I know Intermediate Probability Theory and Statistics (distributions, convergence concepts, characteristic functions, etc.) and am pretty good at these. I would love to know more about Probability Theory. Unfortunately, although I am aware that Measure Theory is an integral tool for advanced Probability Theory, I have not taken a full course of Measure Theory. However, I intend to study Measure Theory as soon as possible.
I am listing some topics in Probability Theory that seem interesting to me. I would be obliged if you could tell me whether I can start (at least) on these topics without Measure Theoretic knowledge. I would be grateful if you can inform me which topics exactly I can do without Measure theory.
- Branching Processes
- Ergodic Theory
- Martingales
- Percolation Theory
- Random Graphs
- Random Walks on Graphs
- Non-Commutative Probability
- Theory of Extremes and Point Processes
- Theory of Large Deviations
- Extreme Value Theory
- Brownian Motion
- Diffusions
Thank you for your time!! It would be a great confidence booster to know that I can start studying some of these subjects without Measure Theoretic knowledge. I am in any way going to learn Measure Theoretic Probability within a few months.
To a large extent, it depends on whether you are interested in learning important properties and facts or whether you want to do proofs and eventually original research, whether you are interested primarily in theory or in applications.
Many of the properties and much of what is done in applications is accessible without measure theory and many important applications are done (sometimes beyond what is known for sure theoretically) by simulation.
For example, while it is true that ergodic theory is mainly a topic in measure theory, a lot of the applications of Markov Chain Monte Carlo (MCMC) assume ergodicity without theoretical backing. (On a case by case basis, numerical and graphical methods are used to verify apparent convergence over the long run, and one hopes for the best.) Of course, this is an open area of research, and theoretical results will almost surely be based on measure theory.
Many concrete applications of Brownian Motion are done by simulation, and some principles can be demonstrated by simulation. Some of the more intricate properties cannot even be properly understood without measure theory.
Maybe you are asking if you can meaningfully explore any of these topics without measure theory, in order to confirm whether you are really interested in them. To a large extent, the answer is yes for most of these topics.
The rest of the answer is that you should learn measure theory as soon as possible, because this list alone is a good indication you would find the topic worthwhile. Why not get started on measure theory ASAP, concurrently explore what you can from your list. Each endeavor will reinforce the other.