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Dawson D. — Introduction to Markov Chains
Dawson D. — Introduction to Markov Chains

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Название: Introduction to Markov Chains

Автор: Dawson D.


In the past 15 years the potential theory of Markov chains with countable state space has been energetically developed by numerous authors. In particular the boundary theory of Markov chains first investigated by J.L. Doob, W. Feller and G.A. Hunt receives much current attention. These lectures aim to present the basic ideas of this theory to graduate students who had previous exposure to basic probability theory. The lectures were given with special emphasis on the probabilistic basis and significance of potential theory. Since boundary theory was one of the main objectives, more attention was given to transient chains than to recurrent chains. Sections 1 through 4 present the basic concepts of Markov chains. Section 5 is a brief treatment of recurrent chains, Sections 7 through 9 form an introduction to potential theory and Section 10 is an introduction to the Martin boundary including a discussion of the boundaries for random walks and the Polya urn scheme. The lecture notes do not form a complete exposition of the theory but are meant as an introduction to more detailed treatments such as is found in the book of Kemeny, Snell, and Knapp and to the current literature. I would like to thank the Canadian Mathematical Congress and in particular Dr. John McNamee and Professor Ron Руке for their encouragement and assistance with the presentation of the lectures and the preparation of these notes.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1970

Количество страниц: 82

Добавлена в каталог: 26.09.2009

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Предметный указатель
A.s. invariant events      5
Annihilation      3
Annihilation state      3
Aperiodic state      11
Arcsine law      5 17
Balayage      35
Balayage, potential      36
Balayage, principle of      36
Bernoulli random walk      17
Bernoulli random walk, transient      18
Birth and death chain      27
Birth and death chain, recurrent      28
Birth and death chain, transient      28
Birth and death chain, two dimensional      29
Birth and death chain, two-sided      29
Boundary      5
Boundary, Feller      61
Boundary, Martin      47
Branching process      18
Branching process, extinction probability      19
Branching process, generating function      19
Branching process, Martin boundary of      72
Branching process, transient      19
Capacity      5 35 38
Chain watched in E      31
Charge, left      33
Charge, right      33
Charge, total      36
Class, aperiodic      13
Class, closed      12
Class, communicating      11 12
Class, maximal      12
Class, minimal      12
Class, recurrent      12
Communicating classes      5 11
Communicating states      11
Dirichlet integral formula      67
Duality      39
Entrance problem      47
Equilibrium potential      37
Equilibrium set      37
Ergodic theorem      5
Escape vector      38
Exit problem      47
Expected number of visits      4 9
Expected number of visits to a transient state      5
Extended chain      41
Extended chain, reverse      43
Extreme function      57
Extreme point      57
Feller boundary      61
Fine boundary function      61
Fine topology      62
Forbidden states      6
Function, concordant      32
Function, excessive      32
Function, extreme      57
Function, fine boundary      61
Function, harmonic      32
Function, minimal      57
Function, regular      32
Function, subregular      32
Function, superregular      32
Galton-Watson process      (see Branching process)
Generating function, branching process      19
Generating function, simple random walk      15
H-chain      45
H-extended chain      46
h-harmonic measure      54
H-process      45
h-transform      45
Harmonic measure      50
Hexagonal random walk      20
Indecomposable chain      9
Initial distribution      3
Invariant $\sigma$-field      5
Invariant events      5
Irreducible chain      9
Left charge      33
Markov chain      1
Markov chain, birth and death      27
Markov chain, birth and death recurrent      28
Markov chain, birth and death transient      28
Markov chain, birth and death two dimensional      29
Markov chain, birth and death two-sided      29
Markov chain, boundary of      5
Markov chain, extended      41
Markov chain, extended reverse      43
Markov chain, finite      5
Markov chain, h-extended      46
Markov chain, h-transformed      45
Markov chain, indecomposable      9
Markov chain, irreducible      9
Markov chain, null recurrent no stationary distribution      24
Markov chain, positive recurrent stationary distribution      24
Markov chain, recurrent limiting behaviour      21
Markov chain, renewal      26
Markov chain, reverse      30
Markov chain, space time      31
Markov chain, transient      20
Markov chain, transient potential theory      32
Markov chain, tree process      53
Markov chain, two-sided      31
Markov chain, watched in E      31
Martin boundary      47
Martin boundary of branching process      72
Martin boundary of Polya’s urn scheme      69
Martin boundary of random walk in $L^{1}$      66
Martin boundary of random walk in $L^{d}$      66
Martin boundary of tree process      53
Martin boundary, entrance      49
Martin boundary, metric for      51
Measure, harmonic      50
Measure, invariant      33
Measure, reference      35
Measure, regular      33
Measure, stationary      33
Measure, subregular      33
Measure, superregular      33
Minimal function      57
Null recurrent state      11
Periodic state      11
Poisson-Martin representation theorem      60
Polya’s urn scheme      6
Polya’s urn scheme, Martin boundary of      69
Positive recurrent state      11
Potential      33
Potential measure      33
Potential, pure      33
Potential, support      36
Potentials      5
Probability, return to state      4
Probability, visit to state      4
Probability, visiting set infinitely often      4 20
Pure potential      33
Random walk      13 16
Random walk, Bernoulli      17
Random walk, Bernoulli transient      18
Random walk, hexagonal      20
Random walk, Martin boundary of      66
Random walk, minimal functions      63
Random walk, recurrent      17
Random walk, simple      14
Random walk, simple generating function      15
Random walk, simple null recurrence      15
Random walk, simple recurrence      14
Random walk, simple transience      14
Random walk, stationary distribution      15
Random walk, transient      17
Ratio limit theorem      25
Recurrence      5 10
Recurrence time      4 7 11
Recurrence time, expectation      4
Reference measure      35
Regular, function      32
Regular, measure      33
Renewal chain      26
Renewal Theorem      21
Return to state, probability of      4
Reverse chains      30
Riesz Decomposition theorem      33
Right charge      33
Simple random walk      14
Simple random walk, generating function      15
Simple random walk, null recurrent      15
Simple random walk, recurrent      14
Simple random walk, transient      14
Space time chains      31
State space      1
State, absorbing      19
State, aperiodic      11
State, communication      11
State, forbidden      6
State, null recurrent      11
State, periodic      11
State, positive recurrent      11
State, recurrent      10 12 21
State, transient      10 11 20
State, transient expected number of visits      5
Stationarity      2
Stationary distribution of a positive recurrent chain      24
Stationary distribution, positive recurrent chain      5
Stationary distribution, random walk      15
Stationary transitions      2
Stochastic matrix      2
Stopping time      7
Stopping time, $\sigma$-field generated by      7
Strong Markov property      7
Subregular, function      32
Subregular, measure      33
Superregular, function      32
Superregular, measure      33
Support of a potential      36
Taboo probability      11
Taboo transition probability      26
Tail $\sigma$-field      4
Transience      5 10
Transient state      11
Transition function      2
Transition matrix      2
Tree process      53
Tree process, Martin boundary of      53
Two dimensional birth and death chain      29
Two-sided birth and death chain      29
Wiener’s test      39
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