In probability theory and mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the law of total variance. It has applications in the analysis of time series. It was introduced by David Brillinger. It is most transparent when stated in its most general form, for joint cumulants, rather than for cumulants of a specified order for just one random variable. In general, we have
κ ( X 1 , … , X n ) = ∑ π κ ( κ ( X i : i ∈ B ∣ Y ) : B ∈ π ) , {\displaystyle \kappa (X_{1},\dots ,X_{n})=\sum _{\pi }\kappa (\kappa (X_{i}:i\in B\mid Y):B\in \pi ),}
where
κ(X1, ..., Xn) is the joint cumulant of n random variables X1, ..., Xn, and the sum is over all partitions π {\displaystyle \pi } of the set { 1, ..., n } of indices, and "B ∈ π;" means B runs through the whole list of "blocks" of the partition π, and κ(Xi : i ∈ B | Y) is a conditional cumulant given the value of the random variable Y. It is therefore a random variable in its own right—a function of the random variable Y.
Examples
The special case of just one random variable and n = 2 or 3 Only in case n = either 2 or 3 is the nth cumulant the same as the nth central moment. The case n = 2 is well-known (see law of total variance). Below is the case n = 3. The notation μ3 means the third central moment.
μ 3 ( X ) = E ( μ 3 ( X ∣ Y ) ) + μ 3 ( E ( X ∣ Y ) ) + 3 cov ( E ( X ∣ Y ) , var ( X ∣ Y ) ) . {\displaystyle \mu _{3}(X)=\operatorname {E} (\mu _{3}(X\mid Y))+\mu _{3}(\operatorname {E} (X\mid Y))+3\operatorname {cov} (\operatorname {E} (X\mid Y),\operatorname {var} (X\mid Y)).}
General 4th-order joint cumulants For general 4th-order cumulants, the rule gives a sum of 15 terms, as follows:
κ ( X 1 , X 2 , X 3 , X 4 ) =
κ ( κ ( X 1 , X 2 , X 3 , X 4 ∣ Y ) )
+ κ ( κ ( X 1 , X 2 , X 3 ∣ Y ) , κ ( X 4 ∣ Y ) )
+ κ ( κ ( X 1 , X 2 , X 4 ∣ Y ) , κ ( X 3 ∣ Y ) )
+ κ ( κ ( X 1 , X 3 , X 4 ∣ Y ) , κ ( X 2 ∣ Y ) )
+ κ ( κ ( X 2 , X 3 , X 4 ∣ Y ) , κ ( X 1 ∣ Y ) ) } ( partitions of the 3 + 1 form )
+ κ ( κ ( X 1 , X 2 ∣ Y ) , κ ( X 3 , X 4 ∣ Y ) )
+ κ ( κ ( X 1 , X 3 ∣ Y ) , κ ( X 2 , X 4 ∣ Y ) )
+ κ ( κ ( X 1 , X 4 ∣ Y ) , κ ( X 2 , X 3 ∣ Y ) ) } ( partitions of the 2 + 2 form )
+ κ ( κ ( X 1 , X 2 ∣ Y ) , κ ( X 3 ∣ Y ) , κ ( X 4 ∣ Y ) )
+ κ ( κ ( X 1 , X 3 ∣ Y ) , κ ( X 2 ∣ Y ) , κ ( X 4 ∣ Y ) )
+ κ ( κ ( X 1 , X 4 ∣ Y ) , κ ( X 2 ∣ Y ) , κ ( X 3 ∣ Y ) )
+ κ ( κ ( X 2 , X 3 ∣ Y ) , κ ( X 1 ∣ Y ) , κ ( X 4 ∣ Y ) )
+ κ ( κ ( X 2 , X 4 ∣ Y ) , κ ( X 1 ∣ Y ) , κ ( X 3 ∣ Y ) )
+ κ ( κ ( X 3 , X 4 ∣ Y ) , κ ( X 1 ∣ Y ) , κ ( X 2 ∣ Y ) ) } ( partitions of the 2 + 1 + 1 form )
+ κ ( κ ( X 1 ∣ Y ) , κ ( X 2 ∣ Y ) , κ ( X 3 ∣ Y ) , κ ( X 4 ∣ Y ) ) . {\displaystyle {\begin{aligned}&\kappa (X_{1},X_{2},X_{3},X_{4})\\[5pt]={}&\kappa (\kappa (X_{1},X_{2},X_{3},X_{4}\mid Y))\\[5pt]&\left.{\begin{matrix}&{}+\kappa (\kappa (X_{1},X_{2},X_{3}\mid Y),\kappa (X_{4}\mid Y))\\[5pt]&{}+\kappa (\kappa (X_{1},X_{2},X_{4}\mid Y),\kappa (X_{3}\mid Y))\\[5pt]&{}+\kappa (\kappa (X_{1},X_{3},X_{4}\mid Y),\kappa (X_{2}\mid Y))\\[5pt]&{}+\kappa (\kappa (X_{2},X_{3},X_{4}\mid Y),\kappa (X_{1}\mid Y))\end{matrix}}\right\}({\text{partitions of the }}3+1{\text{ form}})\\[5pt]&\left.