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 ...
QC_Pod's user avatar
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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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Joint cumulants of repeated random variables

I'm trying to prove that: $$\kappa_{k_1,\dots,k_n}(X_1,\dots,X_n)=\kappa_{\underbrace{1,\dots,1}_{k_1+\dots+k_n}}(\underbrace{X_1,\dots,X_1}_{k_1},\dots,\underbrace{X_n,\dots,X_n}_{k_n})$$ where $X_1,\...
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What did Rota mean by "one can define cumulants relative to any sequence of binomial type"?

$\newcommand{\E}{\mathbb{E}}$ Near the end of "Finite Operator Calculus" (1976), G.C. Rota writes: Note that one can define cumulants relative to any sequence of binomial type, e.g. the ...
Daigaku no Baku's user avatar
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Do semicircular families exist?

Let $(A,\varphi)$ be a $^*$-probability space, i.e., $A$ is a unital $^*$-algebra, and $\varphi$ is a $^*$-preserving unital linear functional. Let $(c_{i,j})_{i,j\in I}$ be a positive definite ...
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Definition of cumulant generating function

The following question comes from M. Kardar's "Statistical Physics of Particles". Let $x$ be a random variable with pdf $p(x)$. Kardar defines the "characteristic function" as the ...
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Moment and Cumulant Notation

I recently started working with moments and cumulants for some statistical analysis and among my research I can’t figure out some of the notation. It will be written as: M2,1 and C4,3 I know M ...
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Reference on the cumulant generating function (basic properties)

The cumulant generating function $K(t)$ of a random variable is defined as $$K(t) = \log \mathbb{E} [e^{Xt}]$$ for any $t$ such that the exponential moment is finite. In Wikipedia, it is said that: ...
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Is every strictly convex smooth function on a half space a cumulant generating function for some random variable

Fix $a \in \mathbb{R}$ and let $\Psi(a, \infty) \mapsto \mathbb{R}$ be strictly convex and infinitely differentiable. Does there exist a probability measure $P$ on $\mathbb{R}$ such that $\Psi$ is the ...
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Expression for the cumulant generating function

Given the moment generating function of a random variable $X$, $$M_X (t) = \mathbb{E}[e^{tX}],$$ one defines the cumulant generating function, $$K(t) = \log \mathbb{E}[e^{tX}].$$ One then states that ...
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Moment/cumulant problem complex normal law.

I know that $N = (N_1, N_2)$, where $N_1, N_2$ follow normal distributions on $\mathbb{R}$ is characterized by its moments. I am also aware of Carleman condition for the moments problem for real ...
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Prove homogeneity of cumulants: If a random variable X is independent, show $\kappa_n(\alpha X) = \alpha^n \kappa_n(X)$

If we have an independent random variable X and a scalar constant alpha, show that $\kappa_n(\alpha X) = \alpha^n \kappa_n(X)$ holds by using the the Maclaurin series is given as $K_X(t) = \sum_{n=1}^\...
That1WasTaken's user avatar
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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 ...
Jean-Armand Moroni's user avatar
2 votes
1 answer
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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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Does zero cumulants imply independence?

Question: Suppose we have two random variable $X$, $Y$ that follow non-Gaussian distribution, and we are given that: $$\operatorname { cum }(X, Y)=\operatorname { cum }(X, X, Y)=\operatorname { cum }(...
graphitump's user avatar
1 vote
1 answer
225 views

Free cumulant generating function

The moments and cumulants of a real-valued random variable X is related via the cumulant-moment formula: $$m_n = \sum_{\pi\in P(n)}\prod_{B\in\pi}c_{|B|},\qquad\qquad(1)$$ where $m_n(X)=E(X^n)$ is the ...
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Why is the third derivative of cumulant generating function = skewness?

From what I know, for random variable $X$, skewness is defined as $$\mathbb{E}\left(\frac{X-\mathbb{E}(X)}{\sigma}\right)^3$$ or $$\frac{\mathbb{\mathbb{E}}(X^3)-3\mathbb{E}(X)\sigma^2-\mathbb{E}(X)^3}...
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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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Continuity and differentiability of the cumulant-generating function

Let $X_1, X_2, \dots$ be i.i.d. random variables with $E[X_i] = \mu$. Define the moment-generating function by $\varphi(\theta) = E[\exp(\theta X_i)]$ for $\theta \geq 0$. Also, let $\theta_+ = \sup\{\...
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Why is the second derivative of Cumulant Generating Function positive?

Wikipedia states (without reference) that the cumulant generating function of a random variable has the property Its first derivative ranges monotonically in the open interval from the infimum to the ...
NonalcoholicBeer's user avatar
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Cumulant generating function of continuous uniform distribution

With $f(x)=\frac{1}{b-a}$, I got the MGF, $M(t) = \frac{e^{bt}-e^{at}}{t(b-a)}$, and the cumulant generating function, $K(t) = \ln (e^{bt}-e^{at}) - \ln [t(b-a)]$. However, when I tried to find the ...
VincentN's user avatar
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Cumulant of sum of correlated random variables?

Let $X,Y$ be two random variables. We denote by $[X^k]$ and $[Y^k]$ the $k$'th order cumulants of $X$ and $Y$, respectively. I'm interested in computing the $k$'th order cumulant of $Z = X+Y$. If $X,...
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Relative magnitude of moments of a probability density to its cumulants

Consider a probability density $f(X)$ of some random variable $X\leq0$, with moments $\mu_n'=\left< X ^n \right>$ and cumulants $\kappa_n$. I aim to proof $$ \Delta_n := (-1)^n \left(\mu_n'- ...
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1 answer
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Find the first four cumulants of T = XY when X and Y are independent standard normal random variables.

