Papers › Infinite Divisibility of Information
Infinite Divisibility of Information
Cheuk Ting Li
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We study an information analogue of infinitely divisible probability distributions, where the i.i.d. sum is replaced by the joint distribution of an i.i.d. sequence. A random variable X is called informationally infinitely divisible if, for any n≥1, there exists an i.i.d. sequence of random variables Z₁,…,Zₙ that contains the same information as X, i.e., there exists an injective function f such that X=f(Z₁,…,Zₙ). While there does not exist informationally infinitely divisible discrete random variable, we show that any discrete random variable X has a bounded multiplicative gap to infinite divisibility, that is, if we remove the injectivity requirement on f, then there exists i.i.d. Z₁,…,Zₙ and f satisfying X=f(Z₁,…,Zₙ), and the entropy satisfies H(X)/n≤H(Z₁)≤1.59H(X)/n+2.43. We also study a new class of discrete probability distributions, called spectral infinitely divisible distributions, where we can remove the multiplicative gap 1.59. Furthermore, we study the case where X=(Y₁,…,Yₘ) is itself an i.i.d. sequence, m≥2, for which the multiplicative gap 1.59 can be replaced by 1+5√((logm)/m). This means that as m increases, (Y₁,…,Yₘ) becomes closer to being spectral infinitely divisible in a uniform manner. This can be regarded as an information analogue of Kolmogorov's uniform theorem. Applications of our result include independent component analysis, distributed storage with a secrecy constraint, and distributed random number generation.
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