{\begin{matrix}&{}+\kappa (\kappa (X_{1},X_{2}\mid Y),\kappa (X_{3},X_{4}\mid Y))\\[5pt]&{}+\kappa (\kappa (X_{1},X_{3}\mid Y),\kappa (X_{2},X_{4}\mid Y))\\[5pt]&{}+\kappa (\kappa (X_{1},X_{4}\mid Y),\kappa (X_{2},X_{3}\mid Y))\end{matrix}}\right\}({\text{partitions of the }}2+2{\text{ form}})\\[5pt]&\left.{\begin{matrix}&{}+\kappa (\kappa (X_{1},X_{2}\mid Y),\kappa (X_{3}\mid Y),\kappa (X_{4}\mid Y))\\[5pt]&{}+\kappa (\kappa (X_{1},X_{3}\mid Y),\kappa (X_{2}\mid Y),\kappa (X_{4}\mid Y))\\[5pt]&{}+\kappa (\kappa (X_{1},X_{4}\mid Y),\kappa (X_{2}\mid Y),\kappa (X_{3}\mid Y))\\[5pt]&{}+\kappa (\kappa (X_{2},X_{3}\mid Y),\kappa (X_{1}\mid Y),\kappa (X_{4}\mid Y))\\[5pt]&{}+\kappa (\kappa (X_{2},X_{4}\mid Y),\kappa (X_{1}\mid Y),\kappa (X_{3}\mid Y))\\[5pt]&{}+\kappa (\kappa (X_{3},X_{4}\mid Y),\kappa (X_{1}\mid Y),\kappa (X_{2}\mid Y))\end{matrix}}\right\}({\text{partitions of the }}2+1+1{\text{ form}})\\[5pt]&{\begin{matrix}{}+\kappa (\kappa (X_{1}\mid Y),\kappa (X_{2}\mid Y),\kappa (X_{3}\mid Y),\kappa (X_{4}\mid Y)).\end{matrix}}\end{aligned}}}
Cumulants of compound Poisson random variables Suppose Y has a Poisson distribution with expected value λ, and X is the sum of Y copies of W that are independent of each other and of Y.
X = ∑ y = 1 Y W y . {\displaystyle X=\sum _{y=1}^{Y}W_{y}.}
All of the cumulants of the Poisson distribution are equal to each other, and so in this case are equal to λ. Also recall that if random variables W1, ..., Wm are independent, then the nth cumulant is additive:
κ n ( W 1 + ⋯ + W m ) = κ n ( W 1 ) + ⋯ + κ n ( W m ) . {\displaystyle \kappa _{n}(W_{1}+\cdots +W_{m})=\kappa _{n}(W_{1})+\cdots +\kappa _{n}(W_{m}).}
We will find the 4th cumulant of X. We have:
κ 4 ( X ) =
κ ( X , X , X , X ) =
κ 1 ( κ 4 ( X ∣ Y ) ) + 4 κ ( κ 3 ( X ∣ Y ) , κ 1 ( X ∣ Y ) ) + 3 κ 2 ( κ 2 ( X ∣ Y ) )
+ 6 κ ( κ 2 ( X ∣ Y ) , κ 1 ( X ∣ Y ) , κ 1 ( X ∣ Y ) ) + κ 4 ( κ 1 ( X ∣ Y ) ) =
κ 1 ( Y κ 4 ( W ) ) + 4 κ ( Y κ 3 ( W ) , Y κ 1 ( W ) ) + 3 κ 2 ( Y κ 2 ( W ) )
+ 6 κ ( Y κ 2 ( W ) , Y κ 1 ( W ) , Y κ 1 ( W ) ) + κ 4 ( Y κ 1 ( W ) ) =
κ 4 ( W ) κ 1 ( Y ) + 4 κ 3 ( W ) κ 1 ( W ) κ 2 ( Y ) + 3 κ 2 ( W ) 2 κ 2 ( Y )
+ 6 κ 2 ( W ) κ 1 ( W ) 2 κ 3 ( Y ) + κ 1 ( W ) 4 κ 4 ( Y ) =
κ 4 ( W ) λ + 4 κ 3 ( W ) κ 1 ( W ) λ + 3 κ 2 ( W ) 2 + 6 κ 2 ( W ) κ 1 ( W ) 2 λ + κ 1 ( W ) 4 λ =
λ E ( W 4 ) (the punch line -- see the explanation below). {\displaystyle {\begin{aligned}\kappa _{4}(X)={}&\kappa (X,X,X,X)\\[8pt]={}&\kappa _{1}(\kappa _{4}(X\mid Y))+4\kappa (\kappa _{3}(X\mid Y),\kappa _{1}(X\mid Y))+3\kappa _{2}(\kappa _{2}(X\mid Y))\\&{}+6\kappa (\kappa _{2}(X\mid Y),\kappa _{1}(X\mid Y),\kappa _{1}(X\mid Y))+\kappa _{4}(\kappa _{1}(X\mid Y))\\[8pt]={}&\kappa _{1}(Y\kappa _{4}(W))+4\kappa (Y\kappa _{3}(W),Y\kappa _{1}(W))+3\kappa _{2}(Y\kappa _{2}(W))\\&{}+6\kappa (Y\kappa _{2}(W),Y\kappa _{1}(W),Y\kappa _{1}(W))+\kappa _{4}(Y\kappa _{1}(W))\\[8pt]={}&\kappa _{4}(W)\k