&=& \ln \mu(\tilde{x}) = n \kappa(\tilde\xi)\vline_{\tilde\xi=\tilde{x}}\text{,}\end{aligned}$$ where $\tilde\xi$ and $\mu(\tilde\xi)$ are, respectively, the reciprocal dual variable[^1] of $\xi$ and its moment-generating function, and likewise for $x$, while $\kappa(\cdot)$ stands for the cumulant-generating function. From now on we consistently denote duals of random variables by the tilde.

By expanding the cumulant-generating function of $\xi$ in Taylor series at $\tilde\xi=0$, $${\label{eq:Taylor}}
    \kappa(\tilde\xi) = \sum_{i=0}^{n} \kappa_i(\xi) \frac{\tilde\xi^i}{i!}
        = \kappa_2(\xi) \tilde\xi^2 / 2 + \mathcal{O}(\tilde\xi^{4}),$$ we use the fact that, for a symmetric random variable, all cumulants of odd orders $\kappa_1(\xi)$, $\kappa_3(\xi)$, and so on, must vanish, whereas the normalization of a probability density function always requires $\kappa_0(\cdot)=0$. The Taylor series [Eq. (\[eq:Taylor\])]{}, truncated at the second term is a cumulant-generating function of a Gaussian probability distribution $$\kappa_G(\tilde\xi) = \kappa_2(\xi) \tilde\xi^2/2,$$ which can be used to approximate $\kappa(\tilde\xi) \approx \kappa_G(\tilde\xi)$ up to the third order statistics; cf. [Eq. (\[eq:Taylor\])]{}. Hence, by virtue of [Eq. (\[eq:kappa\])]{} we further obtain, $${\label{eq:gaussn}}\kappa(\tilde{x}) \approx n \kappa_2(\xi) \tilde{x}^2/2.$$ Not all orders of truncated power series are valid cumulant-generating functions. The second order approximation, however, always corresponds to a Gaussian probability distribution.

Notice that we have not invoked the Central Limit Theorem so far. Now we do this, in order 