The Random Variable at the Margin is Gaussian
We justify modeling marginal utility as a Gaussian random variable.
Cramer’s Decomposition Theorem
I recently learned a really cool tidbit called the Cramer’s Decomposition Theorem.
If C is a Gaussian random variable and can be written as the sum of two independent random variables A and B, then A and B must be Gaussian random variables as well.
Recall that the sum of two random variables corresponds to the convolution of their pdfs. This means that if the output of our convolution is Gaussian, then the inputs to the convolution must each be Gaussian as well!
This is a surprisingly strong statement. It is not the trivial statement that given a Gaussian random variable, we are able to find a sum where both summands are Gaussian. Rather, it is the statement that if there are two independent random variables that sum to a Gaussian, then they must both be Gaussian.
Gobbledygook
This is the part where I speculate, in a completely non-rigorous and probably incorrect manner, about why Cramer’s Decomposition justifies a decision to model random variables at the margin with a Gaussian. I use the word margin the same way it is used in economics, which is not to be confused with marginal distributions and the like.
Suppose we have N non-Gaussian i.i.d. random variables for a large N. Then, we know from the CLT that the sample mean is approximately Gaussian. This means the sum of the samples is also approximately Gaussian. We can write it as a sum of two independent random variables, where the first summand is the sum we have now, and the second summand is the marginal sample \(X_\delta\).
\[\sum_{i=1}^{N} X_i = \sum_{i=1}^{N-1} X_i + X_\delta, \qquad X_\delta = X_N.\]We can apply an “approximate” version of Cramer’s Decomposition to conclude that \(X_\delta\) must be approximately Gaussian as well. Despite the fact that we defined all the \(X\)’s to be explicitly non-Gaussian (and completely arbitrary), which is an ostensible contradiction!
I think an intuitive way of interpreting this might be to say that given a big enough N, for all practical intents and purposes, if we are interested in the value of the sum, it suffices to measure just the mean and variance of the marginal sample (which are sufficient statistics for a Gaussian).
Example
Say we are modeling the GDP of a country as the sum of many many i.i.d. business transactions. Then, even though we don’t know anything about the omnipotent algorithm controlling business transactions, it suffices to model the effect of a marginal transaction on the GDP as a Gaussian random variable.