For the characteristic function $\mathbf E e^{i t X}$ of a random variable $X$ with $n+1$ finite moments, there is the well known and easy to prove bound on the remainder of the Taylor series


$$\left\lvert\mathbf E e^{i t X}-\sum\_{k=0}^n \frac{(it)^k}{k!}\mathbf E X^k\right\rvert\le\min\left\{\frac{\lvert t\rvert^{n+1}\mathbf E \lvert X^{n+1}\rvert}{(n+1)!},\frac{2\lvert t\rvert^n\mathbf E\lvert X^n\rvert}{n!}\right\}.$$


Can something similar be said for the remainder of the cumulant generating function $\log\mathbf E e^{itX}$ with an error bound in terms of cumulants? I.e., I am hoping for a bound of the form


$$\left\lvert \log\mathbf E e^{itX}-\sum\_{k=0}^n\frac{(it)^k}{k!} \kappa\_k(X)\right\rvert\leq a(t) \kappa\_n(X) + b(t) \kappa\_{n+1}(X),$$ where $\kappa\_k(X)$ is the $k$-th cumulant of $X$. 


I am not even sure whether such an bound even exists -- partly because the cumulants can exhibit certain cancellation effects not present in the moments.



Taylor expansion of cumulant generating function

> 
> Can something similar be said for the remainder of the cumulant
>  generating function $\log\mathbf E e^{itX}$ with an error bound in
>  terms of cumulants?
> 
> 
> 


**Yes and no.**
Your question is whether the union bound for Fourier transform can be somehow generalized to log Fourier transform for the class of $L^{n+1}(\mathbb{R})$ (for simplicity I do not discuss $\mathbb{R}^n$ domain case).


**"Yes part".** 


The result you are asked for is basically a Tauberian-type theorem for log Fourier transform that controls the residue in terms of coefficients($\kappa\_n$ as in OP). For log Laplace transform $\log\mathbf E e^{tX}$, the result is readily available in Theorem 16.1 of [Korevaar].
We write the density $p(\xi)=dP(\xi)$ and the Laplace transform 
$\mathcal{L}dP(\xi)=\mathcal{L}dP(\frac{1}{x})=\int\_{0^{-}}^{\infty}e^{-\frac{v}{x}}dP(x)$ according to Part IV Thm 16.1, we have
$log\mathcal{L}dP(\xi)=log\mathcal{L}dP(\frac{1}{x})\sim(\alpha-1)\frac{x^{\alpha}L(x)}{x}$ as $x\rightarrow\infty$ where $L(x)$ is the Laplace transform. For harmonic $P$, there are some results similar contained near the end (possibly title "random functions" or something like that, do not have the book at hand now) of [Paley-Wiener]. [Safarov] provides some concrete applications under certain assumptions on $P$'s (say 1.20-1.21). In this regard, "complex Tauberian" or "Fourier Tauberian" are the correct keywords to search for.


**"No part".**


Although there are some results in spirit of Tauberian theorems as mentioned above, I do not know there is a result concerning a class as general as $L^{n+1}(\mathbb{R})$. 
As you already noticed there is such a potentiality of singularities that prevents you from bounding the residues, say Schwartz-class measures and their power polynomials. If you translate the bound on moment generating function $\mathbf E e^{i t X}$ onto $\log\mathbf E e^{itX}$ using the relation between moments and cumulants (see, Mobius (inverse) transform [wiki, Rota]), you will still have $\pm$ cancellation in these relations; 
However, there is still arising research concerning bounding the tail behavior of a given density using cumulant expansions [Valz et.al], so I guess this is not completely a dead end.


*This is also (approximately) a question puzzled me a year ago. This answer is my exploration, and I really hope someone more knowledgeable can provide a more complete answer. Hope this helps!*


**Reference**


[Korevaar]Korevaar, Jacob. Tauberian theory: A century of developments. Vol. 329. Springer Science & Business Media, 2013.


[Paley-Wiener]Paley, Raymond Edward Alan Christopher, Norbert Wiener, and Norbert Wiener. Fourier transforms in the complex domain. Vol. 19. New York: American Mathematical Society, 1934.


[Safarov]Safarov, Yu. "Fourier Tauberian theorems and applications." Journal of Functional Analysis 185.1 (2001): 111-128.


[wiki] <https://en.wikipedia.org/wiki/Cumulant>


[Rota]Rota, Gian-Carlo, and Jianhong Shen. "On the combinatorics of cumulants." Journal of Combinatorial Theory, Series A 91.1-2 (2000): 283-304.


[Valz et.al]Valz, Paul D., A. Ian McLeod, and Mary E. Thompson. "Cumulant generating function and tail probability approximations for Kendall's score with tied rankings." The Annals of Statistics (1995): 144-160.


[Bazant]Martin Bazant, Lecture 5: Asymptotics with Fat Tails. <https://ocw.mit.edu/courses/mathematics/18-366-random-walks-and-diffusion-fall-2006/lecture-notes/lec05.pdf>


