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Created by Knut M. Synstadfrom the Noun Project

Invariant Distribution for CTMC

Definition (Invariant Distribution for CTMC)

Let QQ be a Q-Matrix over some state space II, and π:I→[0,∞)\pi:I\to[0,\infty). π\pi is said to be an invariant measure relative to QQ if πQ=0  ⟺  ∑i∈IπiQij=0 \mboxforeachj∈I\pi Q=0 \iff\sum\limits_{i\in I}\pi_{i}Q_{ij}=0 \ \mbox{for each }j\in IIf, in addition, ∑i∈Iπi=1\sum\limits_{i\in I}\pi_{i}=1, π\pi is called an invariant distribution.

Theorem (Invariant Distribution Justification)

Assume the jump chain {Yn:n≥0}\{Y_{n}:n\ge0\} is irreducible and recurrent, and let π:I→[0,∞)\pi:I\to[0,\infty). The following are equivalent:

  1. πQ=0\pi Q=0
  2. πP(s)=π\pi P(s)=\pi for each s≥0s\ge0.

Theorem (Relationship to Jump Chain)

Let π:I→[0,∞)\pi:I\to[0,\infty). The following are equivalent:

  1. π\pi is an invariant measure for QQ
  2. μ=μΠ\mu=\mu\Pi, where μi=πiνi\mu_{i}=\pi_{i}\nu_{i}

Remark

If the jump chain {Yn:n≥0}\{Y_{n}:n\ge0\} is irreducible, and if the invariant distribution exists, then it must be unique.

Linked from

Convergence to Equilibrium