I think the question I recently posted was ambiguous. So I refined it.
Is the product of the same two independent normal distribution also a normal distribution?
My question is about the distribution of returns when I invest an ETF multiple years consequently.
For example, I can assume that annual gains of SPY(S&P500 ETF) follow a normal distribution e.g. $N(10\%, 20\%^2)$.
If I invest 10 years on SPY, what kind of cumulative gain distribution I can see?
I could simulate it by calculating $(1 + N(10\%, 20\%^2))^{10}$ or $N(110\%, 20\%^2)^{10}$ multiple times and plotted histogram in a graph.
It was not a normal distribution which I expected. So I'm curious about the name of the distribution. Here is the graph I plotted. (1, 2, 5, and 10 multiplications) Distributions of multiplications of 1, 2, 5, and 10 the same normal distributions
(ADDED) one more question, If I apply a log function to the result values, will it be a normal distribution? It looks like that. I attached a log graph. A log graph
