My little opinion: Geometric Mean is some how related to the product of random variables, in which can be thought as the sum of the log random variables and then having the exponentiation. While the Central Limit Theorem states that you may approximate the arithmetic mean by normal distribution, by the same analogy you may argue that the geometric mean can be approximated by the log-normal distribution. Of course, if you take the log of the geometric mean, then it is equivalent to the

arithmetic mean. But of course which distribution is better for approximation depends on the nature of your data.