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.

Ergodicity in the mean refers to the ensemble mean being the same as the temporal mean, i.e. measuring one process 1000 times will give the same average as a single measurement of 1000 different processes.

One way for a process to not be ergodic in the mean is when there's some sort of barrier, as sibling comments allude to.

Another is if the overall mean value is picked randomly each time the process starts, but is different each time the process runs. So for example personal monthly expenditures are not ergodic in the mean, because some people are born into circumstances that make them wealthy, and they will on average spend more each month than people not born into such good circumstances.

The ensemble average will tend towards people's average spending, while the temporal average will tend towards each individual's spending.

Errr that last sentence should end with “is what makes this non-ergodic.”