Indecomposable distribution

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In probability theory, an indecomposable distribution is any probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables.

Examples

The simplest examples are Bernoulli distributions: if

:[X = \left\1 & \mathrm\ \mathrm & p, \\0 & \mathrm\ \mathrm & 1-p,\end\right.]

then the probability distribution of X is indecomposable.

Suppose a + b + c = 1, a, b, c ≥ 0, and

:[X = \left\2 & \mathrm\ \mathrm\ a, \\1 & \mathrm\ \mathrm\ b, \\0 & \mathrm\ \mathrm\ c.\end\right.]

This probability distribution is decomposable if

:[\sqrt + \sqrt \le 1 \ ]

and otherwise indecomposable. To see, this, suppose U and V are independent random variables and U + V has this probability distribution. Then we must have

:[\beginU = \left\1 & \mathrm\ \mathrm & p, \\0 & \mathrm\ \mathrm & 1 - p,\end\right.& \mbox &V = \left\1 & \mathrm\ \mathrm & 1, \\0 & \mathrm\ \mathrm & 1 - q,\end\right.\end]

for some p, q ∈ [0, 1]. It follows that

:[a = pq, \, ]

:[c = (1-p)(1-q), \, ]

:[b = 1 - a - c. \, ]

This system of two quadratic equations in two variables p and q has a solution (p, q) ∈ [0, 1]² if and only if

:[\sqrt + \sqrt \le 1. \ ]

Thus, for example, the discrete uniform distribution on the set is indecomposable, but the binomial distribution assigneing respective probabilities 1/4, 1/2, 1/4 is decomposable.

An aboslutely continous indecomposable distribution. It can be shown that the distribution whose density function is

:[f(x) = } x^2 e^]

is indecomposable.

The uniform distribution on the interval [0, 1] is decomposable, since it is the probability distribution of the random variable

:[ \sum_^\infty , ]

where the independent random variables X_n are each equal to 0 or 1 with equal probabilities.

This example shows that a random variable whose probability distribution is infinitely divisible can also be a sum of indecomposable distributions. Suppose a random variable Y has a geometric distribution

:[\Pr(Y = y) = (1-p)^n p\, ]

on . For any positive integer k, there is a sequence of negative-binomially distributed random variables Y_j, j = 1, ..., k, such that Y₁ + ... + Y_k has this geometric distibution. Therefore, this distribution is infinitely divisible. But now let D_n be the nth binary digit of Y, for n ≥ 0. Then the Ds are independent and

:[ Y = \sum_^\infty , ]

and each term in this sum is indecomposable.

References

Lukacs, Eugene, Characteristic Functions, New York, Hafner Publishing Company, 1970.

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