Introduction to cumulants

Cumulants are used generally for derivation of closed-form expressions. Because of their interesting properties they are used as an alternative to the moments of a distri-bution. Cumulants were first introduced by Thorvald N. Thiele, who called them as semi-invariants [66]. They were first called cumulants in a paper by Ronald Fisher and John Wishart [67]. Cumulants are of interest for a variety of reasons and in some cases, theoretical treatments of problems in terms of cumulants are simpler than those using moments.

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3.2.1 Definition of cumulants

The cumulants κ of any random variable S can be defined via the cumulant-generating functionC(t), which is natural logarithm of the moment generating function.

C(t) = ln

where, ln[.] is the natural logarithm and κ_n(S) is the n^th order cumulant of S. The expansion in equation (3.1) is a Maclaurin series. We know that, the n^th order raw moment (i.e. moment about zero),m_n can be obtained by differentiating the moment-generating functionE(e^tS) byn times and evaluating the result att= 0

mn(S) = ∂ⁿ

Similarly, the n^th order cumulant, κn can be obtained by differentiating the cumulant-generating functionC(t) by ntimes and evaluating the result att= 0

κn(S) = ∂ⁿ Cumulant of a random variable is also called a ‘Ursell function’ or ‘connected correlation function’ as it can often be obtained by summing over connected Feynman diagrams (the sum over all Feynman diagrams gives the correlation functions) [68].

3.2.2 Cumulants and moments relation

IfS is a random variable with meanµ and varianceσ², then κ_n(S) can be represented as a n^th degree polynomial containing the firstnraw moments, mn(S)

κ1(S) =m1(S) =µ (3.4)

Chapter 3. Spectral regrowth prediction for FBMC-OQAM systems using cumulant

• |p|is number of parts in the partitionp,

• the sum is over all partitions p of the set 1, ..., mof indices,

• B∈p, i.e. B runs through the whole list of “blocks” of the partitionp.

For example

κ(S₁) =E(S₁) (3.8)

κ(S₁, S₂) =E(S₁S₂)−E(S₁)E(S₂) (3.9) κ(S₁, S₂, S₃) =E(S₁S₂S₃)−E(S₁S₂)E(S₃)−E(S₁S₃)E(S₂)

−E(S₂S₃)E(S₁) + 2.E(S₁)E(S₂)E(S₃) (3.10) It can be noted that equations (3.8), (3.9) are nothing but mean and co-variance re-spectively. Number of total possible partitions in equations (3.8), (3.9) and (3.10) are 1, 2 and 5 respectively. In general, if there are m random variables, then, total number of possible partitions is m^th Bell number Bm. In combinatorial mathematics, the Bell numbers count the number of partitions of a set and they satisfy a recurrence relation involving binomial coefficients

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3.2.4 Properties of cumulants

Cumulants exhibit some characteristic properties as mentioned below

3.2.4.1 Additivity

SupposeS₁, S₂, . . . , S_m aremindependent random variables, each havingn^thcumulant.

Then S=P

i

Si hasn^th cumulant, and

κ_n(S) =X

i

κ_n(S_i) (3.12)

i.e. the n^th cumulant of a sum of independent random variables is simply the sum of then^th cumulants of the summands. It means, the cumulants accumulate and hence the name ‘cumulants’.

3.2.4.2 Homogeneity

SupposeS is a random variable with an^th cumulant. Then for any constant c∈R,cS has a n^th cumulant and

κ_n(cS) =cⁿκ_n(S) (3.13)

3.2.4.3 Semi-variance

The first cumulant is ‘shift-equivariant’ and all the others are ‘shift-invariant’; hence the original name ofsemi−variants. Then, for any constantc

κ_i(S+c) =







κ_i(S) +c, i= 1 κi(S), i≥2

(3.14)

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3.2.4.4 Gaussian random variable

IfS is a Gaussian random variable with mean µand varianceσ², then

κ_i(S) =















µ, i= 1 σ², i= 2 0, i≥3

(3.15)

Since the first and second cumulants of any random variables are its mean and variance (see equations (3.8) and (3.9)), the Gaussian random variables have the simplest possible cumulants.

