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Gil-Pelaez (1951) proves the Fourier inversion formula \begin{align*} F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\ &= \frac{1}{2} - \frac{1}{\pi} \int_0^\infty \Im\left(\frac{e^{-itx}\phi(t)}{t}\right)dt, \end{align*} where $F$ is the cdf of a random variable and $\phi$ the characteristic function, $\phi(t)=\int_\mathbb{R}e^{itx}dF(x)$.

Is it possible to relate the integral (or its principal value) $$\int_0^\infty \Im\left(\frac{e^{-itx}\phi(t)}{t^n}\right)dt$$ for $n\in\mathbb{N}$ to the cdf $F$?

Gil-Pelaez' original proof wouldn't work because it would then involve integrals like $\int_0^\infty \frac{\sin(x)}{x^n}dx$ and $\int_0^\infty \frac{\cos(x)}{x^n}dx$ which don't necessarily exist but perhaps somebody has seen a different proof that is adaptable for powers in the denominator?

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    $\begingroup$ As is pretty clear, the indicated integrals do not exist even in a principal value sense. However, they do exist in a "finite-part" sense, following Hadamard and Riesz, extending the notion of "principal part". But they are not quite the same sort of limits of genuine integrals as the principal part is. Is this of interest? $\endgroup$ – paul garrett Oct 12 at 21:57
  • $\begingroup$ @paulgarrett I actually haven't heard about this notion ("finite-part") before. So, I'll have a look at it in detail (independent of my particular problem). Thanks for the suggestion! $\endgroup$ – Alex Oct 12 at 22:14
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    $\begingroup$ Good. Other keywords involve "meromorphic continuation" ... of families of distributions such as (integrate-against) $1/|x|^s$ and such. $\endgroup$ – paul garrett Oct 12 at 22:48
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Whenever the distribution with characteristic function $\phi$ has a finite mean $a$, we have $\phi(t)=1+iat+o(t)$ (as $t\downarrow0$). So, for any real $x\ne a$, the integrand in your integral is $\sim (a-x)t^{1-n}$ and hence for any $n\ge2$ the integral diverges to $\pm\infty$ in a right neighborhood of $0$. So, your integral does not exist for any $n\ge2$ and any real $x\ne a$, even as a principal value.

However, in this paper or its arXiv version, one can find many formulas of the same flavor as the Gil--Pelaez one, with $t^p$ for however large $p$ in the denominator of the integrand.

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  • $\begingroup$ Thanks a lot for the answer + links to your paper! I'll go through it thoroughly :) $\endgroup$ – Alex Oct 12 at 22:12
  • $\begingroup$ Just a quick thought: suppose we looked at something like $$\int_0^\infty \Im\left(\frac{e^{-itx}\phi(t)}{(t+1)^n}\right)dt,$$ i.e. we don't introduce higher order poles to the integrand. Then, the integral should still exist? Perhaps I find something about this in your paper or perhaps you've seen something like this before? $\endgroup$ – Alex Oct 12 at 22:26
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    $\begingroup$ @Alex : The integral in your comment will of course converge for $n>1$, but I don't think it will signify anything interesting. As stated in the linked paper, all formulas there are based on an idea of homogeneity, and $+1$ in $t+1$ destroys the homogeneity. $\endgroup$ – Iosif Pinelis Oct 12 at 23:54

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