Questions tagged [cumulants]

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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, $(...
3 votes
0 answers
85 views

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) \...
1 vote
1 answer
200 views

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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2 votes
1 answer
378 views

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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1 vote
1 answer
130 views

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 ...
  • 641
2 votes
1 answer
277 views

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 $n$-...
  • 611
2 votes
0 answers
44 views

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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3 votes
3 answers
780 views

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 ...
2 votes
0 answers
89 views

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. ...
1 vote
0 answers
105 views

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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4 votes
0 answers
88 views

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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3 votes
0 answers
69 views

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^{...
0 votes
1 answer
148 views

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 $...
7 votes
0 answers
163 views

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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66 votes
6 answers
22k views

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 ...
4 votes
0 answers
254 views

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 ...
6 votes
1 answer
911 views

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 ...
2 votes
0 answers
397 views

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 ...
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