# The law of large numbers and a functional equation

• Volume: 68, Issue: 2, page 165-175
• ISSN: 0066-2216

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## Abstract

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We deal with the linear functional equation (E) $g\left(x\right)={\sum }_{i=1}^{r}{p}_{i}g\left({c}_{i}x\right)$, where g:(0,∞) → (0,∞) is unknown, $\left(p₁,...,{p}_{r}\right)$ 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.

## How to cite

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Maciej Sablik. "The law of large numbers and a functional equation." Annales Polonici Mathematici 68.2 (1998): 165-175. <http://eudml.org/doc/270172>.

@article{MaciejSablik1998,
abstract = {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.},
author = {Maciej Sablik},
journal = {Annales Polonici Mathematici},
keywords = {functional equation; law of large numbers; Jensen equation on curves; bounded solutions; difference equation; bounded solution; iterative functional equation},
language = {eng},
number = {2},
pages = {165-175},
title = {The law of large numbers and a functional equation},
url = {http://eudml.org/doc/270172},
volume = {68},
year = {1998},
}

TY - JOUR
AU - Maciej Sablik
TI - The law of large numbers and a functional equation
JO - Annales Polonici Mathematici
PY - 1998
VL - 68
IS - 2
SP - 165
EP - 175
AB - 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.
LA - eng
KW - functional equation; law of large numbers; Jensen equation on curves; bounded solutions; difference equation; bounded solution; iterative functional equation
UR - http://eudml.org/doc/270172
ER -

## References

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1. [1] J. A. Baker, A functional equation from probability theory, Proc. Amer. Math. Soc. 121 (1994), 767-773. Zbl0808.39010
2. [2] G. A. Derfel, Probabilistic method for a class of functional-differential equations, Ukrain. Mat. Zh. 41 (10) (1989), 1117-1234 (in Russian).
3. [3] W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York, 1961. Zbl0039.13201
4. [4] J. Ger and M. Sablik, On Jensen equation on a graph, Zeszyty Naukowe Polit. Śląskiej Ser. Mat.-Fiz. 68 (1993), 41-52.
5. [5] W. Jarczyk, On an equation characterizing some probability distribution, talk at the 34th International Symposium on Functional Equations, Wisła-Jawornik, June 1996. Zbl0872.39010
6. [6] M. Laczkovich, Non-negative measurable solutions of a difference equation, J. London Math. Soc. (2) 34 (1986), 139-147.
7. [7] M. Pycia, A convolution inequality, manuscript.

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