# Radius of convergence of cumulant generating function

Recall that for a random variable $$X$$ with a moment generating function $$M_X(t)$$ the cumulant generating function is defined as \begin{align} K_X(t)=\log M_X(t) \end{align}

The Taylor expansion of $$K_X(t)$$ is given \begin{align} K_X(t)= k_1 t+ k_2 \frac{t^2}{2}+ k_3 \frac{t^3}{3!}+.. \end{align} where $$k_i$$ are the cumulants.

My question: Can we determine the radius of convergence of $$K_X(t)$$ if we know the radius of convergence of $$M_X(t)$$?

For example, in these slides, it is claimed both moments generating functions and cumulant generating functions have the same radius of convergence.

• If $M_x(t)$ converges for some $t$, then $K_X(t)$ is simply computed by the given formula $K_X(t) = \log M_X(t)$. And vice versa, $M_X(t)$ can be computed as $\exp K_X(t)$ if $K_X(t)$ converges. So, the slides are correct. Jun 24, 2021 at 2:21
• @MaxAlekseyev See the answer below.
– Boby
Jun 24, 2021 at 12:44
• Well, the above argument is valid with addition that we want $M(t)>0$ (to be in the $\log$ domain) for convergence of $K(t)$. If this inequality always holds, then the two radii are the same; otherwise the smallest by absolute vale zero of $M(t)$ comes into play. Jun 24, 2021 at 13:26
• @MaxAlekseyev But $M(t)>0$ for all real $t$, right? Can you explain this a bit more.
– Boby
Jun 24, 2021 at 13:58
• For all real $t$ within the radius of convergence of $M$. Jun 24, 2021 at 14:28

Of course, this is not correct. As a simplest example, let $$X$$ be a random variable which takes only values $$\{0,\ldots, n\}$$, then the moment generating function is a polynomial of $$e^t$$, of degree $$n$$, therefore its radius of convergence is infinite. Any polinomial with positive coefficients which add to $$1$$ can occur. But $$\log M(t)$$ has finite radius of convergence since a polynomial $$P$$ of degree $$n\geq 1$$ has some zeros in the complex plane. So $$P(e^t)$$ also has zeros, unless $$P$$ is a monomial.
In general, the radius of convergence for $$K(t)$$ is the distance from the origin to the closest zero of $$M(t)$$.