Here's how I approached Let X∼Norm(0,1) and Y∼Norm(0,1) Then their joint pdf is $$ f(x) = \dfrac{\mathrm{e}^{-\frac{x^2}{2}}}{\sqrt{{2\pi}}} * f(y) = \dfrac{\mathrm{e}^{-\frac{y^2}{2}}}{\sqrt{{2\...
rbeginner's user avatar
1 vote
1 answer
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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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"Symbolic" relation between moments and cumulants

In the book "Series Expansion Methods" by Oitmaa, Hamer, and Zheng, Appendix 6, they define a moment $\left<\,\,\right>$ as the average of a set of variables, and then define the cumulant $\left[...
Kai's user avatar
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3 votes
1 answer
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What is a bispectrum analysis?

I am in the atmospheric sciences and when I read papers on non linear interactions I came up with this term - bispectrum. It is not very clear what a 2nd order cumulant is . So assuming I have wind ...
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2 votes
1 answer
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Is a distribution whose mean and variance are its only non-zero cumulants neccesarily Gaussian?

I have a distribution. Its cumulants are all zero except for its mean and variance. Does that necessarily mean that my distribution is Gaussian? Or in other words, are there non-Gaussian ...
Will Dorrell's user avatar
5 votes
1 answer
288 views

Bound for cumulants of bounded random variables

Let $X$ be a random variable taking values in $[-1,1]$. The cumulant generating function is defined as $$ K(t) = \log \mathbb{E} [e^{tX}], $$ and the cumulants of $X$ are $$ \kappa_n = K^{(n)}(0). $...
felipeh's user avatar
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1 answer
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Are even cumulants necessary non-zero infinitely often?

Let $X$ be a random variable with moment generating function $e^{P(t)}$ for a power series $P(t)$. Assume the moment generating function exists in a neighbourhood of 0. Then the cumulant generating ...
Landon Carter's user avatar
4 votes
1 answer
2k views

Cumulants vs. moments

In high order statistics, what is the intuition for the difference between cumulants and moments? What does any of them measure and what is the intuition to use one of them over the other? ...
havakok's user avatar
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Laplace transform of generalized hypergeometric distribution

What is please the Laplace transform (moment generating function $M(t)$) of a generalised hypergeometric distribution shown below $$p_X(x)=K\cdot\frac{(a_1)_x\dots(a_p)_x}{(b_1)_x\dots(b_q)_x}\cdot\...
Rafael 's user avatar
1 vote
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208 views

Cumulants of discrete uniform distribution from Bernoulli numbers

Let $X$ be the random variable uniformly distributed on $\{0,\dots,d-1\}$. This is, $$\operatorname{Prob}(X = i) = \frac{1}{d}$$ for $i \in \{0,\dots,d-1\}$. Computations suggest that the cumulants of ...
Christian's user avatar
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-1 votes
2 answers
704 views

Cumulative distribution function to probability density

I am given the cumulative distribution function $F(x) = 1-e^{\frac{-x^2}{2 \alpha}}$ for $x>0$. $X$ describes how long a component works before it fails. $\alpha$ is a parameter describing the ...
novo's user avatar
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Distribution for third cumulant?

It is well known that a probability distribution whose cumulants are all zero except for the first two is gaussian: $$ f(x,\mu,\sigma^2) \propto\exp(-\frac{(x-\mu)^2}{2\sigma^2}) $$ So I'm wondering ...
The Yomster's user avatar
1 vote
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622 views

Why the second cumulant is variance?

I have trouble understanding the term of second cumulant generating function. By the definition of cumulant generation function, it is defined by the logarithm of moment generating function $M_X(t)=E(...
Chen's user avatar
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1 answer
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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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An explicit expression for tail probability using fourier transform

I am reading a paper about tail probability approximation. However, I got into trouble at the very first formula. The background setting and formula goes like this: $\bar X=\frac{1}{n}\Sigma_{i=1}^...
Sena's user avatar
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4 votes
0 answers
242 views

Why is the S-transform necessary in free probability?

In classical probability, adding two independent random variables corresponds to adding their cumulant generating functions, i.e. the logarithms of their Fourier transforms. In Voiculescu's free ...
keej's user avatar
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moment generating function / characteristic function of Gauss-distribution

I'd like to calculate $E[\exp{\left(-\frac{1}{2}X\right)}]$ where $X$ is Gaussian distribtuted. And i know that the characteristic function is given by: $E[\exp\left(ikX\right)]=G(k)=\exp\left(i\mu k-\...
Martin's user avatar
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0 answers
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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 ...
Daschin's user avatar
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2 votes
2 answers
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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 ...
Beverlie's user avatar
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2 votes
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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 ...
NonalcoholicBeer's user avatar
2 votes
0 answers
112 views

How to calculate the cumulant of a generic function of random variables?

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}^{\...
Machine's user avatar
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The second-order moments of the fourth order cumulant

I'm looking for the second-order moments of the fourth order cumulant. This is the same as finding the the second-order moments of the covariance estimate which could be calculated as follow [1]: ...
Amgad A. Salama's user avatar