Context
I am looking at a random picture in which every pixel is randomly generated through complex dynamics and I am interested in the 1-point statistics of the pixels (basically their probability density, moments and cumulants). To estimate the probability density function I would then draw a normalised histogram of the pixels values and I would be happy with that.
What I want to do
For some theoretical reasons that are linked to large deviation statistics, and for a continuous real random variable $X$, the theoretical object that I would however like to measure/estimate is the cumulant generating function (CGF) $\phi(\lambda)$ defined through the Laplace transform of the probability density function (PDF) $\mathcal{P}(x)$, or equivalently the expectation value for a given $\lambda$ of $e^{\lambda X}$ : $$e^{\phi(\lambda)} = \int {\rm d}x \, e^{\lambda x} \mathcal{P}(x) = \langle e^{\lambda X}\rangle.$$ The latter point makes, through the use of some sort of ergodicity hypothesis, the cumulant generating function something that I can in principle estimate from my pixelated picture.
Questions
From this I have several questions:
Let me assume that all the moments and cumulants exist. They thus should define entirely the distribution of $X$. In principle, $\lambda$ could be any complex values, but then restricting myself to real values around the origin should then still be equivalent to measuring the probability density function because this would be equivalent to estimate all the cumulants which are the successive derivatives in zero of the cumulant generating function. Is this true ? I think this is true but I would like to have some confirmation.
This is my main question: When measuring $\langle e^{\lambda X}\rangle$, this is done for a finite number and range of $\lambda$. However it seems easier to me to think about the constraints that would apply to the measurement of the PDF rather than the CGF. For example the width of the bins of the histogram to estimate the PDF can be estimated from the number of points that I would have in each bin. In other words I can more easily think about the appropriate sampling of the PDF. Also, since the number of pixels is finite, the PDF that I measure is convoluted with a heavy side step function to mimic the cut. Translating the latter point in terms of CGF is relatively easy. I know that the Laplace transform of a heavyside step function is $e^{\lambda x_{\rm min}}$ so the equivalent CGF would be the one that I measure times this exponential damping.
However, translating the sampling of the PDF in terms of sampling and range for the CGF is not straightforward to me. If $\lambda$ was purely imaginary, then I would simply say that the minimum boundary values of $\lambda$ would be equal to the nyquist frequency, but I don’t know how that generalises to any values of $\lambda$, and in particular to real values. I suppose this might be linked to an extension of Shannon theorem to z-transform (equivalent of Fourier series but for the Laplace transform) ? Also, this does not tell me anything on the sampling of the $\lambda$ that I should choose.
Any help but also feedback would be greatly appreciated ! Thanks a lot :)