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Athreya K.B., Ney P.E. — Branching Processes

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Название: Branching Processes

Авторы: Athreya K.B., Ney P.E.

Аннотация:

Athreya (Cornell U.) and Ney (U. of Wisconsin) cover developments in the theory of branching processes in the decade following the publication of Harris's important Theory of Branching Processes, although enough older material is presented to make the work self contained. In the first two chapters they develop the Galton-Watson process, later reducing analogous questions of the treatment of continuous time (Markov and age-dependent) cases to their Galton- Watson counterparts. After these discussions of single type processes, they present a chapter dealing multi-type branching processes. A final chapter explores applications. This is a paperbound edition of a work first published in 1972

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Рубрика: Математика/Вероятность/Стохастические процессы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1972

Количество страниц: 294

Добавлена в каталог: 21.05.2005

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Предметный указатель

Absolute continuity of W      34 52 172 228
Additive property      3 102 184 201 257
Age distribution, stationary      179
Age-dependent branching process      108 137—180 230—238 248—249
Age-dependent branching process, relation to Galton — Watson process      141—144
Ahlfors, L.V.      53
assumptions      3 12 24 40 58 82 105 118 119 132 139 184 185 192 201
Athreya, K.B.      30 34 55 63 80 85 86 99 118 120 153 172 177 193 206 207 209 220 223 224 234 235 249 250—256 267
Backward equation      see Kolmogorov equations
Barucha-Reid, A.T.      239
Basic lemma, critical case      19 113 158 189
Baum, L.F.      160
Bellman, R.      137 147 152n 171 238
Biihler, W.J.      242
Billingsley, P.      211
Binary fission      109 127 239
Birth and death process      109
Blackwell, D.      219
Branching Brownian motion      242—246
Branching diffusions      242—246
Branching random walk      230—238 247—248
Brown, B.J.      56
Brownian motion      242- 246
Cascades      239—242
Chapman — Kolmogorov equation      2 107 257
Chistyakov, V.P.      80 147 148 149 152n 168 171 177
Chover, J.      147 148 152n 159 168 178 226
Chung, K.L.      56 104 119n
Church, J.D.      266 267
Coddington, E.A.      203
Comparison lemma      22
Compound Poisson process      45 126
Conner, H.E.      115 242
Continuous branching function      257—262
Continuous state branching process      257—262 267
Continuous time, Markov branching process      79 102 219
Continuous time, Markov branching process, relation to Gallon — Watson process      110—111 130—135
criticality      8 203 226 256
Darling, D.A.      34 65
Decomposition of processes      47—53 123 143
Density of W      34 37 80 84—87 112 130—131 192
Diffusion      62 242—246 260 261
Directly Riemann integrable      146
Dmitriev, N.      102
Doney, R.A.      180
Doob, J.L.      9 10 121
Dubuc, S.      37 80 93 97 131
Dynkin, E.B.      258
Embeddability problem      130—135
Embedded Galton — Watson process      110—111 130—135 178
Embedded generation process      141—144 228
Embedded urn scheme      219—224
Emigration      266
Environmental process      251
Esty, W.W.      Vlll
Exponential limit law      see limit law exponential
Extinction time      16 63
Extinction, probability of, age-dependent process      143—144 159 226—227
Extinction, probability of, continuous process      205
Extinction, probability of, continuous time Markov process      107—108
Extinction, probability of, Galton — Watson process      7—8 19
Extinction, probability of, multitype Galton-Watson process      186 188 191
Extinction, process in random environments      252—255
Extinction, transformation to q=0      47—53
Family trees      3 105 137
Feller, W.      35 89 104 145 170 177 180 219
Fisher, R.A.      162
Forward equation      see Kolmogorov equations
Foster, J.A.      263 264 265
Freedman, D.A.      221 224
Friedman's urn scheme      see Urn schemes
Frobenius theorem      185
Functional equations      10 12 16 29 33 40 43 46 68 112 115 130 172 192 251 258.
Galton — Watson process      1—101 141—144 205 242—246 250 262
Galton — Watson process, relation to age-dependent process      141—144
Galton — Watson process, relation to continuous state branching process      262
Galton — Watson process, relation to continuous time Markov process      110—111 130—135
Galton, Francis      1 2
General time branching process      see Age-dependent branching process
Generating function      2 3 4—7 8 67 106—107 115—118 137—139 182—183 200—201 231
Generating function, elementary properties      4—6
Generating function, for age-dependent process      137—139
Generating function, for continuous time Markov process      106—107
Generating function, for multitype process      182 183 200—201 255
Generating function, for stationary measure      68—69 72
Generating function, infinitesimal      106 200—201
Generating function, linear fractional      6—7 22 71—72 109 132—134
Generations      54 141—144 241—242
Gnedenko, B.V.      33
Goldstein, M.      1 141 158 169 226
Green function      66 90—93 98
Green function, truncated      94
Harmonic function      66 93 99 100
Harris, T.E.      2 3 6 7 8 9 34 47 59n 64 67 72 74 99 102 105 107 108 111 116 130 131 132 137 139 147 152n 162 171 172 178 179 181 185 186 193 201 238 239 246 249 251
Heathcote, C.R.      263 264 267
Heyde, C.C.      30 55 56
Imai, H.      80
Immigration      10 262—266 267
Infinite line of descent      49
Infinitesimal generating function      106 200—201
Infinitesimal probability      103—104 201
Instability      8 255 266—267
Integral equations      138 145—146 151—152 179 180 225 226 231 232 237 240
Invariance principle      56
Invariant function      see Harmonic function
Invariant measure      see Stationary measure
Iteration, functional      2 107 110 138 249 258
Jagers, P.      172
Janossy G-equation      240
Jirina, M.      181 186 257

