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Revuz D. — Markov chains
Revuz D. — Markov chains

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Название: Markov chains

Автор: Revuz D.


This is the revised and augmented edition of a now classic book which is an introduction to sub-Markovian kernels on general measurable spaces and their associated homogeneous Markov chains. The first part, an expository text on the foundations of the subject, is intended for post-graduate students. A study of potential theory, the basic classification of chains according to their asymptotic behaviour and the celebrated Chacon-Ornstein theorem are examined in detail.

The second part of the book is at a more advanced level and includes a treatment of random walks on general locally compact abelian groups. Further chapters develop renewal theory, an introduction to Martin boundary and the study of chains recurrent in the Harris sense. Finally, the last chapter deals with the construction of chains starting from a kernel satisfying some kind of maximum principle.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2005

Количество страниц: 389

Добавлена в каталог: 19.10.2010

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Предметный указатель
Absorbing sets      47
Adapted random walks      98
Admissible $\sigma$-algebras      13
Almost-sure      19
Aperiodic, aperiodic discrete chains      86
Aperiodic, aperiodic Harris chains      198
Aperiodic, aperiodic random walks      98
Asymptotic, asymptotic events      186
Asymptotic, asymptotic random variables      186
Asymptotic, equivalent asymptotic random variables      187
Balayage, balayage operator      25
Balayage, balayage sequence      68
Balayage, balayage theorem      49
Birkhoff theorem      136
Birth and death chain      48
Brownian motion, resolvent of Brownian motion      77
Brunei's maximal ergodic lemma      145
Canonical, canonical Markov chain      19
Canonical, canonical probability space      16
Capacity, recurrent capacity      277
Capacity, transient capacity      64
Cemetary      16
Chacon — Ornstein theorem      132
Charge      217
Choquet — Deny theorem      161
Co (prefix co)      62
Conditional probability distribution      122
Conservative, conservative contraction      124
Conservative, conservative part      124
Continuous, left and right continuous random walks      103
Contraction, positive contraction      12
Convergence, strong convergence      202
Convergence, uniform convergence      201
Convolution kernel      10
Cyclic, cyclic class      196
Cyclic, cyclic decomposition of a Harris chain      196
Directly Riemann integrable      170
Discrete, discrete chain      80
Discrete, discrete group      80
Dissipative part      124
Doeblin, Doeblin condition      210
Doeblin, Doeblin theorem      144
Duality, kernels in duality      62
Duality, Markov chains in duality      62
Equilibrium, equilibrium function      268
Equilibrium, equilibrium measure      268
Equilibrium, equilibrium potential      56
Ergodic, ergodic conservative contraction      126
Ergodic, ergodic Harris chain      197
Ergodic, ergodic point transformation      138
Ergodic, maximal ergodic lemma      122
Exact minorant      153
Excessive, excessive function      78
Excessive, excessive measure      60
Extremal measures      253
Fatou theorem      247
Feller, Feller kernels      36
Feller, strong Feller kernels      36
Filling scheme      70
Galton — Watson chain      22
Haar measure      7
Harmonic functions      40
Harris chains, chains recurrent in the sense of Harris      91
Homogeneous Markov chains      15
Hopf decomposition      124
Induced, induced chain      27
Induced, induced operator      118
Integral kernel      9
Invariant, invariant events      56
Invariant, invariant functions      126
Invariant, invariant measures      60
Invariant, invariant random variables      56
Invariant, invariant sets      126
Irreducible, $\nu$-essentially irreducible chain      140
Irreducible, $\nu$-irreducible chain      87
Irreducible, irreducible random walk      98
Irreducible, irreducible recurrent discrete chain      82
Kac recurrence theorem      128
Kernel, $\sigma$-finite kernel      8
Kernel, bounded kernel      8
Kernel, Feller kernel      36
Kernel, integral kernel      9
Kernel, positive kernel      8
Kernel, potential kernel      41
Kernel, proper kernel      8
Kernel, recurrent potential kernel      217
Kernel, strong Feller kernel      36
Killed chain      27
Law of Large Numbers      138
LCCB space      6
Linear modulus      122
Markov, homogeneous Markov chain      15
Markov, Markov chain      14
Markov, Markov property      20
Markov, strong Markov property      24
Markovian      12
Martin, Martin recurrent boundary      279
Martin, Martin transient boundary      243
Martingale      3
Maximal ergodic lemma      122
Maximum, complete maximum principle      46
Maximum, maximum principle      46
Maximum, reinforced maximum principle      46
Maximum, reinforced semi-complete maximum principle      339
Maximum, semi-complete maximum principle      339
Mean      205
Mean, mean of a random walk      171
Mean, pure mean      205
Measure preserving point transformation      60
Mixing, strongly mixing point transformation      199
Modulus, linear modulus      122
Moment, first moment of a random walk      171
Monotone class theorem      4
Normal chain      272
Normal chain, weakly normal chain      281
Null Harris chain      197
Operate, group operating on      11
Period of a Harris chain      198
Periodic Harris chain      198
Poincare recurrence theorem      128
Point transformation      11
Poisson, Poisson equation      40
Poisson, Poisson integral representation      233
Poisson, Poisson space      241
Positive, positive contraction      12
Positive, positive Harris chain      197
Potential, potential kernel      41
Potential, potential kernel of a Harris chain      217
Potential, potential kernel of a Harris random walk      301
Quasi-compact      202
R-recurrence      97
Random walk      29
Random walk, symmetric random walk      31
Range of a random walk      158
Ratio-limit theorem      230
Recurrent, recurrent group      112
Recurrent, recurrent in the sense of Harris      91
Recurrent, recurrent random walk      98
Recurrent, recurrent sets      57
Recurrent, recurrent states of a discrete chain      80
Reduced function      49
Regular, regular conditional probability distribution      122
Regular, regular function      234
Regular, regular set      234
Renewal, renewal chain      86
Renewal, renewal theorem      180
Resolvent, proper resolvent      75
Resolvent, resolvent equation      74
Resolvent, sub-markovian resolvent      74
Riesz decomposition      41
Robin's constant      268
Shift operator      19
Space time, space time chain      21
Space time, space time harmonic functions      187
Special, special function      212
Special, special set      212
Spread-out      102
Starting measure      15
Stopped chain      28
Stopping time      23
Stopping time, randomized stopping time      67
Subadditive sequence      152
Subadditive sequence, integrable subadditive sequence      152
Superharmonic functions      40
Supermedian, supermedian function      75
Supermedian, supermedian measures      79
Symmetric, symmetric events      33
Symmetric, symmetric random walk      168
Time, death time      23
Time, first hitting time      23
Time, first return time      23
Time, stopping time      23
Time, time constant      152
Trajectory      17
Transient invariant      29
Transient, transient random walk      98
Transient, transient set      57
Transient, transient state      80
Transition probability      12
Transition probability, (sub-)markovian transition probability      12
Translation on the integers      47
Type, type of a recurrent random walk      297
Type, type of a transient random walk      168
Vague topology      6
Walk, left and right random walk      29
Zero-two laws      191
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