My security road
1. Definition of States
- Communicate: State i and j communicate () if i is reachable from j and j is reachable from i. (Note: a state i always communicates with iteself)
- Irreducible: a Markov chains is irreducible if all states are in the same communication class.
- Absorbing state:
- Closed Set: a set of states S is a closed set if no state outside of S is reachable from any state in S. (This is like absorbing set of states)
- : be the probability that, starting in state i, the process returns (at some point) to the sate i
- Transient: a state is transient if
- Recurrent: a state that is not transient is recurrent, i.e., . There are two types of reurrent states
- Positive recurrent: if the expected time to return to the state if finite
- Null recurrent: if the expected time to return to the sate i is infinite (this requires an infinite number of states)
- Periodic: a state is i period if where k is the smallest number such that all paths leading from state i back to state i has a multiple of k transitions
- Aperiodic: if it has period k =1
- Ergodic: a state is ergodic if it is positive recurrent and aperiodic.
2. Example: Gambler’s Ruin