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Arratia R., Barbour A.D., Tavare S. — Logarithmic Combinatorial Structures: A Probabilistic Approach
Arratia R., Barbour A.D., Tavare S. — Logarithmic Combinatorial Structures: A Probabilistic Approach



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Название: Logarithmic Combinatorial Structures: A Probabilistic Approach

Авторы: Arratia R., Barbour A.D., Tavare S.

Аннотация:

The elements of many classical combinatorial structures can be naturally decomposed into components. Permutations can be decomposed into cycles, polynomials over a finite field into irreducible factors, mappings into connected components. In all of these examples, and in many more, there are strong similarities between the numbers of components of different sizes that are found in the decompositions of "typical" elements of large size. For instance, the total number of components grows logarithmically with the size of the element, and the size of the largest component is an appreciable fraction of the whole.

This book explains the similarities in asymptotic behavior as the result of two basic properties shared by the structures: the conditioning relation and the logarithmic condition. The discussion is conducted in the language of probability, enabling the theory to be developed under rather general and explicit conditions; for the finer conclusions, Stein's method emerges as the key ingredient. The book is thus of particular interest to graduate students and researchers in both combinatorics and probability theory.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 363

Добавлена в каталог: 18.11.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$A_1$      156
$A_2$      156
$A_r$      25
$c(\omega_0)$      237
$CP(\lambda^{(m)})$      225
$CP(\lambda^{(m)}\{f\})$      225
$C^{(b,n)}$      168
$C^{(n)}$      2 11 35
$C^{{\ast}(n)}$      66 149 155
$C_i^{(n)}$      1 11 35
$C_{(r.s)}$      155
$D_k$      226 231
$d_n$      21
$d_W$      226 231
$d_{BW}$      209 226 231
$d_{TV}$      15 67 226
$ESF(\theta)$      60
$E_{ij}$      153
$E_{ij}^\ast$      153
$F^{\star}$      78
$f_\theta$      23 108
$f_\theta^{(r)}$      23 71 113
$f_\theta^{[r]}$      107
$F_{ij}$      153
$F_{ij}^{\ast}$      153
$GEM(\theta)$      107
$g_1$      156
$G_2$      156
$K_m^{(1)}$      243
$K_n^{(2)}$      332
$K_{0n}$      2 18 29 35
$K_{{\upsilon}m}(\cdot)$      35 149
$L_r$      23
$L_r^{(n)}$      23
$m_g$      177
$m_i$      36 52
$O_n$      25
$PD(\theta)$      23 117
$Po(\lambda)$      21
$p^-_i$      154
$p_\theta$      82
$P_{\theta}^{(\alpha)}$      86 282
$p_{\theta}^{(\alpha)}(x)$      86
$Q_n^\ast$      188
$r_i$      153
$r_i^-$      154
$R_{nb}$      188
$r_{{\upsilon}s}^{(m)}$      296
$S_m$      225
$S_n$      10
$S_n^{(k)}$      18
$T_{0n}$      15 35
$T_{bn}^\ast$      126
$T_{{\upsilon}m}(\cdot)$      69 149
$U_j(l)$      201
$u_l(b,s)$      154
$u_l^\ast$      154
$W^{(n)}$      216
$X^\star$      78
$x^{(n)}$      19
$x^{[r]}$      11
$X^{{\ast}(n)}$      218
$X_\theta$      80
$X_\theta^{(\alpha)}$      85
$Y_r$      21
$Y_r^{(n)}$      21
$Z^\ast$      126
$Z^{(b,n)}$      168
$Z_j$      14 26 28 35 125
$Z_j^\ast$      126
$Z_{ij}$      153
$\alpha_\theta$      188
$\bar \theta$      112 155
$\beta_0$      71
$\beta_1$      71
$\beta_2$      166
$\beta_{01}$      167
$\beta_{02}$      197
$\beta_{11}$      167
$\beta_{12}$      256
$\chi$      129 170 318
$\chi_0$      133
$\chi_{i1}^{(\alpha)}$      154
$\chi_{i2}^{(\alpha)}$      154
$\Delta$      66
$\Delta_i$      153
$\epsilon_{il}$      153
$\epsilon_{il}^\ast$      153
$\epsilon_{\{r.s\}}$      155
$\hat E_n$      154
$\hat u_r^\ast$      252
$\hat W^{(n)}$      216
$\hat X^{(n)}$      214
$\hat Z^{(b,n)}$      168
$\hat Z_j$      213
$\hat \phi_r^\alpha$      252
$\kappa(\upsilon,s,m)$      288
$\lambda_{0n}^\ast$      190
$\lambda_{bn}$      187 324
$\mathbf{P}^{bl}$      310
$\mathbf{P}^{bl}_\ast$      310
$\mathcal{F}_{BL}$      206
$\mathcal{G}$      45 200
$\mathcal{G}_\alpha$      73
$\mathcal{M}$      121
$\mathcal{M}_n$      120
$\mathds{1}\{\cdot\}$      11
$\mu_i$      153
