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Pollard D. — Convergence of Stochastic Processes
Pollard D. — Convergence of Stochastic Processes



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Название: Convergence of Stochastic Processes

Автор: Pollard D.

Язык: en

Рубрика: Математика/Вероятность/Стохастические процессы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1984

Количество страниц: 215

Добавлена в каталог: 16.12.2004

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$B_H$      See “Brownian motion stretched
$B_P$      See “P-motion”
$C(\mathcal{F},P)$      156
$C[0,\infty)$      108
$D[0,\infty)$, definition of      107
$D[0,\infty)$, metric of uniform convergence on compacta      108
$D[0,\infty)$, Skorohod metric      123
$E_n$      See “Empirical process”
$E_P$      See “P-bridge”
$J(\delta)$      xiv (See also “Covering integral”)
$N(\delta)$      xiv (See also “Covering number”) $\parallel\cdot\parallel$
$P^\circ_n$      15 26 149
$P_n$      See “Empirical measure”
$U_n$      See “Empirical process uniform”
$\frac89$-argument      24
$\mathcal{B}^P$      156
$\mathcal{P}$      See “Projection $\sigma$-field”
$\mathcal{X}$      See “Function space”
$\pi_S$      See “Projection maps”
$\rho_P$      xiv
$\triangle_n(\xi)$      21
2-means      See “k-means”
Abuse, of notation      44 171
Adapted      176
Aldous, D.      133 137 185 187
Aldous’s condition      133
Alexander, K. S.      38 166
Alexandroff, A. D.      85
Allocation Lemma      77
Almost-sure representation      See “Representation Theorem”
Amato, B. R.      viii
Approximation Lemma      27
Araujo, A.      37
Ash, R. B.      39
Autoregression      12 174
Axiom of Choice      65 86
Barry, D. G.      viii
Bennett, G.      193—194
Bennett’s Inequality      160 192 194
Bernstein’s inequality      193
Bertrand-Retali, M.      38 41
Bias, of density estimator      35 42
Billingsley, P.      36 61 85—86 117—118 136
Binomial coefficient      19
Bollobas, B.      86
Bolthausen, E.      167
Bounded variation      42
Boyce, J. S.      viii
Breiman, L.      4 22 106 118
Bremaud, P.      186
Breslow, N.      186
Brown, B. M.      166 185
Brownian bridge      64
Brownian bridge, denned      3 95
Brownian bridge, existence      101 119
Brownian motion, construction from brownian bridge      103
Brownian motion, defined      95
Brownian motion, modulus of continuity for      146
Brownian motion, stretched-out      178
Brownian motion, tied-down      3
Cadlag      3 89 176
Cauchy sequence      81
Censored data, estimation from      182
Central limit theorem, autoregression      174
Central Limit Theorem, Empirical for distribution functions      97
Central limit theorem, empirical process      157
Central Limit Theorem, Empirical, uniform case      96
Central Limit Theorem, Liapounoff      51
Central Limit Theorem, Lindeberg      52
Central limit theorem, martingale-difference array      171
Central limit theorem, minimization functional      141
Central Limit Theorem, Multivariate      57
Cervonenkis, A. Ya.      37
Chaining      37
Chaining for stochastic processes      142—145
Chaining Lemma      144
Chaining, restricted      160—163
Characteristic function      54
Characteristic function, Continuity Theorem      55 60
Characteristic function, inversion formula      63
Characteristic function, uniform convergence on compacta      59
Characteristic function, uniquely determines distribution      55
Chernoff, H.      165 190
Chi-square, Pearson’s      46
Chibisov, D. M.      85
Chung, K. L.      117
Clustering      See “k-means”
Combinatorial method      13
Compactness in metric spaces      81—82
Compactness Theorem      82
Compactness, sequential      82
Completely regular, point of metric space      67
Completely regular, topological space      68
Conditional variance of increment      171
Conditional variance, process      176
Consistency for autoregression      12
Consistency of k-means      9
Consistency of optimal solutions      12
Continuity set      71
Continuity Theorem, for characteristic functions      55 60
Continuous Mapping Theorem for euclidean space      46
Continuous Mapping Theorem for metric spaces      70
Continuous Mapping Theorem, failure of      66
Convergence in distribution in euclidean space      43
Convergence in distribution of empirical process      97 157
Convergence in distribution of L2 martingale      179
Convergence in distribution, Aldous’s condition for, in $D[0,\infty)$      134
Convergence in distribution, in $D[0,\infty)$, for Skorohod metric      131
Convergence in distribution, in $D[0,\infty)$, for uniform convergence on compacta      108
Convergence in distribution, in metric spaces      65
Convergence in distribution, of uniform empirical process      96
Convergence in distribution, process with independent increments      104 135
Convergence in distribution, via semicontinuity      73
Convergence in distribution, via uniform approximation      70
Convergence Lemma for euclidean space      44—46
Convergence Lemma for metric spaces      68
Convex sets, shattering      17 23
Convolution      35 49 54
Coupling      76—80
Coupling in Compactness Theorem      84
Covering integral as modulus of continuity      147
Covering integral for metric space      143
Covering numbers, $\mathcal{L}^1$ for classes of functions      25
Covering numbers, $\mathcal{L}^2$ for classes of functions      31
Covering numbers, direct      164
Covering numbers, for metric spaces      143
Covering numbers, for polynomial classes      27
Covering numbers, inequalities between      34
Covering numbers, measurability of      198
Covering numbers, random      150
Criterion function, minimization of      28
Criterion function, random      12
Cross-section, measurable      197
Crossword puzzle      76
