Meyn S.P., Tweedie R.L. — Markov Chains and Stochastic Stability :: Электронная библиотека попечительского совета мехмата МГУ

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Meyn S.P., Tweedie R.L. — Markov Chains and Stochastic Stability

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Название: Markov Chains and Stochastic Stability

Авторы: Meyn S.P., Tweedie R.L.

Аннотация:

The area of Markov chain theory and application has matured over the past 20 years. This publication deals with the action of Markov chains on general state spaces. It discusses the theories and the use to be gained, concentrating on the areas of engineering, operations research and control theory. Throughout, the theme of stochastic stability and the search for practical methods of verifying such stability, provide a new and powerful technique which not only affects applications, but also the development of the theory itself. The impact of the theory on specific models is discussed in detail.

Язык:

Рубрика: Математика/Вероятность/Стохастические процессы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 558

Добавлена в каталог: 18.12.2004

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Предметный указатель

, indicator function of B      68
$A\otimes B$ , Kronecker product      404
$A\rightsquigarrow B$ , uniformly accessibility      91
$A\stackrel{a}{\rightsquigarrow} B$ , uniformly accessibility using a      120
, points from which A is inaccessible      91
, states reachable from x at time k by CM(F)      150
, states reachable from x by CM(F)      150
, ladder chain transition probabilities      77
$a_x(n) := P_x(\tau_{\alpha}=n)$       313
$C^{\infty}$ , infinitely differentiable lunctions      29
, controllability matrix      96
$C_V(r) = {x : V(x) \leq r}$ , sublevel set of V      190
$C_{x_0}^k$ , generalized controllability matrix      155
$d(\alpha)$ , period of $\alpha$       114
, output maps for LCM      30
$G_C^{(r)}(x,B)=E_x[\sum_{k=0}^{\sigma_C}1_B(\Phi_k)r^k]$       367
, almost everywhere invariant function      412
      72
      129
$K_a(x, A):=\sum_{n=0}^{\infty}P^n(x, A)a(n)$       119
$K_{a_{\varepsilon}}$ , resolvent kernel      68
$L(x,A) := P_x(\tau_A < \infty)$       70
, martingale derived from g      434
, interpolation of       435
, customers in q. before $n^{th}$ arrival      46
$N_n^{\ast}$ , customers in q. after $n^{th}$ service time      48
, control set      152 156
, supports disturbance/control in CM(F) or NSS(F) models      32
, n-step transition probability      59 67
, Kernel for process on h      292
, probability conditional on $\Phi_0 = x$       16
$Q(x, A) = P_x{\Phi\in A i.o.}$       200
, residual service time      77
, sum of $f(\Phi_i)$ between visits to atom      417
