The law of large numbers and a functional equation

Volume 68 / 1998

Maciej Sablik Annales Polonici Mathematici 68 (1998), 165-175 DOI: 10.4064/ap-68-2-165-175

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.

Authors

  • Maciej Sablik

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