Kolmogorov–Nagumo characterization of quasi-arithmetic means

Builds on Quasi-arithmetic mean Functional equation Interval Continuous function Strictly monotone function

Statement

Let $I$ be a real interval. For each integer $n \ge 1$ , let $M_n \colon I^n \to \R$ be a function. Assume that the sequence $(M_n)$ has these four properties for all $n \ge 1$ .

$M_n$ is continuous and strictly increasing in each variable.
$M_n$ is symmetric: its value does not change when its arguments are permuted.
$M_n(x, x, \dots, x) = x$ for all $x \in I$ .
For all $1 \le k \le n$ and $x_1, \dots, x_n \in I$ , with $m = M_k(x_1, \dots, x_k)$ , one has $M_n(x_1, \dots, x_k, x_{k+1}, \dots, x_n) = M_n(m, \dots, m, x_{k+1}, \dots, x_n)$ , where $m$ is repeated $k$ times.

Then there is a continuous strictly monotone function $\varphi \colon I \to \R$ such that, for all $n$ , $M_n$ is the quasi-arithmetic mean generated by $\varphi$ : $M_n(x_1, \dots, x_n) = \varphi^{-1}\left( \frac{1}{n} \sum_{i=1}^{n} \varphi(x_i) \right).$ Conversely, every quasi-arithmetic mean has properties 1 to 4.

Remarks

The converse is a direct check: property 4 holds because replacing $x_1, \dots, x_k$ by $m$ does not change the sum $\sum \varphi(x_i)$ . For the main direction, one fixes two points $a < b$ of $I$ and defines $\varphi$ on the numbers $M_n(a, \dots, a, b, \dots, b)$ with $k$ copies of $b$ out of $n$ as $k/n$ . Properties 1 to 4 show that this is consistent and strictly increasing on a dense set of points, and continuity extends it. The theorem was proved independently by Kolmogorov and by Nagumo in 1930. Strict monotonicity cannot be weakened to monotonicity: $M_n(x_1, \dots, x_n) = \min_i x_i$ is continuous, increasing in each variable and has properties 2 to 4, but it is not a quasi-arithmetic mean.

Going deeper

For a student

The theorem turns a question about means into a functional equation for the generator $\varphi$ , and the solution is unique up to $\varphi \mapsto a \varphi + b$ . It is used to justify the choice of particular means in statistics, economics and information theory. For example, adding the requirement $M_n(x_1 + c, \dots, x_n + c) = M_n(x_1, \dots, x_n) + c$ narrows the generators to $\varphi(x) = x$ and $\varphi(x) = e^{ax}$ up to affine changes. That narrowing is the route from quasi-arithmetic means to Rényi entropies.

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Continuous function concept
Functional equation concept
Interval concept
Quasi-arithmetic mean concept
Strictly monotone function concept
Kolmogorov–Nagumo characterization of quasi-arithmetic means statement

Tag	`9Q5DPYQ5NP`
URI	`m2://9Q5DPYQ5NP`
Kind	statement, theorem
Writing	W0 stub
MSC	26E60, 39B22
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About	Quasi-arithmetic mean
Uses	Quasi-arithmetic mean, Functional equation, Interval, Continuous function, Strictly monotone function