I just learned how finite $p$-th moment control the distribution of random variables. That is, given any random variable $X$ if $E[|X^p|]<\infty$ (or we say $X\in L^p$) for non-negative integer $p$ then $$E[|X|^p]=\int_0^{\infty}px^{p-1}\mathbb{P}(X>x)dx\tag{*}$$. The finite $p$-th moment tells us distribution of $X$ e.g. finite first moment implies small tails and finite $p$-th moment for large $p$ implies not having tall spike.
However, there are other notions of moment. For example,
- Exponential moment of $X$ is defined as $E[e^{tX}]=\sum_{k=0}^{\infty}\frac{t^k}{k!}E[X^k]$ within some radius of convergence.
- Fourier moment of $X$ is defined as $E[e^{itX}]=\sum_{k=0}^{\infty}\frac{i^kt^k}{k!}E[X^k]$ within some radius of convergence.
- Maybe there are other moments like "log moment" -- $E[\log{tX}]$ or anything else with a Taylor expansion.
My question is: what are the controls of distribution of random variable $X$ under those finite moments? Is there any nice formula like (*) in those cases? I appreciate if there are other interesting finite moments you'd like to introduce.
It seems they are stronger control than finite $p$-th moment since to have finite (1), (2) moment it is necessary to have finite $p$-th moment for all $p$. Any references are appreciated!