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| Emanuel Parzen — Stochastic processes (Classics in Applied Mathematics) |
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| Предметный указатель |
 -statistic see “Kolmogorov — Smimov test”
 -statistic see “Cramer-von Mises test”
Absorbing barrier 231
Absorbing Markov chain 241 246
Absorbing state 231 235 236 239 243 244 246
Absorbing state, definition of 211 229
Absorption probability 229 231—233 236 237 246
Absorption, time before 238
Absorption, time to 238—246 263
Acceleration in chaos 150 152
Accessible states 208 211 223 224
Anderson, T.W. 29 102 192
Aperiodic Markov chain 256—258 270—273
Aperiodic state 255
Approximately normal distribution 40 88 97 99 103 141 142 157 180 181 267—269 295
Approximately normal distribution, joint 98
Approximately normal process 98 101
Arrival times 143 165
Arrow, K.J. 4
Asymptotically normal distribution see “Approximately normal distribution”
Asymptotically normal process see “Approximately normal process”
Averages, ensemble 72—74
Bachelier, L. 29
Backward equation 291—293 300
Bailey, N.T.J. 2
Barnard, G.A. 102
Bartlett, M.S. 1 2 6 299
Battin, R.H. 4
Bernoulli trials see also “Coin tossing” 15 50
beta function 61
Bharucha-Reid, A.T. 2 202
Billingsley, P. 192
Binomial distribution 13 15 17 21 45—49
Birkhoff, G.D. 73
Birnbaum, A. 137
Birnbaum, Z.W. 102
Birth and death process see also “Pure birth process and Pure death process” 276 299 300 304 306
Birth and death process, homogeneous 299 300
Birth and death process, non-homogeneous 299 300 306
Birth process, pure see “Pure birth process”
Blackwell's theorem 183
Blackwell, D. 183
Borel function 10n
Borel sets 9 10n
Branching process 58 199 201 207 216 237 261
Breakdown, system see “Failure system”
Brockmeyer, E. 283
Brownian motion see also “Wiener process” 2 7 27—29 80 95 96 98 129
Busy channels or servers 144—149
Campbell's theorem 149 150 155
Cascade process see “Branching process”
Cauchy distribution 21 151
Censored Poisson distribution 283
Central limit theorem 19 101 141 267 269
Cesaro mean, convergence in 74 75 247 249
Chandrasekhar, S. 33 152
Chapman — Kolmogorov equation 194 195 203—205 208 222 250 276 278 290 292
Chapman, D. 142
Characteristic function, definition of 11
Characteristic function, infinitely divisible 124
Characteristic function, joint 13 22 25 27 37 53 77 98 116
Characteristic function, joint, normal 88 89 98
Characteristic function, of a stochastic process with (stationary) independent increment 123 124
Characteristic function, stable 151
Characteristic functions, table of 13 14
Chebyshev inequality for 85 86
Chebyshev inequality for, definition of 70
Chebyshev inequality for, derivative of 84—87 91
Chebyshev inequality for, ergodic 75
Chebyshev inequality for, general discussion of 66 71 74 77 96 108
Chebyshev inequality for, in harmonic analysis 108—116
Chebyshev inequality for, integral of 87 108
Chebyshev inequality for, normally distributed 91 94
Chebyshev's inequality 18 67
Chebyshev's inequality, for stochastic processes 85
Chi-square distribution 14 15 24 136 137
Chi-square test 135
Chung, K.L. 136 221 270 277
coin tossing see also “Bernoulli trials” 45 46 205 206
Communicating classes 208—211 222—225 235 245 256
Communicating classes, classification of 222 223
Communicating states 208 224—227 244 245 256 262—264 267 276
Communication theory, statistical 1 2 84 103 104 109 197 198
Compound exponential distribution 60
Compound Poisson distribution 57
Compound Poisson process 99 128—130 146 156
Conditional characteristic function, definition of 56 57
Conditional expectation 41
Conditional expectation, definition of, continuous case 51—53
Conditional expectation, definition of, discrete case 42—44
Conditional expectation, definition of, joint 58
Conditional expectation, properties of 62—65
Conditional mean see “Conditional expectation”
Conditional moments, definition of 54—56 58
Conditional probability density function, definition of 51
Conditional probability distribution function, definition of 41—43
