Deriving the Kelly Criterion to Maximise Profits

A word that is good to know here is ergodic [0]. Which I must admit to not really understanding although it is something like the average system behaviour being equivalent to a typical point's behaviour. If a process is non-ergodic then E[X] is usually not as helpful as it seems in formulating a strategy.

[0] https://en.wikipedia.org/wiki/Ergodic_process

Some math/finance nerds made a whole YouTube channel about ergodicity, which I've been really enjoying: https://youtu.be/VCb2AMN87cg

Nassim Taleb also talks about this quite a lot: https://youtu.be/91IOwS0gf3g

TL;DR: while a single investment may be ergodic, portfolio management (the math behind weighting successive and concurrent investments/bets) is not, as it has a strong dependence on all prior states.

this comment may be confusing and I doubt this will help much but:

Ergodicity is less about memorylessness and more about the constraints on transitions into this or that state. A system is ergodic if "anything that can be an outcome, eventually will happen".

An example that may be useful to aid in understanding… Casinos are non ergodic.

A million players each placing a single bet will have an expectation of losing the house edge.

A single player placing a million bets has an expectation of $0.

The fact that the aggregate and the single entity Experience different expectations despite both placing a million bets is what makes this ergodic.

An illustrative example to explain ergodicity. Consider the following game. Players start with $100. At every turn, a fair coin is flipped. If tails, the amount of player's money is increased by 50%. If heads, the amount of player's money is decreased by 40%. To play or not to play, that is the question.