Cumulant

In probability theory and statistics, a random variable X has an expected value μ = E(X) and a variance σ² = E((X − μ)²). Probability theory is the branch of Mathematics concerned with analysis of random phenomena Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of These are the first two cumulants: μ = κ₁ and σ² = κ₂.

The cumulants κ_n are defined by the cumulant-generating function which is the g(t) below:

$g(t)=\log(E (e^{t\cdot X}))=\sum_{n=1}^\infty\kappa_n \frac{t^n}{n!}=\mu t + \sigma^2 \frac{ t^2}{2} + \cdots.$

The cumulants are then given by derivatives (at zero) of g(t):

κ₁ = μ = g' (0),

κ₂ = σ² = g' '(0),

κ_n = g⁽ⁿ⁾ (0).

A distribution with given cumulants κ_n can be approximated through the Edgeworth series. The Gram-Charlier A series and the Edgeworth series, named in honor of Francis Ysidro Edgeworth, are series that approximate a Probability distribution

The cumulants of a distribution are closely related to distribution's moments as described later. Working with cumulants can have an advantage over using moments because for independent variables X and Y,

$g_{X+Y}(t)=\log(E(e^{t\cdot (X+Y)}))=\log(E(e^{tX})\cdot E(e^{tY}))=\log(E(e^{tX}))+\log(E(e^{tY}))=g_{X}(t)+g_{Y}(t) \,.$

so that each cumulant of a sum is the sum of the corresponding cumulants of the addends. Addition is the mathematical process of putting things together

Some writers^[1]^[2] prefer to define the cumulant generating function via the characteristic function

$h(t)=\log(E (e^{i t X}))=\sum_{n=1}^\infty\kappa_n \cdot\frac{(it)^n}{n!}=\mu it - \sigma^2 \frac{ t^2}{2} + \cdots.\,$

This characterization of cumulants is valid even for distributions whose higher moments do not exist. In Probability theory, the characteristic function of any Random variable completely defines its Probability distribution.

1 Cumulants of some discrete probability distributions
2 Cumulants of some continuous probability distributions
3 Some properties of cumulants
4 Joint cumulants
- 4.1 Conditional cumulants and the law of total cumulance
5 History
6 Formal cumulants
7 Bell numbers
8 Cumulants of a polynomial sequence of binomial type
9 Free cumulants
10 See also
11 References
12 External links

Cumulants of some discrete probability distributions

The constant random variable X = 1. WikipediaWikiProject Probability#Standards for a discussionof standards used for probability distribution articles such as this one The derivative of the cumulant generating function is g '(t) = 1. The first cumulant is κ₁ = g '(0) = 1 and the other cumulants are zero, κ₂ = κ₃ = κ₄ = . . . = 0.

The constant random variables X = μ. Every cumulant is just μ times the corresponding cumulant of the constant random variable X = 1. The derivative of the cumulant generating function is g '(t) = μ. The first cumulant is κ₁ = g '(0) = μ and the other cumulants are zero, κ₂ = κ₃ = κ₄ = . . . = 0. So the derivative of cumulant generating functions is a generalisation of the real constants. Generalization is a foundational element of Logic and human reasoning.

The Bernoulli distributions, (number of successes in one trial with probability p of success). In Probability theory and Statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete Probability The special case p = 1 is the constant random variable X = 1. The derivative of the cumulant generating function is g '(t) = ((p⁻¹−1)·e^−t + 1)⁻¹. The first cumulants are κ₁ = g '(0) = p and κ₂ = g ' '(0) = p·(1−p) . The cumulants satisfy a recursion formula $\scriptstyle\kappa_{n+1}=p\cdot(1-p)\cdot\tfrac{d\kappa_n}{dp}.\,$

The geometric distributions, (number of failures before one success with probability p of success on each trial). In Probability theory and Statistics, the geometric distribution is either of two Discrete probability distributions the probability The derivative of the cumulant generating function is g '(t) = ((1−p)⁻¹·e^−t−1)⁻¹. The first cumulants are κ₁ = g '(0) = p⁻¹−1, and κ₂ = g ' '(0) = κ₁·p⁻¹. Substituting p = (μ+1)⁻¹ gives g '(t) = ((μ⁻¹+1)·e^−t−1)⁻¹ and κ₁ = μ.

The Poisson distributions. In Probability theory and Statistics, the Poisson distribution is a Discrete probability distribution that expresses the probability of a number of events The derivative of the cumulant generating function is g '(t) = μ·e^t. All cumulants are equal to the parameter: κ₁ = κ₂ = κ₃ = . . . =μ.

