The Annals of Probability

Nested classes of $C$-decomposable laws

John Bunge

Full-text: Open access

Abstract

A random variable $X$ is C-decomposable if $X =_D cX + Y_c$ for all $c$ in $C$, where $_c$ is a random variable independent of $X$ and $C$ is a closed multiplicative subsemigroup of [0, 1]. $X$ is self-decomposable if $C = [0, 1]$ . Extending an idea of Urbanik in the self-decomposable case, we define a decreasing sequence of subclasses of the class of $C$-decomposable laws, for any $C$. We give a structural representation for laws in these classes, and we show that laws in the limiting subclass are infinitely divisible. We also construct noninfinitely divisible examples, some of which are continuous singular.

Article information

Source
Ann. Probab., Volume 25, Number 1 (1997), 215-229.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404286

Digital Object Identifier
doi:10.1214/aop/1024404286

Mathematical Reviews number (MathSciNet)
MR1428507

Zentralblatt MATH identifier
0873.60005

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 60F05: Central limit and other weak theorems

Keywords
Class $L$ distribution decomposability semigroup infinite Bernoulli convolution infinitely divisible measure normed sum self-decomposable measure

Citation

Bunge, John. Nested classes of $C$-decomposable laws. Ann. Probab. 25 (1997), no. 1, 215--229. doi:10.1214/aop/1024404286. https://projecteuclid.org/euclid.aop/1024404286


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References

  • Chung, K. L. (1974). A Course in Probability Theory. Academic Press, New York.
  • Grincevi cius, A. (1974). On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines. Theory Probab. Appl. 19 163-168.
  • Ilinskii, A. (1978). c-Decomposability of characteristic functions. Lithuanian Math. J. 18 481- 485.
  • Jurek,(1992). Operator exponents of probability measures and Lie semigroups. Ann. Probab. 20 1053-1062.
  • Jurek,and Mason, J. (1993). Operator-Limit Distributions in Probability Theory. Wiley, New York.
  • Linnik, J. and Ostrovski i, I. (1977). Decomposition of Random Variables and Vectors (Transl. Math. Monograph 48). Amer. Math. Soc., Providence, RI.
  • Lo eve, M. (1945). Nouvelles classes de lois limites. Bull. Soc. Math. France 73 107-126. (In French.)
  • Lo eve, M. (1963). Probability Theory, 3rd ed. Van Nostrand, Princeton.
  • Lukacs, E. (1970). Characteristic Functions, 2nd ed. Charles Griffin, London.
  • Mi seikis, F. (1972). On certain classes of limit distributions. Litovsk. Mat. Sb. 12 133-152. (In Russian.)
  • Mi seikis, F. (1976). Interrelationship between certain classes of limit distributions. Lithuanian. Math. J. 15 243-246.
  • Mi seikis, F. (1983). Limit distributions of normalized partial sums of a sequence of infinitedimensional random elements. Lithuanian Math. J. 23 78-86.
  • Niedbalska, T. (1978). An example of the decomposability semigroup. Colloq. Math. 39 137-139.
  • Niedbalska-Rajba, T. (1981). On decomposability semigroups on the real line. Colloq. Math. 44 347-358.
  • Sato, K. (1980). Class L of multivariate distributions and its subclasses. J. Multivariate Anal. 10 207-232.
  • Sato, K. and Yamazato, M. (1985). Completely operator-selfdecomposable distributions and operator-stable distributions. Nagoya Math. J. 97 71-94.
  • Siebert, E. (1991). Strongly operator-decomposable probability measures on separable Banach spaces. Math. Nachr. 154 315-326.
  • Siebert, E. (1992). Operator-decomposability of Gaussian measures on separable Banach spaces. J. Theoret. Probab. 5 333-347.
  • Spanier, J. and Oldham, K. (1987). An Atlas of Functions. Hemisphere, New York.
  • Urbanik, K. (1972). L´evy's probability measures on Euclidean spaces. Stud. Math. 44 119-148.
  • Urbanik, K. (1973). Limit laws for sequences of normed sums satisfying some stability conditions. In Multivariate Analysis 3 (P. Krishnaiah, ed.) 225-237. Academic Press, New York.
  • Urbanik, K. (1976). Some examples of decomposability semigroups. Bull. Polish Acad. Sci. Math. 24 915-918.
  • Wintner, A. (1947). The Fourier Transforms of Probability Distributions. Edwards Brothers, Ann Arbor.
  • Wolfe, S. (1983). Continuity properties of decomposable probability measures on Euclidean spaces. J. Multivariate Anal. 13 534-538.
  • Zakusilo, O. (1976). On classes of limit distributions in a summation scheme. Theory Probab. Math. Statist. 12 44-48.
  • Zakusilo, O. (1977). Some properties of random vectors of the form 0 Ai i. Theory Probab. Math. Statist. 13 62-64.
  • Zakusilo, O. (1978). Some properties of the class Lc of limit distributions. Theory Probab. Math. Statist. 15 67-72.