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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.

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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

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