Math 2.0

Probability theory and stochastic processes Theorem

Limiting distribution of a regular transition matrix

Builds on Transition matrixMarkov chainStationary distribution

Statement

Let PP be an n×nn \times n transition matrix of a finite Markov chain with states S1,…,SnS_1, \dots, S_n. Call PP regular if some power PkP^k has only positive entries.

Assume that PP is regular. Then there is a row vector W=(w1,…,wn)W = (w_1, \dots, w_n) with these properties:

  1. every wiw_i is a positive real number, and w1+⋯+wn=1w_1 + \dots + w_n = 1;
  2. as m→∞m \to \infty, every row of PmP^m converges to WW;
  3. WP=WWP = W, so WW is a stationary distribution;
  4. for each ii, the number wiw_i is the limiting probability that the chain is in state SiS_i, whatever the starting state.

Remarks

The vector WW is unique. It is the only probability vector fixed by PP.

Reading path

To understand this page from the ground up, read these 53 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. Stationary distribution concept
  10. Transition matrix concept
  11. Set definition
  12. Ring of integers statement
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. The rationals form a field statement
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. The rationals form an ordered field statement
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Universal set definition
  28. Order on the reals definition
  29. Power set definition
  30. Complement definition
  31. Function definition
  32. Interval definition
  33. Injective function definition
  34. Preimage of a set definition
  35. Sigma-algebra definition
  36. Countable set definition
  37. Event definition
  38. Borel sigma-algebra definition
  39. Probability axioms concept
  40. Probability measure definition
  41. Probability space definition
  42. Conditional probability definition
  43. Random variable definition
  44. Independence of two events definition
  45. Discrete random variable definition
  46. Stochastic process concept
  47. Random variable with a density definition
  48. Conditional independence definition
  49. Expected value definition
  50. Conditional expectation definition
  51. Markov property definition
  52. Markov chain definition
  53. Limiting distribution of a regular transition matrix statement

Further reading

  1. Web https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_kitchensink_markov-chains.ptx#L41, source/s_kitchensink_markov-chains.ptx:41
About this page
TagFA8MNEYG38
URIm2://FA8MNEYG38
Kindstatement, theorem
WritingW1 stated
MSC60J10
StatusThis copy of the site has no build data. Run m2 status FA8MNEYG38.
AboutStationary distribution
UsesTransition matrix, Markov chain, Stationary distribution
Changed4 November 2026

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