Geodesic
The law of large numbers and a functional equation
Annales Polonici Mathematici, Tome 68 (1998) no. 2, pp. 165-175

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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.
DOI : 10.4064/ap-68-2-165-175
Keywords: functional equation, law of large numbers, Jensen equation on curves, bounded solutions, difference equation

Maciej Sablik 1

1 
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Maciej Sablik. The law of large numbers and a functional equation. Annales Polonici Mathematici, Tome 68 (1998) no. 2, pp. 165-175. doi: 10.4064/ap-68-2-165-175

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