The binomial distributions, (number of successes in n independent trials with probability p of success on each trial). WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other The special case n = 1 is a Bernoulli distribution. Every cumulant is just n times the corresponding cumulant of the corresponding Bernoulli distribution. The derivative of the cumulant generating function is g '(t) = n·((p⁻¹−1)·e^−t + 1)⁻¹. The first cumulants are κ₁ = g '(0) = n·p and κ₂ = g ' '(0) = κ₁·(1−p). Substituting p = μ·n⁻¹ gives g '(t) = ((μ⁻¹−n⁻¹)·e^−t+n⁻¹)⁻¹ and κ₁ = μ. The limiting case n⁻¹ = 0 is a Poisson distribution.

The negative binomial distributions, (number of failures before n successes with probability p of success on each trial). In Probability and Statistics the negative binomial distribution is a Discrete probability distribution. The special case n = 1 is a geometric distribution. Every cumulant is just n times the corresponding cumulant of the corresponding geometric distribution. The derivative of the cumulant generating function is g '(t) = n·((1−p)⁻¹·e^−t−1)⁻¹. The first cumulants are κ₁ = g '(0) = n·(p⁻¹−1), and κ₂ = g ' '(0) = κ₁·p⁻¹. Substituting p = (μ·n⁻¹+1)⁻¹ gives g '(t) = ((μ⁻¹+n⁻¹)·e^−t−n⁻¹)⁻¹ and κ₁ = μ. Comparing these formulas to those of the binomial distributions explains the name 'negative binomial distribution'. The limiting case n⁻¹ = 0 is a Poisson distribution.

Introducing the variance-to-mean ratio, $\epsilon=\mu^{-1}\sigma^2=k_1^{-1}k_2$ , the above probability distributions get a unified formula for the derivative of the cumulant generating function:

$g'(t)=\mu\cdot(1+\epsilon\cdot (e^{-t}-1))^{-1}.$

The second derivative is

$g''(t)=g'(t)\cdot(1+e^t\cdot (\epsilon^{-1}-1))^{-1}$

confirming that the first cumulant is κ₁ = g '(0) = μ and the second cumulant is κ₂ = g ' '(0) = μ·ε. In Probability theory and Statistics, the variance-to-mean ratio (VMR, like the Coefficient of variation, is a measure of the dispersion of The constant random variables X = μ have є = 0. The binomial distributions have є = 1 − p so that 0<є<1. The Poisson distributions have є = 1. The negative binomial distributions have є = p⁻¹ so that є > 1. Note the analogy to the eccentricity theory of the conic sections: circles є = 0, ellipses 0 < є < 1, parabolas є = 1, hyperbolas є > 1. In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

Cumulants of some continuous probability distributions

For the normal distribution with expected value μ and variance σ², the derivative of the cumulant generating function is g '(t) = μ + σ²·t. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of The cumulants are κ₁ = μ, κ₂ = σ², and κ₃ = κ₄ = . . . = 0. The special case σ² = 0 is a constant random variable X = μ.

The cumulants of the uniform distribution on the interval [−1, 0] are κ_n = B_n/n, where B_n is the nth Bernoulli number. In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory.

Some properties of cumulants

Invariance and equivariance

The first cumulant is shift-equivariant; all of the others are shift-invariant. In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance. To state this less tersely, denote by κ_n(X) the nth cumulant of the probability distribution of the random variable X. The statement is that if c is constant then κ₁(X + c) = κ₁(X) + c and κ_n(X + c) = κ_n(X) for n ≥ 2, i. A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. e. , c is added to the first cumulant, but all higher cumulants are unchanged.

Homogeneity

The nth cumulant is homogeneous of degree n, i. e. if c is any constant, then

κ n (c X) = c n κ n (X).

Additivity

If X and Y are independent random variables then κ_n(X + Y) = κ_n(X) + κ_n(Y). In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other

A negative result

Given the results for the cumulants of the normal distribution, it might be hoped to find families of distributions for which κ_m = κ_m+1 = . The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields . . = 0 for some m>3, with the lower order cumulants (orders 3 to m -1) being non-zero. There are no such distributions ^[3]. The underlying result here is that the cumulant generating function cannot be a finite order polynomial of order greater than 2.