Conditional probability mass function, definition of 42 43
Conditional probability, definition of, continuous case 51 52
Conditional probability, definition of, discrete case 41 42 44
Conditional probability, properties of, continuous case 51
Conditional probability, properties of, discrete case 45
Conditional variance, definition of 54 55
Confidence interval 136—139
Continuous parameter process, definition of 8
Continuous parameter process, describing a 23
Continuous parameter process, ergodic 72
Continuous parameter process, integral of 79
Continuous parameter process, with Markov property 188 190
Continuous probability laws, table of 14
Continuous random variable, definition of 10
Convergence theorem, dominated see “Dominated convergence theorem”
Convergence, modes of 78
Convergence, modes of, in Cesaro mean 74 75 247 249
Convergence, modes of, in mean square 78 79 83 91
Convolution of distributions 16 17
Correlation coefficients 53 94 96
Correlation coefficients, matrix of 94
Counter, defective 48 50
Counter, defective, Geiger 30 37 48 50 131 164
Counter, defective, non-paralyzable 164 185
Counter, defective, nuclear particle 139 162—166 179 181
Counter, defective, paralyzable 145 186
Counter, defective, type p 164 186
Counting process see also “Integer-valued process” 30 48 133 137 160 162 164
Counting process renewal see “Renewal counting process”
Covariance function, definition of 71
Covariance function, derivative of 84—87 92
Covariance function, for busy servers 149
Covariance function, in harmonic analysis 109—115
Covariance function, of a normal process 91 92 94 96
Covariance function, of a white noise 113—115
Covariance function, of an ergodic process 74—76
Covariance function, sample 109
Covariance kernel, definition of 67
Covariance kernel, exercises of the 76 77 96
Covariance kernel, in harmonic analysis 111 112
Covariance kernel, of a filtered Poisson process 148
Covariance kernel, of a normal process 89 92 95
Covariance kernel, of a Poisson process 67 68
Covariance kernel, of a stationary process 70—72
Covariance kernel, of a Wiener process 67 68
Covariance kernel, of an approximately normal process 101
Covariance kernel, of an ergodic process 74
Covariance kernel, of an integrated Wiener process 81
Covariance kernel, of the derivative of a process 84
Covariance kernel, of the increment process of the Poisson process 68 69
Covariance kernel, role of, in stochastic integral 79—81 87
Covariance matrix 88 89 92
Covariance stationary process, in busy servos problem 149
Covariance, definition of 16
| Cox, D.R. 192
Cramer, H. 89 134
Cramer-von Mises test see also “Goodness of fit test” 100 102 141 143
Customer loss ratio 286 288
Darling, D.A. 29 102
Davenport, W.B.,Jr. 31 150
Davis, D.J. 31
Deadtime, of non-paralyzable counters 164 165 170 186
Deadtime, of nuclear particle counters 163 164 181 182
Deadtime, of paralyzable counters 164—166 186
Death process, pure 299 300
Decision theory 43
Decomposition of Markov chain see “Markov chain decomposition
Defective counter 48 50
Density function, probability 10 14
Derivative of a stochastic process 83 84 87
Difference equations 49 50 239
Difference equations, stochastic 193
Differentiability in mean square 83—85 91—92
Differential equations, stochastic 84 113
Dirac delta function 113
Discrete branching process see “Branching process”
Discrete parameter Markov chain see “Markov chain discrete
Discrete parameter process, definition of 7
Discrete parameter process, describing 22
Discrete parameter process, ergodic 72 73
Discrete parameter process, with Markov property 188
Discrete probability laws, table of 13 15
Discrete random variable, definition of 10
Dishonest Markov process 92
Distribution function, probability 10 72
Dominated Convergence Theorem 128 251
Dominated convergence theorem, proof of 173 174
Donsker, M.D. 102
Doob, J.L. 52 70fn 73 96 97
Dot-dash process 36 38 40
Duration of random walk 239
Ecology 2 32 48
Eigenvalues, theory of 196
Einstein, A. 1 27 29
