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| Emanuel Parzen — Stochastic processes (Classics in Applied Mathematics) |
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| Предметный указатель |
Independent increments, stochastic processes with, definition of 26 27
Independent random variables, criteria for 13 15
Independent stochastic processes, definition of 25
Indicator function, definition of 45
Inessential states 224 243
Infinitely divisible laws 124
Input function 104—108 110—114
Integer-valued process see also “Counting process One- 117 123 160 299
Integer-valued process, definition of 30
Integer-valued process, with independent increments 37 124 125 191
Integer-valued process, with stationary independent increments see also “Poisson process generalized” 118 126 130
Integral of stochastic processes see “Stochastic integral”
Intensity function 168 169
Intensity function of a Poisson process 125
Intensity functions of a Markov chain 289—292 305 306
Intensity of passage 278 289 293 294
Intensity of transition see “Transition intensity”
Inter-arrival times 138 139 168 170 266
Inter-arrival times, definition of 117 132 133 266
Inter-arrival times, exponentially distributed 118 133—135 142 173—177 199 281 283 287
Inter-arrival times, gamma distributed 174—177 181 199
Inter-arrival times, in a renewal counting process 163 165 166 171—183 185 186
Inter-arrival times, in queues 5 199 206 207 258 265 281—283 287
Inter-arrival times, mean 148 185 186 258 265 282
Inter-arrival times, of a non-homogeneous Poisson process 138
Inverse function, definition of 21
Inversion formula 12
Irreducible Markov chain see “Markov chain irreducible”
Jenkins, G.M. 111
Jensen, A. 283
Joint characteristic function see “Characteristic function joint”
Jointly distributed random variables 12
Jointly normal distribution, approximately 98
Jointly normal distribution, n-dimensional 88—94
Jointly normal distribution, two-dimensional 53 54 59 95 96 116
Jointly normal probability density function 54 88 89 94
Kac, M. 29 102
Karlin, S. 4 173 280 299
Kemeny, J.G. 241
Kendall, D.G. 199 221 272 273 304
Key renewal theorem 183 184 186
Khinchin, A.I. (Khintchine, A.Ya.) 73 108 110 125
Kolmogorov — Smirnov test see also “Goodness of fit test” 100—102 141 143
Kolmogorov's backward equation 291—293 300
Kolmogorov's forward equation 291—293 300
Kolmogorov, A.N. 291
Korff, S.A. 164
Lagrange's method 302
Lanning, J.H. 4
Laplace — Stieltjes transform 177 178 180 183
Lattice random variable 182 185 186
Law of Large Numbers 18 267
Ledermann, W. 299
Life testing 137
Linear growth process, non-homogeneous 302
Locking time see “Deadtime”
Loeve, M. 52 70fn 73 136 267
Lomax, K.S. 169
Long-run distribution 250
Long-run distribution, definition of 248
Long-run distribution, determination of 261 262
Long-run distribution, existence of 255—268 265 277 284
Long-run distribution, for queues 284 286
Long-run distribution, of birth and death process 280 282 283 287
Longuet-Higgins, M.S. 40
Lotka, A.J. 201
MacFadden, J.A. 40
Machine failures see “Failures system”
Machlup, S. 39
Maguire, B.A. 102 137 139
Management science 1 4 144 146
Mandelbrot, B. 150
Markov chain 187—199 203—217 220—273 276—306
Markov chain, absorbing 241 246
Markov chain, aperiodic 256—258 270—273
Markov chain, continuous parameter, definition of 190 193
Markov chain, decomposition of 209—211 222
Markov chain, definition of 187—189
Markov chain, discrete parameter, definition of 189 193
Markov chain, finite, absorption time of 239 241 246
Markov chain, finite, decomposition of 222 223
Markov chain, finite, definition of 196
Markov chain, finite, first passage probabilities of 229 253
Markov chain, finite, irreducible 249 251 253 255 257 262 270 271 273 277
Markov chain, finite, stationary distribution of 251
Markov chain, finite, transition probability matrix of 232 246 255 257
Markov chain, homogeneous 208 212
Markov chain, homogeneous, definition of 193 194
Markov chain, homogeneous, imbedded 198 206
Markov chain, homogeneous, intensity functions of 289 292 293
Markov chain, homogeneous, transition probability matrix of 196—198 204 206 207 256
Markov chain, homogeneous, two-state 197 198 207 256 293
Markov chain, imbedded 190 273
Markov chain, imbedded, of the GI/M/1 queue 206 265
Markov chain, imbedded, of the M/G/1 queue 190 198 199 258 260
Markov chain, irreducible 236 249 277
Markov chain, irreducible, classification of 258 260
