<s>[INST] How to take this exponentials Given an expansion of a cumulant function as follows:
$$
\kappa(t) = \frac{t^2}{2} + \frac{\rho_3 t^3}{6\sqrt{n}} + \frac{\rho_4t^4}{24n} +O\left(\frac{1}{n\sqrt{n}}\right), (*)
$$
where $\rho_r=\frac{\kappa_r}{\sigma^r}$ is the standardized cumulant. To obtain the corresponding moment generating function, one only needs to take the exponential of the above formula.
$$
m(t) = e^{\kappa(t)} = \exp\left\{{\frac{t^2}{2} + \frac{\rho_3 t^3}{6\sqrt{n}} + \frac{\rho_4t^4}{24n} +O\left(\frac{1}{n\sqrt{n}}\right)}\right\} = \exp\left\{ \frac{t^2}{2}\left[ 1+\frac{\rho_3t}{3\sqrt{n}}+\frac{\rho_4t^2}{12n} + O\left(\frac{1}{n\sqrt{n}}\right) \right] \right\}.
$$
However, I read the following result:
$$
m(t) = \exp\left(\frac{t^2}{2}\right) \left\{ 1+\frac{\rho_3 t^3}{6\sqrt{n}}+\frac{\rho_4t^4}{24n}+\frac{\rho_3^2t^6}{72n}+O\left(\frac{1}{n\sqrt{n}}\right) \right\}. (**)
$$
I do not see how to get $(**)$ from $(*)$. What trick is used here, please? Thank you!
 [/INST] You wrote $$m(t)  = \exp\left\{ \frac{t^2}{2}\left[ 1+\frac{\rho_3t}{3\sqrt{n}}+\frac{\rho_4t^2}{12n} + O\left(\frac{1}{n\sqrt{n}}\right) \right] \right\}$$ Let us expand it first; so $$m(t)=\exp \Big(\frac{t^2}{2}+\frac{\rho_3t^3}{6\sqrt n}+\frac{\rho_4t^4}{24 n}+\cdots \Big)=\exp \Big(\frac{t^2}{2}\Big)\times\exp \Big(\frac{\rho_3t^3}{6\sqrt n}+\frac{\rho_4t^4}{24 n}+\cdots \Big)$$ Now us Taylor series for the second part $$e^y=1+y+\frac{y^2}{2}+O\left(y^3\right)$$ and make $$y=\frac{\rho_3t^3}{6\sqrt n}+\frac{\rho_4t^4}{24 n}$$
I am sure that you can take from here.
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