, partial sum of $g(\Phi_k)$       410
, interpolation of       435
$t_y(n):= _{\alpha}P^n(\alpha,y)$       313
$T_{ab}$ , coupling time      317
$u(n) := P_{\alpha}(\Phi_n=\alpha)$       313
      43
$U(n):=\sum_{k=0}^{\infty}P(Z_k=n)$       178
$U(x, A):=\sum_{n=1}^{\infty}P^n(x, A)$       70
$U(z):= \sum_{n=0}^{\infty}u(n)z^n$       178
$U_A(x, B) := \sum_{n=1}^{\infty}{_A}P^n(x, B)$       73
$U_C^{(r)}(x,B)=E_x[\sum_{k=1}^{\tau_C}1_B(\Phi_k)r^k]$       364
, resolvent kernel      290
$U_{\alpha}^{(r)}(x,f)=E_x[\sum_{n=1}^{\tau_{\alpha}}f(\Phi_n)r^n]$       357
, forward recurrence time chain      44
, forward recurrence time process      75
, backward recurrence process      75
, minimal solution to (V2)      266
$V_{\delta}^+(n)$ , forward recurrence $\delta$ -skeleton      75
$V_{\delta}^-(n)$ , backward recurrence $\delta$ -skeleton      75
$W = \{W_n\}$ , increment process      24
$\alpha$ , atom      100
$\ast$ , convolution operator      74
$\bar{A}$ , points from which A is accessible      91
$\bar{A}(m)$ , points reaching A in m steps      91
$\bar{P}_k(x,.)$ , Cesaro average of       285
$\check{P}(x_i,A)$ , the split transition lunction      103
$\check{\alpha}$ , the atom in $B(\check{X})$       104
$\check{\Phi}$ , the split chain      103
$\check{\pi}$ , split invariant measure      240
$\Delta V(x) = \int P(x,dy)V(y) — V(x)$ , drift operator      174
$\Delta_k$ , derivative process      166
$\delta_x(A) = P^0(x,A)$ , Dirac measure      67
$\eta_A:=\sum_{n=1}^{\infty}1\{\Phi_n=A\}$       70
$\Gamma$ , distribution of disturbance variable      24
$\gamma^2_g$ , limiting variance in the CLT      411
$\hat{g}$ , solution to the Poisson equation      431
$\lambda^{\ast}$ , split measure on $B(\check{X})$       103
$\Lambda_i^{\ast}(x,A)$ , zero-level l. c. transition probabilities      77
$\leftrightarrow$ , communicates with      82
$\mathbb{C}$ , complex plane      140
$\mathbb{R}$ , real line      516
$\mathbb{R}^n$ , n-dimensional Euclidean space      6
$\mathbb{Z}$ , integers      61
$\mathbb{Z}_+$ , non-negative integers      3
$\mathcal{B}(X)$ , $\sigma$ -field of subsets ol X      55
$\mathcal{B}(\mathbb{R})$ , Borel $\sigma$ -field on $\mathbb{R}$       516
$\mathcal{B}^+(X)$ , sets with $\psi(A)>0$       89
$\mathcal{C}(X)$ , continuous bounded lunctions on X      128 520
$\mathcal{C}_0(X)$ , continuous lunctions vanishing at $\infty$       523
$\mathcal{C}_C(X)$ , continuous lunctions with compact support      143
$\mathcal{F}_n^{\Phi}:=\sigma(\Phi_0,...,\Phi_n)$       69
$\mathcal{F}_{\zeta}^{\Phi}:=\{A\in \mathcal{F}:\{\zeta=n\}\cap A\in\mathcal{F}_n^{\Phi}, n\in\mathbb{Z}_+\}$       72
$\mathcal{G}^+(\gamma)$ , distributions with Laplace — Stieltjes transform convergent in $[0,\gamma]$       389
