The law of large numbers and a functional equation

• Autorzy: Maciej Sablik
• Języki publikacji: EN

Abstrakty

EN

We deal with the linear functional equation
(E) $g(x) = ∑^r_{i=1} p_i g(c_i x)$,
where g:(0,∞) → (0,∞) is unknown, $(p₁,...,p_r)$ is a probability distribution, and $c_i$'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.

Słowa kluczowe

EN

functional equation; law of large numbers; Jensen equation on curves; bounded solutions; difference equation

Kategorie tematyczne

• 39A10: Difference equations, additive
• 39B22: Equations for real functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1998
• Tom: 68
• Numer: 2
• Strony: 165-175

Daty

wydano

1998

otrzymano

1997-03-04

poprawiono

1997-06-15

Twórcy

autor

Maciej Sablik

• Institute of Mathematics, Silesian University, Bankowa 14, 40 036 Katowice, Poland

Bibliografia

• [1] J. A. Baker, A functional equation from probability theory, Proc. Amer. Math. Soc. 121 (1994), 767-773.
• [2] G. A. Derfel, Probabilistic method for a class of functional-differential equations, Ukrain. Mat. Zh. 41 (10) (1989), 1117-1234 (in Russian).
• [3] W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York, 1961.
• [4] J. Ger and M. Sablik, On Jensen equation on a graph, Zeszyty Naukowe Polit. Śląskiej Ser. Mat.-Fiz. 68 (1993), 41-52.
• [5] W. Jarczyk, On an equation characterizing some probability distribution, talk at the 34th International Symposium on Functional Equations, Wisła-Jawornik, June 1996.
• [6] M. Laczkovich, Non-negative measurable solutions of a difference equation, J. London Math. Soc. (2) 34 (1986), 139-147.
• [7] M. Pycia, A convolution inequality, manuscript.