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.

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