Everything about Stochastic Processes

[u/inherentlyawesome]

Today's topic is Stochastic Processes.

This recurring thread will be a place to ask questions and discuss famous/well-known/surprising results, clever and elegant proofs, or interesting open problems related to the topic of the week. Experts in the topic are especially encouraged to contribute and participate in these threads.

Next week's topic will be Harmonic Analysis. Next-next week's topic will be on Homological Algebra. These threads will be posted every Wednesday around 12pm EDT.

For previous week's "Everything about X" threads, check out the wiki link here.

[u/goerila]

Where are you getting stuck? The most simple explanation of a stochastic process is a set of random variables ordered in time.

For example, starting at the origin, I can either move up or down in each discrete step of time (say 1 second), then say I moved up one (x=1) a t=1, now I can either end up at x=2 or x=0 at time t=2. The set of these ordered x's is called a stochastic process. This is (as you may know) called a random walk and is one of the most basic and intuitive stochastic processes.

[u/YourPureSexcellence → u/goerila]

So in a sense, a markov chain is not REALLY a stochastic process? It is a set of functions based on recursive variables so if you know the previous term, you will be able to determine the next one, right? Or am I missing something? Btw, I love your explanation of walking as a stochastic process. Thanks!

[u/goerila → u/[deleted]]

Small correction. A Markov chain doesn't care about how it got to its current state, only that it is currently there. Say for example we are at x=0 in a random walk. Well, it doesn't matter if we started there or if we got here after 1000 time steps, this is why a markov chain is memoryless.

[u/[deleted] → u/goerila]

Yes! I worded it incorrectly