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The variance assigns a number to each of certain probability distributions on Borel subsets of $\mathbb R$. It has the properties of

(1) shift-invariance, i.e. if $X$ is a random variable with a probability distribution with a finite second moment and $c$ is a constant then $\operatorname{var}(c+X) = \operatorname{var}(X),$

(2) second-degree homogeneity, i.e. $\operatorname{var}(cX) = c^2 \operatorname{var}(X),$

(3) additivity, i.e. if $X_1,\ldots,X_n$ are independent random variables then $\operatorname{var}(X_1+\cdots+X_n) = \operatorname{var}(X_1) + \cdots + \operatorname{var}(X_n)$.

Among functionals with these three properties, the only ones that are polynomial functions of the raw moments $\operatorname E(X^n),$ $n=1,2,3,\ldots$ are the scalar multiples of the variance.

What of functionals that are not polynomial functions of the raw moments? The square of the mean absolute deviation has the first two properties but not the third. (I think the lack of the third property is why Abraham de Moivre introduced the variance three centuries ago.)

  • If we don't restrict our search to polynomial functions of the raw moments, and also don't restrict it to any kind of functions of the moments, are there some other functionals with these three properties?

  • What if we allow the weakening of the second property by replacing $c^2$ with some other function of $c$ and also don't insist on functions of the raw moments. Can we then have (1) and (3)? Is it only with cumulants that we then have (1) and (3), or are there some other such functions?

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    $\begingroup$ This nice answer of yours may be helpful for further context. $\endgroup$
    – Tobias Fritz
    Commented Aug 12, 2023 at 7:27
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    $\begingroup$ To be precise, we should specify the set of distributions we work with. Is this the set of all distributions with finite expectation and variance? $\endgroup$
    – Fedor Petrov
    Commented Aug 12, 2023 at 19:26
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    $\begingroup$ "Width of the support" satisfies (1) and (3), and also (2) with $c^2$ replaced by $c$. Technically, "variance if compactly supported, infinity otherwise" satisfies (1), (2) and (3), and there are infinitely many similarly constructed examples. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Aug 12, 2023 at 21:00
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    $\begingroup$ The central limit theorem shows that the variance is unique (up to scalars) amongst all statistics obeying (1)-(3) which are also continuous with respect to any topology in which the central limit theorem holds. $\endgroup$
    – Terry Tao
    Commented Aug 13, 2023 at 4:53
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    $\begingroup$ Yes. Your axioms (1)-(3) show that the functional of a random variable $X$ is equal to that of $(X_1+\dots+X_n)/\sqrt{n}$ where $X_1,\dots,X_n$ are copies of $X$ minus its mean. By CLT, this converges to a gaussian with the same variance as $X$. Assuming continuity, this shows that the functional of $X$ is equal to the functional of that gaussian, which is proportional to the variance by (2). $\endgroup$
    – Terry Tao
    Commented Aug 13, 2023 at 18:17

1 Answer 1

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To address your second question, the functional $v_p$ for $p\in(0,2)$ given by the formula $$v_p(X):=-\int_0^\infty\frac{\ln|f_X(t)|}{t^{p+1}}\,dt$$ will have your properties 1 and 3, and also property 2 with $|c|^p$ in place of $c^2$; here $f_X$ is the characteristic function of a random variable $X$ and $\ln0:=-\infty$. Also, $v_p(X)$ will always be $\ge0$. (As in the other examples, we will have $v_p(X)=\infty$ for some random variables $X$.)

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  • $\begingroup$ And the same idea — evaluating the Riemann–Liouville fractional derivative of $\ln|f_X|$ at zero — works for larger values of $p$ as long as $X$ has sufficiently many moments. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Aug 13, 2023 at 20:15
  • $\begingroup$ @MateuszKwaśnicki : Thank you for your comment. $\endgroup$
    – Iosif Pinelis
    Commented Aug 13, 2023 at 21:56
  • $\begingroup$ @MateuszKwaśnicki : Here may be a problem, though: Even if $f_X$ is smooth, $|f_X|$ does not have to differentiable even once. $\endgroup$
    – Iosif Pinelis
    Commented Aug 14, 2023 at 14:58
  • $\begingroup$ At $t=0$, $|f_X(t)|$ is as smooth as $f_X(t)$, I guess. But I did not take time to work out the details, so I may be missing something. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Aug 15, 2023 at 10:34
  • $\begingroup$ @MateuszKwaśnicki : Can you say what exactly you meant by " the Riemann–Liouville fractional derivative of $\ln|f_X|$ at zero"? $\endgroup$
    – Iosif Pinelis
    Commented Aug 15, 2023 at 15:03

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