Math 2.0

Probability theory and stochastic processes Definition

Transition matrix

Builds on Markov chain

Definition

Let SS be a nonempty countable set. A transition matrix, or stochastic matrix, on SS is a family P=(pij)i,j∈SP = (p_{ij})_{i, j \in S} of real numbers with pij≥0for all i,j∈Sand∑j∈Spij=1for each i∈S.p_{ij} \ge 0 \quad \text{for all } i, j \in S \quad \text{and} \quad \sum_{j \in S} p_{ij} = 1 \quad \text{for each } i \in S. The transition matrix of a time-homogeneous Markov chain (Xn)(X_n) on SS has pij=P(Xn+1=j∣Xn=i)p_{ij} = P(X_{n+1} = j \mid X_n = i).

Remarks

Each row of PP is a probability distribution on SS: the law of the next state, given the present state ii. A product of two stochastic matrices is stochastic. The nn-th power PnP^n gives the nn-step probabilities P(Xm+n=j∣Xm=i)P(X_{m+n} = j \mid X_m = i). If the row vector λ\lambda is the law of X0X_0, then λPn\lambda P^n is the law of XnX_n. A finite stochastic matrix has the eigenvalue 11, with the eigenvector (1,…,1)𝖳(1, \dots, 1)^{\mathsf{T}}, and all its eigenvalues have absolute value at most 11.

Reading path

To understand this page from the ground up, read these 51 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

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Stochastic_matrix, Definition and properties
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