Crowley, J.      186
C[0, 1]      1 90
DeHardt, J.      36
Dellacherie, C.      176 179 185 195—197
Delta method      63 189
Denby, L.      37
Density, convolution      55
Density, estimation      35
Differentiability, in quadratic mean      140 151
Direct approximation      8 164
Discrimination, linear; polynomial; quadratic      17
Distribution function, pointwise convergence of      43 46 53
Donsker, M. D.      117—118
Donsker’s theorem      See “Central Limit Theorem Empirical”
Doob — Meyer decomposition      176
Doob, J. L.      3 117
Dudley, R. M.      vii 37—39 85—86 117—118 166—168 193 200
Durbin, J.      118
Durst, M.      39
D[0, 1]      3 89—90
Eggleston, H. G.      200
Empirical measure, bivariate      12
Empirical measure, defined      6
Empirical process, central limit theorem for      96—97
Empirical process, defined      2 95 97 140
Empirical process, measurability of      65 199
Envelope, for classes of functions      24 151
Equicontinuity Lemma      150
Equicontinuity of class of functions      74
Equicontinuity, stochastic      139
Erdos, P.      118
Ergodicity      9 12
Exponential bound      See “Tail probability”
Exponential Bounds      16
Extreme-value distribution      128
Feller, W.      53 63 193
Fidi projection      92 (See also “Projection maps”)
Fidis (finite-dimensional distributions)      3 90 92
Function space $\mathcal{X}$      155 199
Functional in function space      156
Functional, first passage time      109 124
Functional, jump      179
Functional, terminology      2 7
Functions, infinitely differentiable      49
Gaenssler, P.      vii 36—37 40 118 167
Gaussian process, characterization      106
Gaussian process, indexed by functions      146
Gihman, I.I.      4 117—118 136—137 166 185
Gill, R. D.      186
Gine, E.      37 40 166—167
Glaisher, J. W. L.      61
Glivenko — Cantelli theorem, classical      6—7 13
Glivenko — Cantelli theorem, generalized, for classes of functions      25
Glivenko — Cantelli theorem, generalized, for classes of sets      18 22
Gnedenko, B. V.      37
Goodness-of-fit statistic      2
Goodness-of-fit statistic with estimated parameters      99 159
Goodness-of-fit statistic, Kolmogorov’s, limiting distribution of      113
Goodness-of-fit statistic, Neyman’s      47
Graph, of real-valued function      27
Grid, approximations constructed from      91 124
Hajek, J.      86 117
Hall, P.      185
Halmos, P. R.      39
Hansel, G.      86
Hartigan, J. A.      viii 36 166
Heuristic argument, Doob’s      4 64
Hewitt — Savage, zero-one law of      22
Heyde, C. C.      185
Hoeffding, W.      193—194
Hoeffding’s Inequality      16 26 31 150 164 191
Huber, P. J.      38 165 168
Independent increments, convergence in distribution      104 135
Independent increments, definition      103
Innovations      174
Integration      16
Interpolation in $D[0,\infty)$      124
Interpolation in a function space      158
Interpolation in D[0, 1]      92
Interpolation, continuous      101
Jacobs, K.      86
Jacobsen, M.      186
k-means, central limit theorem      153
k-means, consistency      9 30
k-means, method of      9
Kac, M.      117—118
Kaplan — Meier estimator      182
Kelley, J. L.      39
Kernel, smoothing      35
Kernel, translates of      42
Kildea, D. G.      166
Kolchinsky, V. I.      38 166
Kolmogorov, A. N.      99 117 136 166
Kurtz, T. G.      137
Laplace transform, for hitting time      112
Le Cam, L.      37—38 85 166—167 200
Levy, P.      61
Liapounoff condition      51
Liapounoff, A. M.      61 63
Lindeberg condition      52 106
Lindeberg condition for martingale-difference arrays      171
Lindeberg, J. W.      61
Lindelof’s theorem      68 72 87
Lindvall, T.      118
Lipschitz condition      45 74 101
Liptser, R. S.      185
Little links      163
Location, center of      28
Loeve, M.      118
M-estimator, generalized      12
M-estimator, stochastic equicontinuity      165
Major, P.      86
Mann, H. B.      189
Marginal stacks, with coupling      77
Maritz, J. S.      166
Markov property for empirical process      96
Marriage lemma      77
Martin, R. D.      37
Martingale, $L^2$      176
Martingale, $L^2$, convergence to gaussian process      179
Martingale, continuous time      176
Martingale, difference array      171
Martingale, reversed      39
Martingale, reversed, empirical measure as      22
Martingale, stopped      180
Maximal inequality      15
Maximal inequality by chaining      144
Maximal inequality for $L^2$-martingales      177
Maximal inequality for cadlag processes      94
Maximum likelihood      138
McKean, H. P.      146
McLeish, D. L.      185
Measurability, for random elements of metric spaces      64—66
Median, central limit theorem for      53 98
Median, consistency      7
Median, spatial, central limit theorem for      152
Median, spatial, consistency of      28
Metivier, M.      185
Metric, bounded-Lipschitz      74
Metric, entropy      166
Metric, Prohorov      75 79 88
Metric, Skorohod, on $D[0,\infty)$      123—124
Meyer, P-A.      176 179 185 195—197
Minimization of random criterion function      140
Modulus function for $D[0,\infty)$      125
Multinomial distribution      46
Neveu, J.      22
Neyman, J.      47
Nolan, D. A.      viii
Norm      14 24 156
Normal distribution, characteristic function of      63
Normal distribution, multivariate      29
Normal distribution, tail probabilities      191
Optimal centers      10
Outer measure      70
Oxtoby, J. C.      65 86
P-bridge      149
P-motion      147
Parr, W.      118
Parthasarathy, K. R.      85 136
Partial sum process      106 109
Permissible, classes of functions      24
Permissible, classes of sets      17
Permissible, definition      196
Philipp, W.      166 200
Pickands, J.      136
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