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Markov Chains

Alex Carr Markov Chains Year 2 Inglés 47 fichas

Markov Chains

Fichas del conjunto (47)

  1. 1Theorem 1.1.1 - Markov ChainTheorem 1.1.1 - Markov Chain
  2. 2Theorem 1.1.2 - Markov PropertyTheorem 1.1.2 - Markov Property
  3. 3Theorem 1.13Theorem 1.13
  4. 4Theorem 1.2.1 - For distinct states i and j, the following are equivalentTheorem 1.2.1 - For distinct states i and j, the following are equivalent
  5. 5Theorem 1.3.2 - The vector of hitting probabilitiesTheorem 1.3.2 - The vector of hitting probabilities
  6. 6Theorem 1.3.5 - The vector of mean hitting timesTheorem 1.3.5 - The vector of mean hitting times
  7. 7Theorem 1.4.2 - Strong Markov PropertyTheorem 1.4.2 - Strong Markov Property
  8. 8Lemma 1.5.1Lemma 1.5.1
  9. 9Lemma 1.5.2Lemma 1.5.2
  10. 10Theorem 1.5.3 - The following dichotomy holds (transient or recurrent)Theorem 1.5.3 - The following dichotomy holds (transient or recurrent)
  11. 11Theorem 1.5.4 - Let C be a communicating class:Theorem 1.5.4 - Let C be a communicating class:
  12. 12Theorem 1.5.5 - Every recurrent class:Theorem 1.5.5 - Every recurrent class:
  13. 13Theorem 1.56 - Every finite close class:Theorem 1.56 - Every finite close class:
  14. 14Theorem 1.5.7 - Suppose P is irreducible and recurrent:Theorem 1.5.7 - Suppose P is irreducible and recurrent:
  15. 15Theorem 1.7.1 - Let Xn be Markov(Lambda, P) and suppose that lambda is invariant for P, then:Theorem 1.7.1 - Let Xn be Markov(Lambda, P) and suppose that lambda is invariant for P, then:
  16. 16Theorem 1.7.2Theorem 1.7.2
  17. 17Theorem 1.7.5 - Let P be irreducible and recurrent, then:Theorem 1.7.5 - Let P be irreducible and recurrent, then:
  18. 18Theorem 1.7.6 - Let P be irreducible and let lambda be an invariant measure for P with lambdak = 1, then:Theorem 1.7.6 - Let P be irreducible and let lambda be an invariant measure for P with lambdak = 1, then:
  19. 19Theorem 1.7.7 - Let P be irreducible, then the following are equivalent:Theorem 1.7.7 - Let P be irreducible, then the following are equivalent:
  20. 20Lemma 1.8.2 - Suppose P is irreducible and has an aperiodic state i, then:Lemma 1.8.2 - Suppose P is irreducible and has an aperiodic state i, then:
  21. 21Theorem 1.8.3 - Convergence to equilibriumTheorem 1.8.3 - Convergence to equilibrium
  22. 22Theorem 1.8.4 - Let P be irreducible:Theorem 1.8.4 - Let P be irreducible:
  23. 23Theorem 1.8.5 - Let P be irreducible of period d and let C0,C1,...Cd-1 be the partition obtained in theorem 1.8.4:Theorem 1.8.5 - Let P be irreducible of period d and let C0,C1,...Cd-1 be the partition obtained in theorem 1.8.4:
  24. 24Theorem 1.9.1 - Let P be irreducible and have an invariant distribution pi:Theorem 1.9.1 - Let P be irreducible and have an invariant distribution pi:
  25. 25Lemma 1.9.2 - If P and lambda are in detailed balance:Lemma 1.9.2 - If P and lambda are in detailed balance:
  26. 26Theorem 1.9.3 - Let P be an irreducible stochastic matrix and let lambda be a distribution. Suppose that Xn is a Markov(lambda,P). Then the following are equivalent:Theorem 1.9.3 - Let P be an irreducible stochastic matrix and let lambda be a distribution. Suppose that Xn is a Markov(lambda,P). Then t...
  27. 27Theorem 1.10.1 (Strong law of large numbers) - Let Y1,Y2,... be a sequence of iid non-negative random variables with E(Y1) = mu. Then:Theorem 1.10.1 (Strong law of large numbers) - Let Y1,Y2,... be a sequence of iid non-negative random variables with E(Y1) = mu. Then:
  28. 28Theorem 1.10.2 (Ergodic theorem)Theorem 1.10.2 (Ergodic theorem)
  29. 29Transition MatrixThis is P
  30. 30Initial distributionThis is lambda
  31. 31Pi(A)P(A | X0 = i)
  32. 32Closed ClassA class which you cannot escape from
  33. 33AbsorbingA state i is absorbing if {i} is a closed class
  34. 34IrreducibleA chain where the entire chain is a closed class is irreducible. If you can find a repeating path through all the states in a chain, it is irreducible
  35. 35Hitting timeMaximum time expected for the Markov chain to travel between any two statesHitting time
  36. 36Absorption ProbabiliyWhen A is a closed class, this on the left is called the absorption probabilityAbsorption Probabiliy
  37. 37Notation for hitting time and expected hitting timeNotation for hitting time and expected hitting time
  38. 38Stopping timeLook up proper definition
  39. 39Recurrent and TransientA recurrent state is one you keep coming back to forever and a transient you eventually leave foreverRecurrent and Transient
  40. 40Passage TimeLook up
  41. 41InvariantP is invariant if:Invariant
  42. 42Expected time spent between each visit to kExpected time spent between each visit to k
  43. 43Positive and null recurrentIf the expected return time mi is finite it is positive recurrent and if it fails we call it null recurrentPositive and null recurrent
  44. 44AperiodicIf pii^(n) >0 for all sufficiently large n
  45. 45PeriodGreatest common divisor of the set:Period
  46. 46Time-reversalTime-reversal
  47. 47A stochastic matrix P and a measure lambda are said to be in detailed balance if:A stochastic matrix P and a measure lambda are said to be in detailed balance if:

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Year 2 Markov Chains Fichas sobre Markov Chains, creado por Alex Carr el 28/11/2022.
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Alex Carr
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