3.2.5 Cumulants of a random process

Ifs(t) is a random process then, we denote its i^th order cumulant as

κ_is(τ) =κ{s(t), s(t+τ₁), ..., s(t+τi−1)} (3.16) whereτ =(τ₁, τ₂, ...., τi−1) andτ₁, τ₂, ...., τi−1 are delays applied tos(t). Similarly, itsi^th order moment at delay τ is defined as

mis(τ) =m{s(t)s(t+τ1)...s(t+τi−1)} (3.17) General definitions and properties of cumulants can be found in [69]. For e.g. if s(t) is zero-mean, then

κ_4s(τ₁, τ₂, τ₃) =κ{s(t), s(t+τ₁), s(t+τ₂), s(t+τ₃)}

=E[s(t)s(t+τ₁)s(t+τ₂)s(t+τ₃)]−E[s(t)s(t+τ₁)]E[s(t+τ₂)s(t+τ₃)]

−E[s(t)s(t+τ₂)]E[s(t+τ₁)s(t+τ₃)]−E[s(t)s(t+τ₃)]E[s(t+τ₁)s(t+τ₂)]

=m_4s(τ₁, τ₂, τ₃)−m_2s(τ₁)m_2s(τ₃−τ₂)−m_2s(τ₂)m_2s(τ₃−τ₁)

−m_2s(τ₃)m_2s(τ₂−τ₁)

(3.18) Although it may apparently be seem that cumulants are more tedious than moments, they possess important properties, which do not hold for moments.

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Gaussian case: If s(t) is a Gaussian random process with mean µ and variance σ², then κ_is(τ)∀k >3,∀τ. But, for moments in general, we don’t have m_is(τ) = 0,∀τ for i >3, even if s(t) is Gaussian. Therefore, a cumulant in a higher-order than second can be used for the measurement of non-Gaussianity or non-normality of time sequence. By puttingκ4s(τ1, τ2, τ3) = 0, into the 4^th order cumulant in equation (3.18), we obtain

m4s(τ1, τ2, τ3) =m2s(τ1)m2s(τ2−τ3) +m2s(τ2)m2s(τ1−τ3)+ m2s(τ3)m2s(τ1−τ2) (3.19) This is one form of the Gaussian moment theorem [70] based on which references [71–73]

derive the closed-form expressions. As cumulants are the higher-order generalization of covariance functions; they can naturally be used as a tool to derive closed-form expression for the auto-covariance functions

Independent and identically distributed case: In probability theory and statis-tics, a sequence or other collection of random variables is independent and identically distributed (i.i.d.), if each random variable has the same probability distribution as the others and all are mutually independent. So, ifs(t) is an i.i.d. process then itsi^th order cumulant is a multidimensional Dirac functions [74], given by

κis(τ) =κis(0).δ(τ),∀i,∀τ (3.20) whereδ is the Kronecker delta function. However, for i >2,m_is(τ) is not a delta func-tion in general. Ifs(t) is zero mean and i.i.d. with varianceσ², then it is also wide-sense stationary (WSS). This property implies that cumulants, but not moments, characterize independence.

For a WSS processs(t), its PSD can be defined as the FT of its auto-covariance function κ2s(t):

S2s(f) =

+∞

Z

−∞

κ2s(τ)e^{−j2πf τ}dτ (3.21)

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Symmetrical case: Ifs(t) is a zero-mean random process having a symmetrical dis-tribution, then

κ_is(τ) = 0,∀iodd,∀τ (3.22)

This property also holds for moments, i.e. mis(τ) = 0,∀i odd.

Complex case: Ifs(t) is a complex random process, then we can conjugate any copy of sin the cumulant expression and obtain different versions of thei^th-order cumulant.

Fori= 2 for example, we define the auto-covariance function as κ2s(τ) =κ{s(t), s^∗(t+τ)}

=κ{s^∗(t), s(t+τ)}

(3.23)