Ensemble averages 72—74
Epstein, B. 137
Ergodic Markov chain see “Markov chain irreducible positive
Ergodic process 74 75
Ergodic theorem 72 73
Ergodicity, geometric 272 273
Erlang's loss formula 283
Erlangian probability distribution 199
Essential states 224
Estimation, of ensemble averages 72 83
Estimation, of the spectral density function 111 116
Estimation, of transition probability matrix 207
Estimation, parameter 3 6 35 143 268 298
Estimation, parameter, methods of 135
Estimation, parameter, with confidence intervals 136—139
Estimation, parameter, with method of moments 134 139
Evans, R.D. 31 164
Event, random, definition of 8
Evolutionary process 69
Excess life, asymptotic distribution of 184
Excess life, distribution of 171—173
Excess life, mean of 179 186
Expectation see “Mean”
Exponential distribution 14 21 31 62 285
Exponential distribution, characterization of 123 282
Exponential distribution, compound 60 61
Exponential distribution, of excess life 173
Exponential distribution, of inter-arrival times 98 118 133 135 137 142 163 173—177 199 281 283 287
Exponential distribution, of lifetime 31 60 61 163
Exponential distribution, of service time 82 148 149 190 199 206 281 286 287
Exponential distribution, of the sample spectral density function 116
Extinction probabilities 201 305
Extrapolation 3 43 113 114
Extreme value distribution 169
F-distribution 14 136
F-probability law 14 136
F-test 139
Failure function, conditional rate of 168 169
Failure process 289
Failures, system 5 31 137 162 168 184 192 205 247 286 289
Family of events 8 9
Fatou's lemma 250 273
Feller, W. 57 126 164 172 196 202 203 221 235 270 291
Filter 104—107 110—115
Filtered Poisson process see “Poisson process filtered”
Finite Markov chain see “Markov chain finite”
First entrance, method of 214 215
First passage probabilities 213 226 227 229 231 242 247 259 266
First passage probabilities, an inequality on 224 225
First passage probabilities, definition of 213
First passage time 189 213 246 265
First passage time, mean 242—244 253
First passage time, mean, recurrent case 259 264
Forward equation 291—293 300
Foster, F.G. 260
Fourier integral 108—112
Fourier series 108
Fourier transform 12 106—109
Frequency response function 105—108 110—113
Friedman, B. 89
Fubini's theorem 273 275
Fuller, A.T. 37
Functional equations, solution of 121—123
Furry process 296 297 299 302 306
Furry, W. 297
Galaxies, distribution of 129 130
Galton, F. 201
Gambler's ruin 229 231 234 235
Gamma distribution 14 20 57 61 132 135 199 285
Gamma distribution, of inter-arrival times 162 175 177 181 199
Gamma distribution, of waiting times 133 134 174 285
Gamma function 62
Gaussian distribution see “Normal distribution”
Geiger counter see “Counter Geiger”
Generalized Poisson process see “Poisson process generalized”
Geometric distribution 13 15 21 132 139 284 299 303
Geometric distribution, characterization of 123
Geometric ergodicity 272 273
Gilbert, E.N. 150
Girshick, M.A. 136
Good, I.J. 150 152
Goodman, L.A. 192
Goodness of fit test 99 102 135
Greenwood, M. 57
Grenander, U. 6
Groll, P.A. 135
Gumbel, E.J. 169
Gupta, S.S. 135
Haar, D. ter see “ter Haar”
Hagstroem, K.G. 7
Halstroem, H.L. 283
Hannan, E.J. 6
Hardy, G.H. 218
Harmonics 105 106
Harris, T.E. 2
Hazard function 168 169
Helstrom, C.W. 4
Homogeneous Markov chain see “Markov chain homogeneous”
Homogeneous Markov process 189
Homogeneous Poisson process 124—126
Hypothesis testing 3 135 137 139 142—144
Identically distributed random variables, definition of 10
Identically distributed stochastic processes, definition of 25
Imbedded Markov chain see “Markov chain imbedded”
Impulse response function 105 113 156
Increment process of a Poisson process 68 87
Independent increments, integer-valued process with 37 124 125 191
Independent increments, of the pure birth process 296
Independent increments, stochastic processes with 28 47 65 81 117 130 296
Independent increments, stochastic processes with, covariance kernel of 68
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