$$ \frac{I_N(x)}{I_0(x)}\sim \frac{1-\frac{4N^2-1}{8x}+\mathcal{O}(\frac{N^4}{x^2})}{1-\frac{1}{8x}+\mathcal{O}(\frac{1}{x^2})}$$
Now if $x>>N^2>>1 \quad (1)$ $$ \frac{I_N(x)}{I_0(x)}\sim 1-\frac{4N^2}{8x}+\mathcal{O}(\frac{N^4}{x^2})= 1-\frac{N^2}{2x}+\mathcal{O}(\frac{N^4}{x^2}) $$
Now compare with the first order expansion of $\exp$: $\exp(-N^2/2x)\sim1-\frac{N^2}{2x}+\mathcal{O}(\frac{N^4}{x^2})$
Therefore $$ \frac{I_N(x)}{I_0(x)}\approx \exp(-N^2/2x) $$
under assumoption (1) and the error is $\mathcal{O}(\frac{N^4}{x^2})$