According to E. Lukacs, "Characteristic functions", 2nd ed., we say that F(x) is a stable probability distribution if for all b1,b2>0 and all real c1,c2 we can find b>0 and real c such that:
F( (x-c1)/b1 ) * F( (x-c2)/b2 ) = F( (x-c)/b ),
where * denotes convolution, i.e. convolution of the distribution with itself is again the same distribution.
Is there a description of a generalization of this property for a family of distributions F_A(x), dependent on parameter A, for which we would have:
F_A1( (x-c1)/b1 ) * F_A2( (x-c2)/b2 ) = F_A( (x-c)/b ),
i.e. so that convolution of two distributions would produce another distribution from the same family?
You should check out the work of John Nolan at American University. Distributions of which their independent sum belongs again to the same family is called a stable family. Unfortunately, their pdf 'sand cdf's are not available in a closed form. Numerical routines are needed to evaluate them. The Gaussian family is a stable family and is I believe the only exception with a closed form pdf.
Add the Cauchy and Lévy distributions, which are stable, and have densities.
You can see the books of Zolotorev (One-dimensional stable distributions) and Feller (Probability theory and its applications Vol. 2).
The stable family describes by its characteristic function φ(t) and except the Gaussian, Cauchy and Lévy distributions this family has not closed form for probability density functions and of course distribution functions.
You can refer to the following book.
Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance,
By Gennady Samoradnitsky, M.S. Taqqu.
The stable distribution is important as an attractor. The family parameterised by Nolan is accessible in R using package stabledist.
Your looser concept of "formula stable" could probably apply to a system with Bayesian updating of an uncertain skew parameter. The distribution family being a mixture of the above and likely to be an intractable 2d infinite integral let alone analytic.
EDIT: Bayes with a "conjugate prior" is tractable. Your "family" can be thought of as prior hyper-parameter values.
See "beta-binomial distribution" for a family distributions that can be convolved in the way you want. All such examples will converge to a stable distribution by definition if you add enough iid r.v.s together.
Other such families do exist, e.g. NIG "normal-inverse Gaussian" family is closed under convolution and has a known pdf.
Is there a way to define approximately stable law ?
The sum of two Lognornal distributions may be approximately by another Lognormal.
Thank you all for the answers. My question is not about stable distribution itself. I was wondering if there is developed a similar theory for family/families of functions that would be closed (the family) with respect to convolution. Such problem arises when considering random variables X[K] = sum_{k=1}^K X_k, where X_k are i.i.d rvs. Characteristic function of X[K] = (ch.f. of X[1])^K, so their probability distributions must be also closely related.
look up "infinitely divisible" -- there is a one to one correspondence between infinitely divisible laws and levy processes
http://en.wikipedia.org/wiki/Infinite_divisibility_(probability)
I've gone through infinite divisibility already (it's well described in Lukacs' book), that's not the case here.
1) I am considering a particular kind of probability distributions, which are not infinitely divisible. Perhaps infinitely divisible probability distributions have a property which could be "copied" to my problem, but I haven't seen such one yet.
2) The problem is a kind of "inverse" to infinite divisibility. Infinite divisibility is about decomposition of a random variable into any arbitrary number of other ("unknown") random variables. I do the oposite - I am given a set of "basic" random variables and I am investigating properties of their compositions.
Not true--infinitely divisible laws are r.v. that can always be expressed as a sum of two iid r.v.s from the same family, such a normal of any mean and any variance as these are additive, and gamma of any shape with scale 1 as the shapes are additive, or poissons of arbitrary rates as these are additive. Your inverse problem, as. I understand it, is to find laws whose convolutions belong to the same family. The solution _is_ the class of infinitely divisible laws.
Adam, how are you? You have such an interest taste for perfect work and I will like you to keep it up. Your question in the first place did not capture essential components you need to help you achieve the interesting objective you will like to achieve. Essential details like the infinite divisibility probability and even the relation your problem has with "inverse" to infinite divisibility though the last paragraph of the question gives a hint to that effective.
In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of random variables is the convolution of their corresponding probability mass functions or probability density functions respectively. You can read from the link below to to see how you will go about this "http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter7.pdf".
You can also read Limit theorems and infinitely divisible distributions. Part I.
by Peter Major from The Mathematical Institute of the Hungarian Academy of Sciences to know how to combine the so called infinite divisibility distribution and convolution and even the stable distribution. "In probability theory, a random variable is said to be stable (or to have a stable distribution) if it has the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters".
As you can see there are a whole lot of reading one will have to do in order to get the kind of answer you need or there is for your problem.
Below are few reference for your consideration. I am very interested in the work so keep me updated. Good luck.
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
Mathematical Reviews (MathSciNet): MR898871
Zentralblatt MATH: 0617.26001
Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions. Springer, New York.
