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Emanuel Parzen Stochastic processes (Classics in Applied Mathematics)
Emanuel Parzen  Stochastic processes (Classics in Applied Mathematics)









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: Stochastic processes (Classics in Applied Mathematics)

: Emanuel Parzen

:

This introductory textbook explains how and why probability models are applied to scientific fields such as medicine, biology, physics, oceanography, economics, and psychology to solve problems about stochastic processes. It does not just show how a problem is solved but explains why by formulating questions and first steps in the solutions.
Stochastic Processes is ideal for a course aiming to give examples of the wide variety of empirical phenomena for which stochastic processes provide mathematical models. It introduces the methods of probability model building and provides the reader with mathematically sound techniques as well as the ability to further study the theory of stochastic processes.
Originally published in 1962, this was the first comprehensive survey of stochastic processes requiring only a minimal background in introductory probability theory and mathematical analysis. Stochastic Processes continues to be unique, with many topics and examples still not discussed in other textbooks. As new fields of applications (such as finance and DNA analysis) become important, researchers will continue to find the fundamental and accessible topics explained in this book essential background for their research.


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$D_{n}$-statistic      see Kolmogorov Smimov test
$W_{n}$-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 231233 236 237 246
Absorption, time before      238
Absorption, time to      238246 263
Acceleration in chaos      150 152
Accessible states      208 211 223 224
Anderson, T.W.      29 102 192
Aperiodic Markov chain      256258 270273
Aperiodic state      255
Approximately normal distribution      40 88 97 99 103 141 142 157 180 181 267269 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      7274
Bachelier, L.      29
Backward equation      291293 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 4549
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 2729 80 95 96 98 129
Busy channels or servers      144149
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 203205 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      8487 91
Chebyshev inequality for, ergodic      75
Chebyshev inequality for, general discussion of      66 71 74 77 96 108
Chebyshev inequality for, in harmonic analysis      108116
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      208211 222225 235 245 256
Communicating classes, classification of      222 223
Communicating states      208 224227 244 245 256 262264 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 128130 146 156
Conditional characteristic function, definition of      56 57
Conditional expectation      41
Conditional expectation, definition of, continuous case      5153
Conditional expectation, definition of, discrete case      4244
Conditional expectation, definition of, joint      58
Conditional expectation, properties of      6265
Conditional mean      see Conditional expectation
Conditional moments, definition of      5456 58
Conditional probability density function, definition of      51
Conditional probability distribution function, definition of      4143
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      136139
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 162166 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      8487 92
Covariance function, for busy servers      149
Covariance function, in harmonic analysis      109115
Covariance function, of a normal process      91 92 94 96
Covariance function, of a white noise      113115
Covariance function, of an ergodic process      7476
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      7072
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      7981 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      164166 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      8385 9192
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      7274
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      136139
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      171173
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 173177 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      104107 110115
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      242244 253
First passage time, mean, recurrent case      259 264
Forward equation      291293 300
Foster, F.G.      260
Fourier integral      108112
Fourier series      108
Fourier transform      12 106109
Frequency response function      105108 110113
Friedman, B.      89
Fubini's theorem      273 275
Fuller, A.T.      37
Functional equations, solution of      121123
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      124126
Hypothesis testing      3 135 137 139 142144
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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