Markov chain, irreducible, definition of 235 276
Markov chain, irreducible, geometrically ergodic 272
Markov chain, irreducible, in queueing theory 258
Markov chain, irreducible, non-current 235—237 249 253 258 260 261 265
Markov chain, irreducible, null-recurrent 258 260 265
Markov chain, irreducible, periodic 256 258 263
Markov chain, irreducible, positive recurrent 245 252 253 256 258 260—262 264—266 277
Markov chain, irreducible, recurrent 225 235 236 242—244 246 252 258 260 270
Markov chain, irreducible, stationary distribution of 251 252 258 265 277
Markov chain, irreducible, with a long-run distribution 255 256 257 258 265 277
Markov chain, irreducible, with doubly stochastic transition probability matrix 255 256 262
Markov chain, multiple 191 192
Markov chain, non-homogeneous 288
Markov chain, two-state 197 207 213 293 256
Markov chain, with stationary transition probabilities see “Markov chain homogeneous”
Markov process 117 187—189 191 192
Markov process, classification of 188
Markov process, definition of 187 188
Markov process, homogeneous 189
Markov process, pathological 292
Markov process, with stationary transition probabilities 189
Markov property 194 214
Martingale 65
McGregor, J.L. 280 299
Mean recurrence time see “Recurrence time mean”
Mean square differentiability 83—85 91—92
Mean square error 114
Mean square, convergence in 78 79 80 91
Mean value function 68 72 76 79—83 85 92 95
Mean value function, definition of 67
Mean value function, derivative of 83 156
Mean value function, of a covariance stationary process 71 94
Mean value function, of a homogeneous Poisson process 68 125 143
Mean value function, of a non-homogeneous Poisson process 125 138 156 158 297
Mean value function, of a normal process 67 95 101
Mean value function, of a renewal counting process 170 177—181 183 185
Mean value function, of a stochastic process with stationary independent increments 77
Mean value function, of the increment process of the Poisson process 68 69
Mean, definition of 13
Means, table of 13 14
Method of first entrance 214 215
Middleton, D. 4
Miller, G.A. 191
Molina, E.C. 281
Moment function, second 179
Moment-generating function, definition of 11
Moments 17 18
Montroll, E.W. 29
Multiple Markov chain 191 192
Multiplicative process see “Branching process”
Negative binomial distribution 13 15 17
Negative binomial distribution, related to pure birth processes 297
Negative binomial distribution, related to renewal counting processes 179
| Negative binomial distribution, related to the compound Poisson and geometric distributions 57 58
Negative binomial distribution, related to the generalized Poisson distribution 127
Neumann, J. von see “von Neumann”
Neyman, J. 1 71 130
Non — Markov chain 205
Non — Markov process 203
Non-homogeneous birth and death process 299 300 306
Non-homogeneous linear growth process 302
Non-homogeneous Markov chain 288
Non-homogeneous Poisson process see “Poisson process non-homogeneous”
Non-paralyzable counter 164 165 170 181 182 185
Non-recurrent class 221—225 235
Non-recurrent Markov chain 235—237 249 253 258 260 261 265
Non-recurrent state 221—229 235 238 239 241—247 252 263
Non-recurrent state, definition of 221
Non-return state 208—211
Normal distribution 14 15 17 19—21 54 60 91 93 95 96 141 169
Normal distribution, asymptotically see “Approximately normal distribution”
Normal distribution, in stochastic processes 26 66 76 77 192
Normal distribution, jointly see “Jointly normal distribution”
Normal distribution, of a normal process 88
Normal distribution, related to a filtered Poisson process 151
Normal probability density function, jointly 54 88 89 94
Normal process 66 88—99 101 102 113 114 116
Normal process, approximately 98 101
Normal process, definition of 89 90
Nuclear particle counter 139 162—166 179 181
Null-recurrent Markov chain 258 260 265
Null-recurrent state 245
Occupation time 189 221 265 268
Occupation time, definition of 211 212
Olkin, I. 18
One-minus-one process see also “Two-valued process” 36 37 77
Operations research 1 4 144 146
Order statistics 140 141 143
Ornstein — Uhlenbeck process 96 97 115
Ornstein, L.S. 97
Osborne, M.F.M. 29
Output function 104—108 110—114
Owen, A.R.G. 168
Paralyzable counter 145 158 164—166 181 182 186
Parameter estimation see “Estimation parameter”
Parzen, E. 8fn 72 74 111
Passage intensity 278 289 293 294
Pathological Markov process 292
Pearson, E.S. 102 137 139
Period, of a communicating class 256 262 263