$\mu_i^\ast$      153
$\mu_{0n}^\ast$      191
$\nu_i$      153
$\nu_i^\ast$      153
$\omega(\cdot)$      22
$\omega_0$      237
$\omega_g$      180
$\omega_\theta(\cdot)$      104
$\partial$      45 200
$\phi_l^\alpha$      154
$\phi_{\{r.s\}}$      155
$\Psi^\ast$      72 177
$\Psi^{(n)}$      72 177
$\rho(\cdot)$      22
$\rho_i$      153
$\rho_i^\ast$      153
$\rho_\alpha$      185
$\theta$      3 65 125
$\theta_i$      70
$\tilde C$      155
$\tilde X^{(n)}$      219
$\tilde X^{{\ast}(n)}$      220
$\tilde Z$      153
$\tilde Z[a,b]$      155
$\tilde Z^\ast$      155
$\tilde \theta$      125
$\xi_j$      19
Additive functions      200—224
Additive semigroup, arithmetic      45 200—224
Additive semigroup, Erdos — Wintner theorem      203
Additive semigroup, Erdos — Wintner theorem, rate      209
Additive semigroup, Kubilius Main Theorem      214
Additive semigroup, Kubilius Main Theorem, functional      216
Additive semigroup, Kubilius Main Theorem, functional rate      217
Additive semigroup, regular variation, convergence      220
Additive semigroup, regular variation, functional rate      221
Additive semigroup, slow variation, convergence      214
Additive semigroup, slow variation, functional rate      217
Age-ordering      24
Aldous, D.J.      40 43 175
Approximation of $m^{-1}T_{{\upsilon}m}(Z)$ by $X_\theta$, Kolmogorov      282
Approximation of $m^{-1}T_{{\upsilon}m}(Z)$ by $X_\theta$, Wasserstein      282
Approximation of $m^{-1}T_{{\upsilon}m}(Z^\ast)$ by $X_\theta$ in distribution      277—282
Approximation of $m^{-1}T_{{\upsilon}m}(Z^\ast)$ by $X_\theta$, Kolmogorov      280
Approximation of $m^{-1}T_{{\upsilon}m}(Z^\ast)$ by $X_\theta$, Wasserstein      279
Approximation of $m^{-1}T_{{\upsilon}m}(Z^\ast)$ by $X_\theta^{(\alpha)}$ in distribution      282—283
Approximation of $m^{-1}T_{{\upsilon}m}(Z^\ast)$ by $X_\theta^{(\alpha)}$, ${\upsilon}/m~\alpha$      282
Approximation of $T_{0m}(Z)$ by $T_{0m}(Z^\ast)$, Kolmogorov      276
Approximation of $T_{0m}(Z)$ by $T_{0m}(Z^\ast)$, total variation      272
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{0m}(Z^\ast)$, Kolmogorov, any $\upsilon$      274
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{0m}(Z^\ast)$, Wasserstein, any $\upsilon$      277
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{{\upsilon}m}(Z^\ast)$ in distribution      267—277
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{{\upsilon}m}(Z^\ast)$, interval probabilities      292
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{{\upsilon}m}(Z^\ast)$, point probabilities      288—297
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{{\upsilon}m}(Z^\ast)$, point probabilities, from interval probabilities      289
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{{\upsilon}m}(Z^\ast)$, point probabilities, large $\upsilon$      294
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{{\upsilon}m}(Z^\ast)$, point probabilities, main theorem      293
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{{\upsilon}m}(Z^\ast)$, point probabilities, ratios      296
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{{\upsilon}m}(Z^\ast)$, point probabilities, uniform bound      294
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{{\upsilon}m}(Z^\ast)$, total variation, large $\upsilon$      267
Approximation of $T_{{\upsilon}m}(Z)$ by $T_{{\upsilon}m}(Z^\ast)$, Wasserstein, large $\upsilon$      268
Approximation of $T_{{\upsilon}m}(Z)$ by $X_\theta$ in distribution      267—282
Approximation of $T_{{\upsilon}m}(Z)$ by $X_\theta$, point probabilities      285—299
Approximation of $T_{{\upsilon}m}(Z)$ by $X_\theta^{(\alpha)}$ in distribution      282—283
Approximation of $T_{{\upsilon}m}(Z^\ast)$ by $X_\theta$, point probabilities      297—299
Approximation, Brownian      171—175
Approximation, Brownian, group order      199
Approximation, discrete      149
Approximation, global      71 97
Approximation, global, all components      168
Approximation, global, large components      167
Approximation, global, small components      165
Approximation, local      71 151
Approximation, local, large components      169