Joffe, A.      15 38n 45 186—191 227
Kaplan, N.      256
Karlin, S.      38 n 41 42 68 79 88 99 116 118 120 130—135 185 194 207 224 249 250—256 267
Katz, M.      160
Kemeney, J.G.      66 70 93 98
Kendall, D.G.      2 64 125 127 129 130 136 219
Kesten, H.      20 24 68 73 74 87 88 90—93 94 148 193 209 211 227
Kharlamov, B.P.      242
Kingman, J.F.C.      73 113 115
Knapp, A.W.      66 70 93 98
Kolmogorov equations      103 106 139 200—201 260
Kolmogorov's rock-crushing problem      241
Kolmogorov, A.      3 15 19 33 102 105 113 241 256
Kuczma, M.      131
Kurtz, T.      193
Lamperti, J.      56 58 61 62 125 126 178 257—262
Law of the iterated logarithm      56 129
Levinson, N.      24 139 172 203
Levy, P.      207
Limit law, exponential      20 73 113 169 191 227 256
Limit law, for $m=\infty$ case      34 65
Limit law, for critical case      18 19—20 61—62 73 113 169 180 191—192 227 256 265
Limit law, for subcritical case      16 18 45 63 64—65 114 170 171 178 186—188 226 227 256
Limit law, for supercritical case      9 16 24 30 33—34 63 80—81 85 86 99 100 112 123—124 136 172 177 179 192 228 256
Limit law, global      80 85 100
Limit law, local      73—82 98 99 100
Linear fractional generating function      6—7 22 71—72 109 132—134
Lipow, C.W.      89
Lipschitz continuity of density      84
Loeve, M.M.      216
Malthusian case      162—167 170 178 226 232—233
Malthusian parameter      146 177 225
Malthusian population growth      9 111 162
Markov branching process      see Continuous time Markov branching process
Martin boundary      69 219
Martingale      9 31 111 119 193—194 207 209
Martingale, methods      246—249
McGregor, J.      38n 41 42 68 79 88 116 130—135 207 224 267
Mean      see Moments
Mean matrix      184 202 222 225
Minimal process      104 119 201
Mode, C.J.      181 226
Moments, for age-dependent process      143—144 150—158
Moments, for branching random walk      232—238
Moments, for continuous state branching process      259—260
Moments, for continuous time Markov process      108—109 111
Moments, for Galton — Watson process      4
Moments, for multitype branching process      184—185
Moments, for multitype continuous time Markov process      202—205
Monotone ratio lemma      12
Mullikin, T.W.      181 189 191
Multitype branching process      181—228 265
Multitype, age-dependent      225—227 228
Multitype, continuous time Markov      199 209
Neuts, M.F.      64
Neveu, J.      66
Ney, P.      20 58 61 62 68 73 74 80 85 86 87 88 90—93 94 99 147 148 152n 159 168 178 189 192 226 230 232 233 234 235 236 240
Non-explosion hypothesis      104 177 201 222 228 261
Non-singular process      184
Nucleon cascades      239—242
Pakes, A.G.      267
Papangelou, F.      11 15
Pinsky, M.A.      267
Poisson process      125 130 263
Polya's urn scheme      see Urn schemes
Positive regular      184 202
Potential theory      66—101
Q-process      56—60 62 64
Queueing process, relation to branching process      64
Random environments      249—256 267
Ratio theorem      15
Renewal function      66 144 see
Renewal theory      144—150 232 237
Reproductive age value      249
Ryan, T.A., Jr.      163 170 226
Samuels, M.L.      235 242
Savits,T.H.      105 139 201 256
Selivanov, B.      1 38n
Seneta, E.      11 15 18 24 30 33 34 38n 41 65 179 228 257 264 265 267
Sevastyanov, B.A.      102 116 159 263
Sims, J.      94
Slack, R.S.      19 20 100
Slowly varying      63
Smith, W.L.      249—250 252 254
Snell, J.L.      66 70 93 98
Snow, R.N.      226
Space time boundary      98—99
Spectral representation      41—42 79 178 267
Spitzer, F.      20 22 38n 45 58 61 66 68 69 73 74 87 88 90—93 94 186—187 189—191 227 232
Split times      118—123 154 206—209 221
Stationary age distribution      179
Stationary distribution      66
Stationary measure      66 73 87—89 91 100
Stigum, B.P.      24 34 193 209 211 227
Sub-exponential case      168—169 171 235—236
Sub-exponential class      147—150
Total number of particles      180
Transience      7—8 186 266—267
Type-space      246 249
Urn schemes      219—224 228
Vere-Jones, D.      15 257
Vinogradov, O.P.      147 163
Wainger, S.      147 148 152 168 178
Watanabe, S.      125 242 245 246 257
Watson, H.W.      2
Weiner, H.J.      226 227
Wilkinson, W.E.      249—250 252 254
Williamson, J.A.      64 264 265
Yaglom theorem      18 114 170 186—187 256
Yaglom, A.M.      15 20 113
Yule process      109 127—130 136

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