# Questions tagged [cumulants]

The cumulants tag has no usage guidance.

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### What did Rota mean by "one can define cumulants relative to any sequence of binomial type"?

This question is cross-posted from MSE.$\newcommand{\E}{\mathbb{E}}$
Near the end of "Finite Operator Calculus" (1976), G.C. Rota writes:
Note that one can define cumulants relative to any ...

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### Is there a determinantal point process proof of the Keating-Snaith formula for the cumulants of the log characteristic polynomial of a random matrix?

For $U$ a unitary $N \times N$ matrix, randomly distributed according to Haar measure, we have the complex-valued random variable $\log (\det (1-U))$. The real part and imaginary parts of $\log (\det (...

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### Was this proposition on cumulants of compound Poisson distributions known before I put it into a Wikipedia article?

The $n$th cumulant $\kappa_n$ of a probability distribution for $n\ge2$ is functional that is a polynomial in the first $n$ moments of the distribution, that has the properties of $(1)$ homogeneity, $(...

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### Cumulants as coefficients of degree-n polynomial and conditions for real roots

In a $(n-1)$-degree polynomial, $P_{n-1}(z) = c_{n-1} z^{n-1} + c_{n-2} z^{n-2} + \ldots + c_0$, defined by,
\begin{equation}
P_{n-1}(z) = \sum\limits_{i=1}^n \, \prod\limits_{j=1,j\neq i}^n (z-x_j)
\...

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### How is 4th order cumulant of a complex random vector defined?

Suppose that ${\bf x} \in\mathbb C^n$ is a complex random vector, we know the mean vector and covariance matrix of $\bf x$ are defined as follows:
$${\bf m}_{\bf x} = \mathbb{E} ({\bf x}) \\
{\bf C}_{\...

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### Radius of convergence of cumulant generating function

Recall that for a random variable $X$ with a moment generating function $M_X(t)$ the cumulant generating function is defined as
\begin{align}
K_X(t)=\log M_X(t)
\end{align}
The Taylor expansion of $...

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### Is this (somewhat specific) moment problem treated somewhere?

Suppose I have a measure $\mu$ over $\mathbb R_+$ given by its moments $\mu_0,...,\mu_n$, defined as :
$$\mu_k = \int x^{k} \partial\mu(x),\; k \in 1,...,n$$
Using Faà di Bruno's formula, I can ...

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1
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### Bounds on cumulants in terms of moments

I am interested in finding bounds on cumulants in terms of moments.
For example, this paper alludes to the bound
\begin{align}
|\kappa_n|\le n^n E[|X-E[X]|^n]
\end{align}
where $\kappa_n$ is the $n$-...

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### Cumulant of functions of weakly dependent random variables

Suppose $X_1,\dots,X_4$ are Gaussian random variables with mean and variance
$$\mathbf E X_i = 0,\quad \mathbf E X_i^2=1.$$
Furthermore suppose that the random variables have a certain weak ...

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### Logarithm of the Fourier transform?

I've found this paper on the logarithm of the discrete fourier transform which proves that
$$
log F = 1/4 i \pi (I - (1 +i)F + F^2 - (1 - i)F^3)
$$
where $F$ is the unitary discrete Fourier ...

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### Arithmetic structure of non-zero cumulants

It is known that any non-Gaussian distribution must have infinitely many non-zero cumulants (Marcinkiewicz). I was wondering if something stronger is known about the structure of non-zero cumulants. ...

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### Existence of a Laplace transform that takes specific values on the integers

The classical Marcinkiewicz theorem (1939) states that if a random variable $X$ has a Laplace transform/characteristic function of the form $\mathbb{E}(e^{tX})=e^{P(t)} $ with $P$ a polynomial, then ...

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### Random variables whose expectations are cumulants

In my research I stumbled about the following class of random variables: Let $X_0,X_1,\dots$ be random variables on a common probability space with finite moments of all orders. We then define
\...

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### Finding analytic expressions for the cumulants of a correlated random variable

I am working with cumulants of a distribution. I have an example of how the second cumulant may be simplified from:
$k_2 = p\alpha^2\left\{\left(\sum a_i\right)^2 - 2\sum_{i<j}a_ia_j\left(1-\rho^{...

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### Can an unskewed distribution be expressed as product of a normal and another distribution?

Let $x$ be a continuous random variable with zero mean and zero skew. What are the conditions under which we can say that $x$ can be expressed as the product $z y$ where $z$ is a standard normal and $...

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### Joint cumulants of $Z_2^n$ characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus \...

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### What is a cumulant really?

A cumulant is defined via the cumulant generating function
$$ g(t)\stackrel{\tiny def}{=} \sum_{n=1}^\infty \kappa_n \frac{t^n}{n!},$$
where
$$
g(t)\stackrel{\tiny def}{=} \log E(e^{tX}).
$$
Cumulants ...

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### Is connected correlation/cumulant expansion additive?

Say X is a free field or a Gaussian random variable.
Then I want to analyse the connected correlation, $<(X + a (X^2 - \langle X^2 \rangle))^n>_c$
I think that for $n \geq 4$ there are no ...

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### cumulant problem

A couple of days after I posted this to stackexchange, no one's answered:
I take the problem of cumulants to be this: given a sequence $(\kappa_1,\kappa_2,\kappa_3,\ldots)$ of real numbers, is it the ...

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### Generalizations of Gram-Charlier and Edgeworth series?

I am looking for references pertaining to, and/or help in deriving, generalizations of the Gram-Charlier and Edgeworth series for non-Gaussian reference probability distributions.
I would like to ...