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)?

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?

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, then only ones that are polynomial functions of the raw moments $\operatorname E(X^n),$ $n=1,2,3,\ldots$ is 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)?

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)?