6
votes
Accepted
Was this proposition on cumulants of compound Poisson distributions known before I put it into a Wikipedia article?
This is a simple fact, below is a short proof. It is certainly very-well known for quite some time, a sample reference is formula (6.6) in
Cacoullos T. (1989) Generating Functions. Characteristic ...
- 17.2k
6
votes
What did Rota mean by "one can define cumulants relative to any sequence of binomial type"?
I am not quite sure which generalization Rota had in mind, but this is a generalization that has found applications in the literature on stochastic processes:
Given a probability distribution function ...
- 188k
5
votes
What is a cumulant really?
There are two articles that answer your question in different levels of detail.
First What are Cumulants? and then Cumulants are universal homomorphisms
into Hausdorff groups.
What I can say in one ...
4
votes
Accepted
Is there a determinantal point process proof of the Keating-Snaith formula for the cumulants of the log characteristic polynomial of a random matrix?
Reposting a comment as an answer: There are some papers of Soshnikov which are in this direction. See for instance Central Limit Theorem for local linear statistics in classical compact groups and ...
- 2,151
4
votes
Is there a determinantal point process proof of the Keating-Snaith formula for the cumulants of the log characteristic polynomial of a random matrix?
This alternative proof of the Keating-Snaith formula avoids the Selberg integral and may be what you after: The characteristic polynomial of a random unitary matrix: a probabilistic approach, by Paul ...
- 188k
4
votes
Radius of convergence of cumulant generating function
Of course, this is not correct. As a simplest example, let
$X$ be a random variable which takes only values $\{0,\ldots, n\}$, then
the moment generating function is a polynomial of $e^t$, of degree $...
- 91.8k
2
votes
What did Rota mean by "one can define cumulants relative to any sequence of binomial type"?
Rota and Shen published "On the Combinatorics of Cumulants" in 2000, but no mention of factorial cumulants is made. (Beware: my notation is different from their's. I use the subscript, or ...
- 10.5k
2
votes
Is this (somewhat specific) moment problem treated somewhere?
$\newcommand\ka\kappa$$\newcommand\R{\mathbb R}$Without loss of generality, $\mu_0=1$, so that $\mu$ is a probability measure. Let $\ka$ be a measure on $\R$ with moments $\ka_0,\dots,\ka_n$ and $|\ka|...
- 128k
2
votes
Accepted
Bounds on cumulants in terms of moments
Let $k_n:=\kappa_n$, $a_n:=E(X-EX)^n$, $b_n:=E|X-EX|^n$, so that $|a_n|\le b_n$. We have to show that
$$|k_n|\le n^n b_n$$
for natural $n$. For $n=1,2$ this is obvious. The key is the recursion
$$k_n=...
- 128k
1
vote
Logarithm of the Fourier transform?
Let us start with the identity $F^4=I$. As a result, we have formally
\begin{align}
\ln F&=\ln(I+F-I)=\sum_{k\ge 1}\frac{(-1)^{k-1}(F-I)^k}{k}
\\&=
\sum_{1\le k\le 3}\frac{(-1)^{k-1}(F-I)^k}{k}...
- 16.2k
1
vote
Can an unskewed distribution be expressed as product of a normal and another distribution?
The characteristic function of $x$ is $\varphi(s) = \mathbb E[e^{isx}] = \mathbb E[e^{-s^2 y^2/2}]$. The fact that $\varphi(s) \to 0$ as $s \to \pm \infty$ tells you that $x$ is continuous.
- 54.2k
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