We deal with the linear functional equation (E) , where g:(0,∞) → (0,∞) is unknown, is a probability distribution, and ’s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli’s Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.
@article{bwmeta1.element.bwnjournal-article-apmv68z2p165bwm,
author = {Maciej Sablik},
title = {The law of large numbers and a functional equation},
journal = {Annales Polonici Mathematici},
volume = {69},
year = {1998},
pages = {165-175},
zbl = {0911.39010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv68z2p165bwm}
}
Maciej Sablik. The law of large numbers and a functional equation. Annales Polonici Mathematici, Tome 69 (1998) pp. 165-175. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv68z2p165bwm/
[000] [1] J. A. Baker, A functional equation from probability theory, Proc. Amer. Math. Soc. 121 (1994), 767-773. | Zbl 0808.39010
[001] [2] G. A. Derfel, Probabilistic method for a class of functional-differential equations, Ukrain. Mat. Zh. 41 (10) (1989), 1117-1234 (in Russian).
[002] [3] W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York, 1961. | Zbl 0039.13201
[003] [4] J. Ger and M. Sablik, On Jensen equation on a graph, Zeszyty Naukowe Polit. Śląskiej Ser. Mat.-Fiz. 68 (1993), 41-52.
[004] [5] W. Jarczyk, On an equation characterizing some probability distribution, talk at the 34th International Symposium on Functional Equations, Wisła-Jawornik, June 1996. | Zbl 0872.39010
[005] [6] M. Laczkovich, Non-negative measurable solutions of a difference equation, J. London Math. Soc. (2) 34 (1986), 139-147.
[006] [7] M. Pycia, A convolution inequality, manuscript.