# Questions tagged [cumulants]

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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 ...
115 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 $n$-...
40 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 ...
410 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 ...
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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. ...
104 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 ...
80 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 \...
64 views