Cumulants and moments

The moment generating function is:

$1+\sum_{n=1}^\infty \frac{\mu'_n t^n}{n!}=\exp\left(\sum_{n=1}^\infty \frac{\kappa_n t^n}{n!}\right) = \exp(g(t)).$

so the cumulant generating function is the logarithm of the moment generating function. In Probability theory and Statistics, the moment-generating function of a Random variable X is M_X(t=\operatorname{E}\left(e^{tX}\right The first cumulant is the expected value; the second and third cumulants are respectively the second and third central moments (the second central moment is the variance); but the higher cumulants are neither moments nor central moments, but rather more complicated polynomial functions of the moments. In Probability theory and Statistics, the k th moment about the Mean (or k th central moment In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of

The cumulants are related to the moments by the following recursion formula:

$\kappa_n=\mu'_n-\sum_{k=1}^{n-1}{n-1 \choose k-1}\kappa_k \mu_{n-k}'.$

The nth moment μ′_n is an nth-degree polynomial in the first n cumulants, thus:

$\mu'_1=\kappa_1\,$

$\mu'_2=\kappa_2+\kappa_1^2\,$

$\mu'_3=\kappa_3+3\kappa_2\kappa_1+\kappa_1^3\,$

$\mu'_4=\kappa_4+4\kappa_3\kappa_1+3\kappa_2^2+6\kappa_2\kappa_1^2+\kappa_1^4\,$

$\mu'_5=\kappa_5+5\kappa_4\kappa_1+10\kappa_3\kappa_2 +10\kappa_3\kappa_1^2+15\kappa_2^2\kappa_1 +10\kappa_2\kappa_1^3+\kappa_1^5\,$

$\mu'_6=\kappa_6+6\kappa_5\kappa_1+15\kappa_4\kappa_2+15\kappa_4\kappa_1^2 +10\kappa_3^2+60\kappa_3\kappa_2\kappa_1+20\kappa_3\kappa_1^3+15\kappa_2^3 +45\kappa_2^2\kappa_1^2+15\kappa_2\kappa_1^4+\kappa_1^6.\,$

The coefficients are precisely those that occur in Faà di Bruno's formula. Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition Faà di Bruno's formula is an identity in Mathematics generalizing the Chain rule to higher derivatives named in honor of Francesco Faà di Bruno (1825&ndash1888

The "prime" distinguishes the moments μ′_n from the central moments μ_n. In Probability theory and Statistics, the k th moment about the Mean (or k th central moment To express the central moments as functions of the cumulants, just drop from these polynomials all terms in which κ₁ appears as a factor:

$\mu_1=0\,$

$\mu_2=\kappa_2\,$

$\mu_3=\kappa_3\,$

$\mu_4=\kappa_4+3\kappa_2^2\,$

$\mu_5=\kappa_5+10\kappa_3\kappa_2\,$

$\mu_6=\kappa_6+15\kappa_4\kappa_2+10\kappa_3^2+15\kappa_2^3.\,$

Cumulants and set-partitions

These polynomials have a remarkable combinatorial interpretation: the coefficients count certain partitions of sets. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " A general form of these polynomials is

$\mu'_n=\sum_{\pi}\prod_{B\in\pi}\kappa_{\left|B\right|}$

where

π runs through the list of all partitions of a set of size n;

"B ∈ π" means B is one of the "blocks" into which the set is partitioned; and

|B| is the size of the set B.

Thus each monomial is a constant times a product of cumulants in which the sum of the indices is n (e. In Mathematics, the word monomial means two different things in the context of Polynomials The first meaning is a product of powers of Variables g. , in the term κ₃ κ₂² κ₁, the sum of the indices is 3 + 2 + 2 + 1 = 8; this appears in the polynomial that expresses the 8th moment as a function of the first eight cumulants). A partition of the integer n corresponds to each term. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French The coefficient in each term is the number of partitions of a set of n members that collapse to that partition of the integer n when the members of the set become indistinguishable.

Joint cumulants

The joint cumulant of several random variables X₁, . . . , X_n is

$\kappa(X_1,\dots,X_n) =\sum_\pi (|\pi|-1)!(-1)^{|\pi|-1}\prod_{B\in\pi}E\left(\prod_{i\in B}X_i\right)$

where π runs through the list of all partitions of { 1, . . . , n }, and B runs through the list of all blocks of the partition π. For example,

$\kappa(X,Y,Z)=E(XYZ)-E(XY)E(Z)-E(XZ)E(Y)-E(YZ)E(X)+2E(X)E(Y)E(Z).\,$

The joint cumulant of just one random variable is its expected value, and that of two random variables is their covariance. In Probability theory and Statistics, covariance is a measure of how much two variables change together (the Variance is a special case of the covariance If some of the random variables are independent of all of the others, then the joint cumulant is zero. If all n random variables are the same, then the joint cumulant is the nth ordinary cumulant.

