How does diversification actually lead to less variance in a portfolio? I'm looking for a formal reason why this is the case. There are a number of explanations I have been able to find, but they make the assumption that stocks are wiener processes - that is they assume that the daily changes in a stock's price are normally distributed. In this model of stocks, diversification leads to less variance in portfolio prices because the average of some normally distributed random variables is always a normal variable with less variance.
For example, if a and b are distributed normally with mean 0 and variance 1, then their average is normally distributed with mean 0 and variance 1/Sqrt(2). If a and b represented daily price differences for stocks, then this means we could reduce the variance of the daily price difference by spreading our investment out among multiple such stocks.
This is essentially what is shown on the Wikipedia page for the theory of diversification.
Changes in stock price are Cauchy Distributed - that is stocks are better modeled as Levy processes. The Cauchy distribution doesn't exhibit the behavior above which is used to justify diversification. Instead, averaging out a number of Cauchy random variables does nothing to reduce the amount of spread in the distribution. Put more formally, if a and b are identically distributed Cauchy random variables, then their average is a Cauchy random variable with the same parameters. In this way, spreading out your investment across many different levy processes will not result in a levy process with a less variation and there is no reason then to diversify.
I might be missing something pretty simple, however it seems kinda strange to me how mischaracterized diversification seems to be in everything I've read. Is there a paper or something which covers the topic better and doesn't assume that stocks are Wiener processes?
As a bit or warning, I'm clearly not a quant. I just happen to know math and was kinda curious about stocks.