Infinite Divisibility of Information

We study an information analogue of infinitely divisible probability distributions, where the i.i.d. sum is replaced by joint distribution sequence. A random variable $X$ called informationally if, for any notation="LaTeX">$n\ge 1$ , there exists sequence variables notation="LaTeX">$Z_{1},\ldots,Z_{n}$ that contains same as i.e., injective function notation="LaTeX">$f$ such notation="LaTeX">$X=f(Z_{1},\ldots,Z_{n})$ . While does not exist discrete variable, we show can be divided into arbitrarily many identical pieces with a multiplicative penalty to entropy, is, if remove injectivity requirement on then and satisfying entropy satisfies notation="LaTeX">$H(X)/n\le H(Z_{1})\le 1.59H(X)/n+2.43$ bits. Furthermore, case notation="LaTeX">$X=(Y_{1},\ldots,Y_{m})$ itself sequence, notation="LaTeX">$m\ge 2$ which gap 1.59 notation="LaTeX">$1+5\sqrt {(\log m)/m}$ This means notation="LaTeX">$m$ increases, notation="LaTeX">$(Y_{1},\ldots,Y_{m})$ becomes closer being spectral in uniform manner. regarded Kolmogorov’s theorem. Applications our result include independent component analysis distributed storage secrecy constraint.