The combinatorial meaning of the expression of moments in terms of cumulants is easier to understand than that of cumulants in terms of moments:

$E(X_1\cdots X_n)=\sum_\pi\prod_{B\in\pi}\kappa(X_i : i \in B).$

For example:

$E(XYZ)=\kappa(X,Y,Z)+\kappa(X,Y)\kappa(Z)+\kappa(X,Z)\kappa(Y) +\kappa(Y,Z)\kappa(X)+\kappa(X)\kappa(Y)\kappa(Z).\,$

Another important property of joint cumulants is multilinearity:

$\kappa(X+Y,Z_1,Z_2,\dots)=\kappa(X,Z_1,Z_2,\dots)+\kappa(Y,Z_1,Z_2,\dots).\,$

Just as the second cumulant is the variance, the joint cumulant of just two random variables is the covariance. In Probability theory and Statistics, covariance is a measure of how much two variables change together (the Variance is a special case of the covariance The familiar identity

$\operatorname{var}(X+Y)=\operatorname{var}(X) +2\operatorname{cov}(X,Y)+\operatorname{var}(Y)\,$

generalizes to cumulants:

$\kappa_n(X+Y)=\sum_{j=0}^n {n \choose j} \kappa(\,\underbrace{X,\dots,X}_{j},\underbrace{Y,\dots,Y}_{n-j}).\,$

Conditional cumulants and the law of total cumulance

Main article: law of total cumulance

The law of total expectation and the law of total variance generalize naturally to conditional cumulants. See also Cumulant In Probability theory and mathematical Statistics, the law of total cumulance is a generalization to Cumulants The proposition in Probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, the smoothing theorem In Probability theory, the law of total variance or variance decomposition formula states that if X and Y are Random variables on the The case n = 3, expressed in the language of (central) moments rather than that of cumulants, says

$\mu_3(X)=E(\mu_3(X\mid Y))+\mu_3(E(X\mid Y)) +3\,\operatorname{cov}(E(X\mid Y),\operatorname{var}(X\mid Y)).$

The general result stated below first appeared in 1969 in The Calculation of Cumulants via Conditioning by David R. Brillinger in volume 21 of Annals of the Institute of Statistical Mathematics, pages 215-218.

In general, we have

$\kappa(X_1,\dots,X_n)=\sum_\pi \kappa(\kappa(X_{\pi_1}\mid Y),\dots,\kappa(X_{\pi_b}\mid Y))$

where

the sum is over all partitions π of the set { 1, . In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " . . , n } of indices, and

π₁, . . . , π_b are all of the "blocks" of the partition π; the expression κ(X_{π_k}) indicates that the joint cumulant of the random variables whose indices are in that block of the partition.

History

Cumulants were first introduced by the Danish astronomer, actuary, mathematician, and statistician Thorvald N. Thiele (1838 - 1910) in 1889. An actuary is a business professional who deals with the financial impact of risk and uncertainty A mathematician is a person whose primary area of study and research is the field of Mathematics. Thorvald Nicolai Thiele (December 24 1838 &ndash September 26 1910 was a Danish Astronomer, Actuary and Mathematician, most notable for his Thiele called them half-invariants. They were first called cumulants in a 1931 paper, The derivation of the pattern formulae of two-way partitions from those of simpler patterns, Proceedings of the London Mathematical Society, Series 2, v. The London Mathematical Society ( LMS) is the leading mathematical society in England. 33, pp. 195-208, by the great statistical geneticist Sir Ronald Fisher and the statistician John Wishart, eponym of the Wishart distribution. Sir Ronald Aylmer Fisher, FRS ( 17 February 1890 – 29 July 1962) was an English Statistician, Evolutionary John Wishart ( 28 November 1898 &ndash 14 July 1956) was a Scottish agricultural statistician. In Statistics, the Wishart distribution, named in honor of John Wishart, is a generalization to multiple dimensions of the Chi-square distribution, The historian Stephen Stigler has said that the name cumulant was suggested to Fisher in a letter from Harold Hotelling. Stephen Mack Stigler is Ernest DeWitt Burton Distinguished Service Professor at the Department of Statistics of the University of Chicago[http //chronicle Harold Hotelling ( Fulda Minnesota, September 29, 1895 &ndash December 26, 1973) was a mathematical statistician and very influential In another paper published in 1929, Fisher had called them cumulative moment functions.

Formal cumulants

More generally, the cumulants of a sequence { m_n : n = 1, 2, 3, . . . }, not necessarily the moments of any probability distribution, are given by

$1+\sum_{n=1}^\infty m_n t^n/n!=\exp\left(\sum_{n=1}^\infty\kappa_n t^n/n!\right)$

where the values of κ_n for n = 1, 2, 3, . . . are found formally, i. e. , by algebra alone, in disregard of questions of whether any series converges. All of the difficulties of the "problem of cumulants" are absent when one works formally. The simplest example is that the second cumulant of a probability distribution must always be nonnegative, and is zero only if all of the higher cumulants are zero. Formal cumulants are subject to no such constraints.

