Divisibility, decomposability, stability

Ways of slicing randomness

🏗 all of these are about sums; but presumably we can construct this over other algebraic structures of distributions, e.g. max-stable processes.

For now, some handy definition disambiguation.

Infinitely divisible

The Lévy process quality.

A probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of any arbitrary natural number of independent and identically distributed random variables. i.e. The distribution \(F\) is infintely divisible if, for every positive integer \(n\), there exist \(n\) i.i.d. RVs whose sum

\[X_1 + \dots + X_n = S_n \sim F\]

Decomposable

The distribution of \(X\) is decomposable if there are 2 or more non-constant RVs, not necessarily in the same family, whose sum is equal in distribution. Not a strong property, but the cases where an RV fails to possess even this are curious. 🏗

Self-decomposable

Decomposable, but the components must be in the same family. (so presumably this is about what we have declared to be the family.)

Stable

A distribution or a random variable is said to be stable if a linear combination of two independent copies of a random sample has the same distribution, up to location and scale parameters. (So a stronger property than the Lévy property which allows other parameters than location and scale.)

This induces at least 2 families of infinitely divisible distributions, the discrete and continuous stable family. See (van Harn and Steutel 1993).

A well-known distribution construction: the stable distribution class.

For continuous-valued continuous/discrete indexed stochastic process, \(X(t)\), \(\alpha\)-stability implies that the marginal law of the value of the process at certain times satisfies a stability equation

\[X(a) \simeq W^{1/\alpha}X(b),\] for \(0 < a < b\), \(\alpha> 0\) and \(W\sim \operatorname{Unif}([0,1])\perp X\).

The marginal distributions of such processes are those of the \(\alpha\)-stable processes. For \(\alpha=2\) we have Gaussians and for \(\alpha=1\), the Cauchy law.

Induced processes

All the divisible distributions induce an obvious process - the decomposition of the process into independent increment Markov chains; which is to say, each divisible family induces an associated Lévy process; i.e. the Gaussian distribution induces a Gauss Markov process, and the gamma distribution a gamma process.

Cahoy, Dexter O., Vladimir V. Uchaikin, and Wojbor A. Woyczynski. 2010. “Parameter Estimation for Fractional Poisson Processes.” Journal of Statistical Planning and Inference 140 (11): 3106–20. https://doi.org/10.1016/j.jspi.2010.04.016.

Harn, K. van, and F. W. Steutel. 1993. “Stability Equations for Processes with Stationary Independent Increments Using Branching Processes and Poisson Mixtures.” Stochastic Processes and Their Applications 45 (2): 209–30. https://doi.org/10.1016/0304-4149(93)90070-K.

Harn, K. van, F. W. Steutel, and W. Vervaat. 1982. “Self-Decomposable Discrete Distributions and Branching Processes.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 61 (1): 97–118. https://doi.org/10.1007/BF00537228.

Houdré, Christian. 2002. “Remarks on Deviation Inequalities for Functions of Infinitely Divisible Random Vectors.” The Annals of Probability 30 (3): 1223–37. https://doi.org/10.1214/aop/1029867126.

Janson, Svante. 2011. “Stable Distributions,” December. http://arxiv.org/abs/1112.0220.

Steutel, F. W., and K. van Harn. 1979. “Discrete Analogues of Self-Decomposability and Stability.” The Annals of Probability 7 (5): 893–99. https://doi.org/10.1214/aop/1176994950.