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In which weak sense does the series representation of the log-characteristic function of a probability distribution converge? More specific, if $\mu$ is a probability distribution on the real line and $\phi\colon \mathbb R\to\mathbb R$ is a test function, under which conditions on $\phi$ and $\mu$ do we have the convergence: $$\int_{\mathbb R}\log\left(\int_{\mathbb R} e^{i t x} \,d \mu(x)\right) \phi(t)\,d t = \sum_{n=0}^\infty \frac{\kappa_n}{n!} \int_{\mathbb R} (it)^n \phi(t)\,dt, $$

Here $\kappa_n$ are the Taylor coefficients of the log-characteristic function, i.e., the cumulants of $\mu$.

I guess for any such result there is a trade-off between decay and regularity of $\phi$ and the tail-behaviour of $\mu$. I think the effect I am asking for might be quite delicate: Even for bounded distribution $\mu$ we can have that $\kappa_n\sim n!$ and the integral on the rhs. can certainly also grow with $n$, so if such a convergence holds, it might rely on some subtle cancellation.

Out of the two possible regimes I am rather interested in the regime, where one assumes a lot of regularity on the distribution (say, sub-Gaussian decay or compact support) and less regularity on $\phi$ (maybe $C_0^\infty$ and not analytic).


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