Subindependence

In probability theory and statistics, subindependence is a weak form of independence.

Two random variables X and Y are said to be subindependent if the characteristic function of their sum is equal to the product of their marginal characteristic functions. Symbolically:

$${\phi }_{X+Y}(t)={\phi }_X(t)\cdot {\phi }_Y(t).$$

This is a weakening of the concept of independence of random variables, i.e. if two random variables are independent then they are subindependent, but not conversely. If two random variables are subindependent, and if their covariance exists, then they are uncorrelated.

Subindependence has some peculiar properties: for example, there exist random variables Template:Mvar and Template:Mvar that are subindependent, but Template:Mvar and Template:Math are not subindependent when Template:Math and therefore Template:Mvar and Template:Mvar are not independent.

One instance of subindependence is when a random variable Template:Mvar is Cauchy with location Template:Math and scale Template:Mvar and another random variable Template:Math, the antithesis of independence. Then Template:Math is also Cauchy but with scale Template:Math. The characteristic function of either Template:Mvar or Template:Mvar in Template:Mvar is then Template:Math, and the characteristic function of Template:Math is Template:Math.

References

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Further reading

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