If I am asked to find the limiting probability of a Markov chain, what does this pertain to? $\lim \limits_{n \to \infty} P^n$? Where $P$ is the stepping matrix and $n$ is the number of steps.
"What is the limiting probability of success?" For example.
My guess: This may be the number of steps until success is essentially guaranteed.
My guess 2: It is the $n$ that makes $P^n$ becomes stable and doesn't change with further squaring.
My guess 3: This is the probability of success in the long term(steady state).
Is this non-standard terminology perhaps?
Its Guess no. 3:
If you have an N x N transition matrix, then you will need to set up N linear equations, where the variables are the, as yet unknown, transition probabilities. Each equation describes the probability of being in a different state, with one equation per state. So, for State 1 (S1), in a 4 state system, you need to set up the equation:
$\pi_1 = p_{11}\pi_1+p_{21}\pi_2+p_{31}\pi_3+p_{41}\pi_4$ (this is just the law of total probability in different guise), where $\pi_1$ is the steady state probability of being in state 1, and $p_{j1}$ is the transition probability of going from state j to state 1. Since the system is in equlibrium, each $\pi_i$ must be such that all 4 equations balance (and the system is stochastically in equlibrium).