Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T15:15:17.482Z Has data issue: false hasContentIssue false
  • English
  • Français

A Probabilistic Approach to the Convolution Transform

Published online by Cambridge University Press:  20 November 2018

Louis-Paul Rivest
Louis-Paul Rivest*
Affiliation:
Department of Statistics, University of Toronto Toronto, Ontario M5S 1A1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The inversion and the characterization of the convolution transform is derived via the concept of unimodality introduced by Khintchine (1938). This method yields simple and intuitively appealing proofs.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 24 , Issue 2 , 01 June 1981 , pp. 205 - 211
Copyright
Copyright © Canadian Mathematical Society 1981

References

Feller, W. (1959). An Introduction to Probability Theory and its Applications. Volume I. Second edition. John Wiley and Sons.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications. Volume II. Second edition. John Wiley and Sons.Google Scholar
Gnedenko, B. V. and Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, Massachusetts.Google Scholar
Hirschman, I. I. and Widder, D. V. (1955). The Convolution Transform. Princeton University Press.Google Scholar
Karlin, S. (1968). Total Positivity. Stanford University Press.Google Scholar
Khintchine, A. Y. (1938). On Unimodal Distributions. (In Russian). Izv Nauchno-Issled. Inst. Mat. Mekh., Tomsk. Cos. Univ., 2, 1-7.Google Scholar
Levy, P. (1962).Extensions d'un théorème de D. Dugué et M. Girault, Z. Wahrscheinlichkeitstheorie, 1, 159-173.Google Scholar
Olshen, R. A. and Savage, L. J. (1970). A Generalized Unimodality. J. Appl. Prob., 7, 21-34.Google Scholar
Prabhu, N. V. (1965). Stochastic processes. Macmillan, New York.Google Scholar
Rao, C. R. (1973). Linear Statistical Inference and its Applications. Second edition. John Wiley and Sons.Google Scholar
Simmons, G. F. (1972). Differential Equations. McGraw-Hill Book Company.Google Scholar
Widder, D. V. (1971). An Introduction to transform theory. Academic Press.Google Scholar
Williamson, R. T. (1956). Multiply Monotone Functions and Their Laplace Transform, Duke Math. J., 23, 189-207.Google Scholar