$\mathcal{M}$ , space ol Borel probability measures      19
$\mid f\mid_c$ , norm on $\mathcal{C}(X)$       145
$\mid f\mid_V := \sup_{x\in X}\dfrac{\mid f(x)\mid}{V(x)}$       385
$\mu^{Leb}$ , Lebesgue measure on $\mathbb{R}$       93 516
$\mu_k\stackrel{w}{\rightarrow}\mu_{\infty}$ , Weak convergence      143
$\Omega= X^{\infty}$ , sequence space      55
$\Omega_+(C)$ , omega limit set for NSS(F)      157
$\Phi$ Markov chain      3 66
$\Phi^m$ , the m-skeleton chain      68
$\Phi_a$ , sampled chain with transition kernel       119
$\Phi_n$ , Markov chain value at time n      3
$\pi$ , invariant measure      229
$\pi{A}$ , invariant random variable      414
$\psi$ , maximal irreducibility measure      88
$\rho(F)$ , maximum eigenvalue modulus      140
$\rightarrow$ , leads to      82
$\sigma$ -field      515
$\sigma$ -field generated by r.v.      518
$\sigma$ -finite measure      516
$\sigma_A(j)$ , $j^{th}$ hitting time on A      417
$\sigma_A:=\min\{n\geq 0:\Phi_n\in A\}$ , hitting time on A      70
$\Sigma_{\mu}$ , $\sigma$ -field of $P_{\mu}$ -invariant events      412
$\succ$ , absolute continuity      80
$\tau_A(j)$ , $j^{th}$ return time on A      70
$\tau_A:=\min\{n\geq 1:\Phi_n\in A\}$ , return time to A      16 70
$\textbf{W} = \{W_n\}$ , disturbance, noise, innovation process      24
$\textbf{Z} = \{Z_k\}$ , discrete time renewal process      43
$\theta^k$ , $k^{th}$ order shift operator on $\Omega$       69
$\varphi$ , irreducibility measure      87
$\varphi$ -irreducibity      87
$\|\mu\|$ , total variation norm      311
$\|\nu\|_f$ , f-norm      330
$\||P_1-P_2\||_V=\sup_{x\in X}\dfrac{\|P_1(x,.)-P_2(x,.)\|_v}{V(x)}$       382
$_aP^n(x, B) := P_x(\Phi_n\in B,-tau_A \geq n)$       73
Absolute continuity      80
Absorbing set      89
Absorbing set, maximal a.s.      204
Accessible atom      100
Accessible set      91
Adaptive control      38
Adaptive control model, boundedness in probability      303
Adaptive control model, irreducibility      165
Adaptive control model, performance      406
Adaptive control model, simple      39
Adaptive control model, V-Uniform ergodicity      406
Age process      44
Antibody model      471
Aperiodic Ergodic Theorem      309
Aperiodicity      116 118
Aperiodicity, strong a.      116
Aperiodicity, topological a. for states      447
ARMA model      27 28
Ascoli’s Theorem      520
Assumptions for models, (AR1)      27
Assumptions for models, (AR2)      27
Assumptions for models, (ARMA1)      28
Assumptions for models, (ARMA2)      28
Assumptions for models, (CM1)      33
Assumptions for models, (CM2)      151
Assumptions for models, (CM3)      156
Assumptions for models, (CSM1)      52
Assumptions for models, (CSM2)      52
Assumptions for models, (DBL1)      36

