Relationship between tail behavior of a density function and the radius of convergence of its Taylor expansion

Question

Given a continuous probability density function $f(x)$, whose Taylor expansion is $f(x) = \sum\limits_{n=0}^\infty a_n x^n$ with radius of convergence $r$. Can we say something about the relationship between $r$ and the tail behavior of $f$? For example, for the normal density, $r=\infty$ and the it has a somewhat "light" tail. However, for the standard Cauchy density, $r = 1$ and it has a heavy tail. Another example is the logistic distribution, which has $r = \pi$ and a tail heavier than normal but lighter than Cauchy. I guess in general, a density with heavier tail should have smaller $r$. Can we find the explicit relationship?

Answer 1

The only information in the tail of the expansion of f is the coefficients $a_n$ for $n \gg 0$.

The relationship is $$\frac{1}{r} = \limsup_{n \rightarrow \infty} \sqrt[n]{a_n}.$$

For many standard examples the limit exists so that $$\limsup_{n \rightarrow \infty} \sqrt[n]{a_n} = \lim_{n \rightarrow \infty} \sqrt[n]{a_n} = 1/r$$.

In light of whuber's comment on the OP, are you asking about the tails of $f$ as a Taylor series or the tails of the distribution $F(t) = \int_{-\infty}^t f dx$?