Approximation, local, small components      169
Approximation, normal      187
Approximation, normal, group order      199
Approximation, Poisson      186 191 328
Approximation, Poisson-Dirichlet      175—186
Approximation, total variation      226
Arratia, R.      3 7 16 22 23 26 29—31 43 46 47 100 103 104 115 123 125 141 143 166 167 170 171 173—175 187 189 191 199 202
Assemblies      45—46 48—49 54 59 60 70—73 133 139—140
Assemblies, logarithmic      51 70
Asymptotic independence      66 71
Bach, E.      7 31
Balakrishnan, N.      61
Barban, M.B.      29
Barbour, A.D.      7 16 17 20 21 23 26 43 67 100 123 125 166 167 170 173—175 186 187 189—191 199 202 225 309
Barouch, E.      79
Barton, D.      14
Bell numbers      37
Bell, J.P.      134
Bender, E.A.      134
Bergeron, F.      5
Bernoulli, random variables      19 55 58 145 156 186 236
Bernoulli, representation      19 20 101
Best, M.R.      25
Beta random variables      107 119 120 122
Beurling generalized primes      45
Beurling, A.      45
Billingsley, P.      7 28 30 31
Binomial random variables      51
Bollobas, B.      38 39 176
Bovey, J.D.      25
Brenti's representation      187
Brenti, F.      187
Brownian motion      20 216
Buchstab's function $\omega$      22 87—89
Buchstab, A.A.      7 22
Cameron, P.J.      134
Car, M.      163 164 186 302
Card shuffling      44
Cayley, A.      41
Central limit theorem      103
Central limit theorem, functional      20 30 103 171—175 216
Central limit theorem, permutations      19
Central limit theorem, prime factorizations      30
Chen, L.H.Y.      73 225
Chistyakov, V.P.      4
Coagulation      63—64
Coloring      56—57
Completely additive      201
Component spectrum      1 35
Components, fixed number of      101 134
Components, functional limit theorem      216
Components, large      66 71—73 149—151 167 202
Components, largest      130
Components, number of      186—199
Components, small      66 71—73 96—100 151—152 165 202
Components, smallest      128
Conditioning Relation      2 15 26 32 34 48 69 77 97 111—113 125 149 155
Conditioning Relation, general statement      35 65
Conditions, G, A, D, B      156
Conditions, general      155—157
Conventions, general      155
Coupling      181—186
Coupling, Feller      100 181
Coupling, operational time      182
Coupling, Poisson process      181 182
Csoergo, M.      175 217
Daley, D.J.      121
David, F.N.      14
de Montmort, P.-R.      9
Decomposable      35
Delange, H.      29
DeLaurentis, J.M.      20 25 171
Derangement      13
Devroye, L.      61
Diaconis, P.      4 17 39 44 61
Diamond, H.G.      45
Dickman's function p      22 85
Dickman, K.      7 22 85
Dissected representation      153 155 163 167
Distance, bounded Wasserstein      174 187 209 226 231
Distance, Kolmogorov      226 231
Distance, total variation      15 67—70 226
Distance, Wasserstein      185 226 231
Distinct, component sizes      57
Distinct, parts of a multiset      57
Divisor function      158
Donnelly, P.      7 31 103 119 171
Dudley, R.M.      67 207
Elliott, P.D.T.A.      7 29
Engen, S.      107 119
Erdos — Turan theorem, functional rate      199
Erdos — Turan theorem, rate      199
Erdos — Wintner theorem      203
Erdos, P.      7 25 30 115 199
Ewens sampling formula      3 11 60—64 66 77—124 149 155 168 177 181 186 199 202
Ewens, W.J.      61 96
Exponential random variables      118 122
Factorial, falling      11
Factorial, moments      12 59 96
Factorial, rising      19
Feller coupling      16—18 100 181
Feller, W.      16 20 100
Fienberg, S.E.      61
Finite fields, random nonsingular matrices      44
Finite fields, random polynomials      43
Flajolet, P.      5 40 43 44 47
Foata, D.      5 46
Freedman, D.      4
Fristedt, B.      4 37
Functional central limit theorem      30
Functional central limit theorem, general      171—175
Functional central limit theorem, permutations      20 103
Functional central limit theorem, prime factorizations      30
G(n)      200
GEM, density      107 119
GEM, distribution      25 31 107 117—124
GEM, process      117—124
Generalized primes      45
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