Assumptions for models, (DBL2)      36
Assumptions for models, (DS1)      260
Assumptions for models, (DS2)      260
Assumptions for models, (LCM1)      9
Assumptions for models, (LCM2)      9
Assumptions for models, (LCM3)      95
Assumptions for models, (LSS1)      9
Assumptions for models, (LSS2)      9
Assumptions for models, (LSS3)      97
Assumptions for models, (LSS4)      139
Assumptions for models, (LSS5)      140
Assumptions for models, (NARMA1)      33
Assumptions for models, (NARMA2)      33
Assumptions for models, (NSS1)      32
Assumptions for models, (NSS2)      32
Assumptions for models, (NSS3)      156
Assumptions for models, (Q1)      45
Assumptions for models, (Q2)      45
Assumptions for models, (Q3)      45
Assumptions for models, (Q4)      47
Assumptions for models, (Q5)      48
Assumptions for models, (RCA1)      404
Assumptions for models, (RCA2)      404
Assumptions for models, (RCA3)      404
Assumptions for models, (RT1)      44
Assumptions for models, (RT2)      44
Assumptions for models, (RT3)      75
Assumptions for models, (RT4)      75
Assumptions for models, (RW1)      11
Assumptions for models, (RW2)      111
Assumptions for models, (RWHL1)      14
Assumptions for models, (SAC1)      39
Assumptions for models, (SAC2)      39
Assumptions for models, (SAC3)      39
Assumptions for models, (SBL1)      30
Assumptions for models, (SBL2)      154
Assumptions for models, (SETAR1)      31
Assumptions for models, (SETAR2)      142
Assumptions for models, (SETAR3)      223
Assumptions for models, (SLM1)      25
Assumptions for models, (SLM2)      25
Assumptions for models, (SNSS1)      29
Assumptions for models, (SNSS2)      29
Assumptions for models, (SNSS3)      152
Assumptions for models, (SSM1)      49
Assumptions for models, (SSM2)      49
Assumptions for models, (V1)      190
Assumptions for models, (V2)      262
Assumptions for models, (V3)      337
Assumptions for models, (V4)      367
atom      100
Atom, Ergodic      314
Atom, f-Kendall a.      360
Atom, Geo. ergodic a.      357
Autoregression      27
Autoregression, dependent parameter random coeff. a.      36
Autoregression, random coefficient a.      404
Backward recurrence time $\delta$ -skeleton      75
Backward recurrence time chain      44 60
Backward recurrence time process      75
Balayage operator      305
Bilinear model      30
Bilinear model, dependent parameter b.m.      36
Bilinear model, f-regularity and ergodicity for b.m.      347
Bilinear model, Geo. ergodicity for b.m.      380 403
Bilinear model, irreducible T-chains as b.m.      155
Bilinear model, multidimensional b.m.      403
Blackwell Renewal Theorem      348
Borel $\sigma$ -field      516 519
Bounded in probability      145 301
Bounded in probability for T-chains      455
Bounded in probability on average      285
Brownian motion      525
Causality in control      38
Central limit theorem      411
Central Limit Theorem for Martingales      525
Central Limit Theorem, CLT for Autoregressions      442
Central Limit Theorem, CLT for Random walks      442
Central Limit Theorem, functional CLT      431 435
Chapman — Kolmogorov equations      67 68
Chapman — Kolmogorov equations, generalized C-K.e.      120
Closed sets      519
Closure of sets      519
Communication of discrete states      82
Compact set      519
Comparison theorem      337
Comparison Theorem, geometric C.T.      368
Conditional expectation      518
Continuous function      520
Control set      30 32 152 156
Controllability grammian      98
Controllability matrix      155
Controllable      16 95
Converges to infinity      201 207
Convolution      43 74 520
Countably generated $\sigma$ -field      516
Coupling      316
Coupling time      317
Coupling to bound sums      325
Coupling, null chains      448
Coupling, renewal processes      316
Cruise control      3
Cyclic classes      115
Cyclic classes for control models      161
Cyclic classes, content-dependent release rules      50
Dams      48
Dense sets      519
Dependent parameter bilinear model      35
Dependent parameter bilinear model, Geo. ergodicity for the d.p.b.m.      479
Derivative process      166
Dirac probability measure      67
Disturbance      24
Doeblin’s condition      391 407
Dominated Convergence Theorem      518
Drift criteria      174 496 501
Drift criteria for deterministic models      259
Drift criteria for f-moments      331 337
Drift criteria for geometric ergodicity      367
Drift criteria for invariant measures      296
Drift criteria for non-evanescence      215
Drift criteria for non-positivity      276 499
Drift criteria for positivity (Foster’s criterion)      262 499
Drift criteria for positivity for e-chains      298
Drift criteria for recurrence      190
Drift criteria for recurrence, converse for Feller chains      215
Drift criteria for transience      189 276
Drift criteria, history dependent      474
Drift criteria, mixed      481
Drift criteria, state dependent      466
Drift operator      174
Dynamical system      19 28
Dynkin’s formula      263
e-chain      144
e-chain, aperiodicity for      458
Eigenvalue condition      140
Embedded Markov chains      7
Equicontinuous functions      520
Equicontinuous Markov chains (e-chains)      144
Ergodic atom      314
Ergodicity      312 500
Ergodicity for e-chains      459
Ergodicity for null chains      446
Ergodicity, f-e.      331
Ergodicity, f-geometric      355 374
Ergodicity, f-norm      330
Ergodicity, history of      329
Ergodicity, strong      407
Ergodicity, uniform      383 390
Ergodicity, V-uniform      382
Error      24

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