What is the definition of the geometric mean of a random variable?

Question

I haven't found any definition online although searching for hours:

So here is the thing: The geometric mean(GM) of an (iid) sample drawn from some random variable is given by:

$$\text{GM}(X_1,...,X_n)=\sqrt[n]{X_1\cdot ...\cdot X_n}$$

and the expected geometric mean for a sample size $n$ is given by

$$\mathbb E(X_1^{1/n}\cdot ... \cdot X_n^{1/n})=\left[E\left(X_1^{1/n}\right)\right]^n.$$

So we can see that the (expected) geometric mean is dependent on the sample size $n$.

However, some sources like wikipedia talk about geometric moments e.g. here:

https://en.wikipedia.org/wiki/Log-normal_distribution#Geometric_moments

However, there is no definition of it; Now I am curious what the official definition of the geometric mean/geometric first moment actually is; Is it:

$$\lim\limits_{n \to \infty}\left[E\left(X_1^{1/n}\right)\right]^n$$

or does it have some other definition?

Answer 1

You can define the geometric mean of a strictly positive random variable readily enough; an easy method would be to take:

$$GM(X)=\exp(E[\log(X)])$$

For a discrete variable you can write the geometric mean as $\prod_i x_i^{p_i}$ where $p_i={p(X=x_i)}$.