# Questions tagged [cumulants]

Use this tag for questions about those quantities that provide an alternative to the moments of a probability distribution.

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### Unbiased Cumulant Estimate - Fifth Cumulant

I am searching the definition of the $5^{th}$ unbiased cumulant estimate. Let $K_j$, be the $j$-th unbiased cumulant estimate of a probability distribution. Let $m_j$ be the sample moments for samples ...
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### Are cumulants the only additive functions of independent random variables?

For a random variable $X$, the cumulant generating function $CGF_X$ is defined as $CGF_X(t)=\log Ee^{tX}$, and the nth cumulant $k_n(X)$ is defined as the coefficient of $t^n/n!$ in the corresponding ...
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### Sums of central moments for a Gaussian distribution with mean zero and standard deviation one

I want to know, whether the following calculations for a vector of variables $A_{i}$ of size N, drawn independently at random from a complex normal distribution, with mean zero and standard deviation ...
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### Bound on cumulant generating function of a weighted sum of uniform random variables

Question Define $\mathbf{a} = (a_1, \ldots, a_p)$ where $p$ is a positive integer and the $a_l$ are i.i.d $\text{Uniform}(-1,1)$ random variables. Fix a unit vector $w \in \mathbb{R}^p$. Consider the ...
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### Quantify the similarity between a polynomial roots and the roots of its derivatives

On $\mathbb{C}[X]$, many theorems and conjectures deal with relations between a polynomial roots and the roots of its derivatives. When looking at a graph, the derivative roots distribution somewhat ...
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### Proving the cumulants to moments formula and combinatorics

I'm trying to prove the following Cumulants to moments transformation formula with induction and recursion relations for the Cumulants. There are other ways to prove this formula, but I would like to ...
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### Under what conditions is $K^{(2)}(t)$ continuous at $t=0$ where $K(t)$ is the cumulant generating function?

For a random variable $X$ let \begin{align} K(t)= \log (E[e^{tX}]) \end{align} This quantity is known as the cumulant generating function. Question: What is some weak sufficient condition for \begin{...
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### Positivity and Negativity of Cumulants

There's few examples of distributions whose cumulants can be computed relatively easily. Normal random variables are of course the easiest: the first cumulant is the mean and the second is the ...
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### How to translate sampling of probability density into sampling of cumulant generating function?

Context I am looking at a random picture in which every pixel is randomly generated through complex dynamics and I am interested in the 1-point statistics of the pixels (basically their probability ...
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### Can cumulants arise from another property of the distribution.?

Recall that cumulants are defined as \begin{align} k_n = \frac{d^n}{d t^n}\log (M_X(t)) |_{t=0}, n \in \mathbb{N}, \end{align} where $M_X(t)$ is the moment generating function of a random variable $X$....
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### Cumulant generating function and largest eigenvalue of operator

I am working on a recent paper (arXiv:1805.02887) about an application of large deviation theory to the statistical mechanics of active matter and am a bit bewildered by a result dropped in appendix F ...
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### Do small cumulants imply that a distribution is well-approximated by a Gaussian?

Suppose I know of all the cumulants $\{\kappa_n\}$ of some probability distribution. Is there any result of the form: Suppose the cumulants $\{\kappa_n\}$ of a probability distribution converge to ...
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### Kurtosis of Binomial Distribution

Let $X \thicksim B(n,p)$ then I would like to evaluate kurtosis and skweness of X. First I want to use the fact that kurtosis $k_3(\dfrac{X-\mu}{ \sigma})=\dfrac{k_3(X)}{\sigma^3}$ and skewness ...
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### 3rd Cumulant Equals to 3rd moment, 4th Cumulant Equalts to 4th moment${}- 3$

Under given standardized random variable $Z = (X-\mu)/\sigma$ Show that $$C_3(Z) = M_3(Z)\;\;\text{and}\;\; C_4(Z) = M_4(Z) -3$$ where $C_r(Z)$ refer to $r$-th cumulant, $M_r(Z)$ refer to $r$-th ...
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### The condition for cumulants to uniquely determine a distribution

Let $(\kappa_i)_{i\ge1}$ be a sequence of constants. What is the sufficient condition for the existence of a unique distribution $X$ with these cumulants? My guess is that, just like moments, it ...
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Consider the example: Let $X$, $Y$ be two joint random variables and $\alpha$, $\beta$ two parameters, then show that \langle X {\rm e}^{\alpha X + \beta Y}\rangle = \langle \langle X {\rm e}^{\...