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Concept
Law of total cumulance
Résumé
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
where
κ(X1, ..., Xn) is the joint cumulant of n random variables X1, ..., Xn, and
the sum is over all partitions of the set { 1, ..., n } of indices, and
"B ∈ pi;" means B runs through the whole list of "blocks" of the partition pi, 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.
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.
For general 4th-order cumulants, the rule gives a sum of 15 terms, as follows:
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.
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:
We will find the 4th cumulant of X. We have:
We recognize the last sum as the sum over all partitions of the set { 1, 2, 3, 4 }, of the product over all blocks of the partition, of cumulants of W of order equal to the size of the block. That is precisely the 4th raw moment of W (see cumulant for a more leisurely discussion of this fact). Hence the cumulants of X are the moments of W multiplied by λ.
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