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Created by Knut M. Synstadfrom the Noun Project

Invariant Distribution ↔ Positive Recurrence

Theorem

Let PP be irreducible. The following are equivalent:

  1. Every state is positive recurrent.
  2. Some state k∈Sk\in S is positive recurrent.
  3. PP has an invariant distribution, π\pi.

Remark

If P is irreducible:

  1. Have three scenarios:
  2. All states are transient
  3. All states are positive recurrent.
  4. All states are null recurrent.
  5. ∃\exists invariant distribution π  ⟺  \pi\iff PP is positive recurrent
  6. ∃\exists invariant distribution π  ⟹  \pi\implies π\pi is unique