Math 2.0

Probability theory and stochastic processes Definition

Stationary distribution

Builds on Markov chainTransition matrix

Definition

Let P=(pij)i,j∈SP = (p_{ij})_{i, j \in S} be the transition matrix of a time-homogeneous Markov chain on a countable state space SS. A stationary distribution of the chain is a family π=(πj)j∈S\pi = (\pi_j)_{j \in S} of real numbers with πj≥0\pi_j \ge 0 for each jj, ∑j∈Sπj=1\sum_{j \in S} \pi_j = 1, and πj=∑i∈Sπi pijfor each j∈S.\pi_j = \sum_{i \in S} \pi_i \, p_{ij} \quad \text{for each } j \in S. In matrix form, πP=π\pi P = \pi for the row vector π\pi.

Remarks

If X0X_0 has the law π\pi, then XnX_n has the law π\pi for each n≥0n \ge 0. An irreducible chain on a finite state space has exactly one stationary distribution. Its entries are positive, and πj=1/Ej[Tj]\pi_j = 1 / E_j[T_j], where TjT_j is the time of the first return to jj. For an irreducible and aperiodic finite chain, P(Xn=j)→πjP(X_n = j) \to \pi_j as n→∞n \to \infty, for each initial law. The simple random walk on ℤ\mathbb{Z} has no stationary distribution.

Reading path

To understand this page from the ground up, read these 52 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Real numbers object
  9. Set definition
  10. Ring of integers statement
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Ordered pair definition
  15. Union definition
  16. Sample space definition
  17. The rationals form a field statement
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Order on the reals definition
  27. Power set definition
  28. Complement definition
  29. Function definition
  30. Interval definition
  31. Injective function definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  34. Countable set definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability space definition
  40. Conditional probability definition
  41. Random variable definition
  42. Independence of two events definition
  43. Discrete random variable definition
  44. Stochastic process concept
  45. Random variable with a density definition
  46. Conditional independence definition
  47. Expected value definition
  48. Conditional expectation definition
  49. Markov property definition
  50. Markov chain definition
  51. Transition matrix definition
  52. Stationary distribution definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Discrete-time_Markov_chain, Stationary distributions
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