Period, of a state 255 256 262 263
Periodic Markov chain 256 258 263
Periodic state 255 256
Perrin, J. 27 29
Persistent state see “Recurrent state”
Plancherel theory 108 109
Poisson counting process see “Counting process”
Poisson distribution 13 15 17 20 21 26 31 97 102 127
Poisson distribution, and a filtered Poisson process 147 148 151—153
Poisson distribution, censored 283
Poisson distribution, compound 57
Poisson distribution, in estimation and testing 136 137 139 141 142 144
Poisson distribution, in queueing 149 190 282
Poisson distribution, of a compound Poisson process 57 129 132
Poisson distribution, of a Poisson process 30—32 34 67 119 124
Poisson distribution, of a pure birth process 296
Poisson distribution, of a renewal counting process 174 176
Poisson distribution, of arrivals 82 134 135 138 144 147 149 152 153 158
Poisson distribution, of particle arrivals at counters 162 170 179 181 182 185 186
Poisson process 8 15 26 31 33—35 69 77 143 288
Poisson process, and the binomial distribution 49
Poisson process, and the uniform distribution 140 141
Poisson process, as a random telegraph signal 36 37
Poisson process, as a renewal process 163 174
Poisson process, asymptotic normality of 103
Poisson process, axiomatic derivation of 118—120 123
Poisson process, compound 99 128—130 146 156
Poisson process, covariance kernel of 68 69
Poisson process, definition of 30
Poisson process, filtered 144—147 297 306
Poisson process, filtered non-homogeneous 156
Poisson process, filtered, extended 152 153
Poisson process, filtered, generalized 156
Poisson process, generalized 32 118 126 127 130 131
Poisson process, generalized, axiomatic derivation of 127
Poisson process, homogeneous 124—126
Poisson process, in a Brownian motion 80 98
Poisson process, in a pure birth process 296 305
Poisson process, in a shot noise process 149
Poisson process, in estimation and testing 135 136 141 143
Poisson process, in particle counter problems 132 138 139 158 163 166
Poisson process, in space 31 32 34
Poisson process, increment process of 68 87
Poisson process, intensity function of 125
Poisson process, mean value function of 68 69 126 143
Poisson process, non-homogeneous 118 124—126 138 143 156 157 297
Poisson process, non-homogeneous, axiomatic derivation of 125 126
Poisson process, sample function of 29
Poisson process, sample mean of 86
Poisson process, truncated 298
Poisson process, under random selection 47 48 50
Poisson process, with non-stationary increments see “Poisson process non-homogeneous”
Poisson process, with stationary increments 124—126
Poisson square wave 77
Poliak, H.O. 150
Positive recurrent Markov chain 245 252 253 260—262 264—266
Positive recurrent state 245 253
Prais, S.J. 257
Pratt, J.W. 18
Prediction 3 43 113 114
Probability density functions, definition of 10
Probability density functions, table of 14
Probability distribution function, definition of 10
Probability distribution function, of a stochastic process 72
Probability function 8—10
Probability function, transition see “Transition probability function”
Probability laws, definition of 10
Probability laws, table of 13—15
Probability mass functions, definition of 10
Probability mass functions, table of 13
Pure birth process 293 295 300
Pure birth process, with constant birthrate 118 296 298
Pure birth process, with immigration 305
Pure birth process, with linear birthrate 296 299 302 306
Pure death process 299 300
Quadratic mean, convergence in 78 79 83 91
Queues 190 191 262 273 281—283 288
Queues, classification of 5
queues, description of 199
Queues, finite server 281—288
Queues, infinite server 144 145 147—149
Queues, sample mean of 82
Queues, single server 198 199 284 288
Queues, single server G1/M/1 206 265
Queues, single server M/G/1 258
Queues, two-server 288
Radon — Nikodyn theorem 52
Random event, definition of 8
Random telegraph signal 37 77 80 86 115
Random telegraph signal, definition of 36
Random variables, definition of 8—10
Random variables, identically distributed 10
Random variables, independent 13 15
Random variables, jointly distributed 12
Random variables, kinds of 10
Random variables, uncorrelated 16
Random walk 204 242 254 280
Random walk, approximating a birth and death process 280
Random walk, definition of 23 229
Random walk, duration of 239 241
Random walk, first passage time in 243 244
Random walk, non-current 225 236 253
Random walk, null-recurrent 259
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