5

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?

Michael Hardy
  • 11,922
  • 11
  • 81
  • 119
  • Partial answer to that second bullet-pointed question: The third central moment satisfies (1) and (3) and is a function of the first three raw moments. So maybe the question should be: How broad is the class of functions that satisfy (1) and (3)? – Michael Hardy Aug 11 '23 at 23:19
  • 1
    This nice answer of yours may be helpful for further context. – Tobias Fritz Aug 12 '23 at 07:27
  • 1
    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? – Fedor Petrov Aug 12 '23 at 19:26
  • 2
    "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. – Mateusz Kwaśnicki Aug 12 '23 at 21:00
  • @FedorPetrov : If a distribution has finite variance, then it is redundant to mention finite mean. I did mention finite second moment, although not until I got into the first item in the list. – Michael Hardy Aug 12 '23 at 21:59
  • 2
    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. – Terry Tao Aug 13 '23 at 04:53
  • @TerryTao : I'm not sure what you mean by "any topology in which the central limit theorem holds." Do you mean the kind of convergence that the conclusion of the theorem asserts? Somehow it seems as if you're referring to something else. – Michael Hardy Aug 13 '23 at 16:58
  • 2
    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). – Terry Tao Aug 13 '23 at 18:17
  • Also, cumulants are the canonical answer to your second question. https://en.wikipedia.org/wiki/Cumulant – Terry Tao Aug 13 '23 at 18:18
  • @TerryTao : Your remark causes me to realize my second question is not exactly what I meant. I will revise it. – Michael Hardy Aug 13 '23 at 19:10
  • @TerryTao : Mateusz Kwaśnicki's comment shows that cumulants are not the only answer to that last question when we restrict attention to distributions with bounded support. – Michael Hardy Aug 13 '23 at 19:13
  • Well, I guess "width of the support" is some kind of infinite-order cumulant. – Mateusz Kwaśnicki Aug 13 '23 at 20:10
  • 1
    @MateuszKwaśnicki : But it has only first-degree homogeneity. – Michael Hardy Aug 14 '23 at 18:52

1 Answers1

4

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 $X$.)

Iosif Pinelis
  • 116,648