Bell numbers

In combinatorics, the nth Bell number is the number of partitions of a set of size n. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects In combinatorial Mathematics, the n th Bell number, named in honor of Eric Temple Bell, is the number of partitions of a set All of the cumulants of the sequence of Bell numbers are equal to 1. The Bell numbers are the moments of the Poisson distribution with expected value 1.

Cumulants of a polynomial sequence of binomial type

For any sequence { κ_n : n = 1, 2, 3, . . . } of scalars in a field of characteristic zero, being considered formal cumulants, there is a corresponding sequence { μ ′ : n = 1, 2, 3, . In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division . . } of formal moments, given by the polynomials above. For those polynomials, construct a polynomial sequence in the following way. In Mathematics, a polynomial sequence is a Sequence of Polynomials indexed by the nonnegative integers 0 1 2 3. Out the polynomial

$\begin{matrix}\mu'_6= & \kappa_6+6\kappa_5\kappa_1+15\kappa_4\kappa_2+15\kappa_4\kappa_1^2 +10\kappa_3^2+60\kappa_3\kappa_2\kappa_1 \\ \\ & {}+20\kappa_3\kappa_1^3+15\kappa_2^3 +45\kappa_2^2\kappa_1^2+15\kappa_2\kappa_1^4+\kappa_1^6\end{matrix}$

make a new polynomial in these plus one additional variable x:

$\begin{matrix}p_6(x)= & (\kappa_6)\,x+(6\kappa_5\kappa_1+15\kappa_4\kappa_2+10\kappa_3^2)\,x^2 +(15\kappa_4\kappa_1^2+60\kappa_3\kappa_2\kappa_1+15\kappa_2^3)\,x^3 \\ \\ & {}+(45\kappa_2^2\kappa_1^2)\,x^4+(15\kappa_2\kappa_1^4)\,x^5 +(\kappa_1^6)\,x^6\end{matrix}$

. . . and generalize the pattern. The pattern is that the numbers of blocks in the aforementioned partitions are the exponents on x. Each coefficient is a polynomial in the cumulants; these are the Bell polynomials, named after Eric Temple Bell. In combinatorial Mathematics, the Bell polynomials Eric Temple Bell ( February 7

This sequence of polynomials is of binomial type. In Mathematics, a Polynomial sequence, ie a sequence of Polynomials indexed by { 0 1 2 3. In fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of formal cumulants.

Free cumulants

In the identity

$E(X_1\cdots X_n)=\sum_\pi\prod_{B\in\pi}\kappa(X_i : i\in B)$

one sums over all partitions of the set { 1, . . . , n }. If instead, one sums only over the noncrossing partitions, then one gets "free cumulants" rather than conventional cumulants treated above. In Combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things its application to the theory of Free probability These play a central role in free probability theory. Free probability is a mathematical theory which studies Non-commutative Random variables The "freeness" property is the analogue of the classical In that theory, rather than considering independence of random variables, defined in terms of Cartesian products of algebras of random variables, one considers instead "freeness" of random variables, defined in terms of free products of algebras rather than Cartesian products of algebras. In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Abstract algebra, the free product of groups constructs a group from two or more given ones

The ordinary cumulants of degree higher than 2 of the normal distribution are zero. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields The free cumulants of degree higher than 2 of the Wigner semicircle distribution are zero. The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the Probability distribution supported on the interval ''R'' the graph of whose This is one respect in which the role of the Wigner distribution in free probability theory is analogous to that of the normal distribution in conventional probability theory.

References

^ Kendall, M. In Mathematics &mdash specifically in Large deviations theory &mdash a rate function is a function used to quantify the probabilities of rare events In Mathematics, a multiset (or bag) is a generalization of a set. In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n G. , Stuart, A. (1969) The Advanced Theory of Statistics, Volume 1 (3rd Edition). Griffin, London. (Section 3. 12)
^ Lukacs, E. (1970) Characteristic Functions (2nd Edition). Griffin, London. (Page 27)
^ Lukacs, E. (1970) Characteristic Functions (2nd Edition), Griffin, London. (Theorem 7. 3. 5)

External links

Eric W. Weisstein, Cumulant at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
cumulant on the Earliest known uses of some of the words of mathematics

Dictionary

-noun

(mathematics) Any of a set of parameters of a one-dimensional probability distribution of a certain form

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