Mathematical Reviews (MathSciNet): MR1224674
Zentralblatt MATH: 0756.60015
Breiman, L. (1965). On some limit theorems similar to the arc-sine law. Theory Prob. Appl. 10, 323--331.
Mathematical Reviews (MathSciNet): MR184274
Chistyakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Prob. Appl. 9, 640--648.
Mathematical Reviews (MathSciNet): MR170394
Chover, J., Ney, P. and Wainger, S. (1973). Degeneracy properties of subcritical branching processes. Ann. Prob. 1, 663--673.
Mathematical Reviews (MathSciNet): MR348852
Digital Object Identifier: doi:10.1214/aop/1176996893
Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529--557.
Mathematical Reviews (MathSciNet): MR847385
Digital Object Identifier: doi:10.1007/BF00344720
Zentralblatt MATH: 0577.60019
Cline, D. B. H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A 43, 347--365.
Mathematical Reviews (MathSciNet): MR904394
Embrechts, P. and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29, 243--256.
Mathematical Reviews (MathSciNet): MR566289
Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stoch. Process. Appl. 13, 263--278.
Mathematical Reviews (MathSciNet): MR671036
Digital Object Identifier: doi:10.1016/0304-4149(82)90013-8
Zentralblatt MATH: 0487.60016
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335--347.
Mathematical Reviews (MathSciNet): MR547833
Goldie, C. M. and Klüppelberg, C. (1998). Subexponential distributions. In A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions, eds R. Adler, R. Feldman and M. S. Taqqu, Birkhäuser, Boston, MA, pp. 435--460.
Mathematical Reviews (MathSciNet): MR1652293
Zentralblatt MATH: 0923.62021
Mantegna, R. N. and Stanley, H. E. (1994). Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. Phys. Rev. Lett. 73, 2946--2949.
Mathematical Reviews (MathSciNet): MR1303317
Digital Object Identifier: doi:10.1103/PhysRevLett.73.2946
Pakes, A. G. (2003). Investigating the structure of truncated Lévy-stable laws. In Science and Statistics: A Festschrift for Terry Speed (Lecture Notes Monogr. Ser. 40), ed. D. R. Goldstein, Institute of Mathematical Statistics, Beachwood, OH, pp. 49--78.
Mathematical Reviews (MathSciNet): MR2004332
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.
Mathematical Reviews (MathSciNet): MR1680267
Zentralblatt MATH: 0940.60005
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Mathematical Reviews (MathSciNet): MR1739520
Sigman, K. (1999). Editorial introduction (Special issue on queues with heavy-tailed distributions). Queueing Systems 33, 1--3.
Mathematical Reviews (MathSciNet): MR1748646
Digital Object Identifier: doi:10.1023/A:1019180230133
Zentralblatt MATH: 0997.60118
Stam, A. J. (1973). Regular variation of the tail of a subordinated distribution. Adv. Appl. Prob. 5, 308--327.
Mathematical Reviews (MathSciNet): MR339353
Teugels, J. L. (1975). The class of subexponential distributions. Ann. Prob. 3, 1000--1011.
Mathematical Reviews (MathSciNet): MR391222
Digital Object Identifier: doi:10.1214/aop/1176996225
Willekens, E. (1987). Subexponentiality on the real line. Res. Rep., Katholieke Universiteit Leuven.
Yakymiv, A. L. (2002). On the asymptotics of the density of an infinitely divisible distribution at infinity. Theory Prob. Appl. 47, 114--122.
Mathematical Reviews (MathSciNet): MR1978697
Cai, J. and Tang, Q. (2004). On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Prob. 41, 117--130.
Mathematical Reviews (MathSciNet): MR2036276
Digital Object Identifier: doi:10.1239/jap/1077134672
Foss, S. and Zachary, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Prob. 13, 37--53.
Mathematical Reviews (MathSciNet): MR1951993
Digital Object Identifier: doi:10.1214/aoap/1042765662
Project Euclid: euclid.aoap/1042765662
Zentralblatt MATH: 1045.60039
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132--141.
Mathematical Reviews (MathSciNet): MR929511
Sgibnev, M. S. (1990). Asymptotics of infinitely divisible distributions in $\bf R$. Siberian Math. J. 31, 115--119.
Mathematical Reviews (MathSciNet): MR1046818
I think you can review weighting distributions and form invariance under different weights
Chover, J., Ney, P. and Wainger, S. (1973). Degeneracy properties of subcritical branching processes. Ann. Prob. 1, 663--673.
Second author -- my thesis advisor --
I think the family of probability distributions you are looking for are the so called pseudo-stable distributions mentioned on page 191 of the book "Chance and stability-Stable distributions and their applications" by Uchaikin V.V. and Zolotarev, V.M.
I think the family of probability distributions you are looking for are the so called pseudo-stable distributions mentioned on page 191 of the book "Chance and stability-Stable distributions and their applications" by Uchaikin V.V. and Zolotarev, V.M.
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Probability