Recently I was reading articles and books about MCMC, and realized that many materials were not taught in my graduate study. To this end, I decide to make a summary of such content, to assist readers and myself to gain deeper understanding of MCMC in the future. I hope to make this topic a series, although I cannot guarantee its completion. This article is the first one in this hypothetical series, and we are going to introduce an important concept, the geometric ergodicity.

The rumor says that Normal distribution is everything.
It will take a long long time to talk about the Normal distribution thoroughly. However, today I will focus on a (seemingly) simple question, as is stated below:
If $X$ and $Y$ are univariate Normal random variables, will $X+Y$ also be Normal?
What’s your reaction towards this question? Well, at least for me, when I saw it I said “Oh, it’s stupid. Absolutely it is Normal.