Gut A. — Stopped Random Walks: Limit Theorems and Applications :: Электронная библиотека попечительского совета мехмата МГУ

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Gut A. — Stopped Random Walks: Limit Theorems and Applications

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Название: Stopped Random Walks: Limit Theorems and Applications

Автор: Gut A.

Аннотация:

Classical probability theory provides information about random walks after a fixed number of steps. For applications it is more natural to consider random walks evaluated after random number of steps. This book offers a unified treatment of the subject and shows how this theory can be used to prove limit theorems for renewal counting processes, first passage time processes, and certain two-dimensional random walks, and how these results are useful in various applications.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1988

Количество страниц: 198

Добавлена в каталог: 19.05.2006

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$(C, \mathcal{C})$       148 157 171—176 177
$(D, \mathcal{D})$       149 154—157 164 172—176 178
$(D, \mathcal{D})$ ,       148 155 164 176
-convergence theorem      166
Ahlberg      6 119
Aleskeviciene      87
Anderson      61
Anscombe      3 8 15 42
Anscombe — Donsker invariance principle      147—149
Anscombe, condition      15 44
Anscombe’s theorem      3 4 15—16 36 44 56 85 137 147 151
Anscombe’s theorem, -analogue of      33
Anscombe’s theorem, multidimensional      150
Arjas      53
Asmussen      4 48 62 109 125
Athreya      53 61
Barlow      122—124
Basu      152
Baum      42 43
Belyayev      122
Berbee      107
Berry      61
Bickel      62
Billingsley      7 18 149 150 152 153 160 166 169 171 174 176
Bingham      103 156 178
Binomial process      1
Blackwell      8 23 26 27 44 52 61 76 87 88 90
Breiman      7
Brown      7
Brownian motion      see “Wiener process”
Burkholder      20 29 167 169 170
Carlsson      62 89 100
Central limit theorem      1 7 8 15 17 18 29 36 38 55 61 72 85 92—96 102 108 111—114 130 137 147 168—169
Central limit theorem, moment convergence in      18 168—169
Central limit theorem, multidimensional      117
Chang      33 163
Choquet      53
Chow      7 17 23 24 33 34 67 92—95 103 106 107 132 137 142—144 146
Chromatography      6 118—121
Chung      4 15 62 65 73 90 100 165 166 168 170
Cinlar      4 48 62
Cluster set      see “Limit points set
Combinatorial methods      4 46
composition      149 151 176
Continuity of function(al)s      164 174—175
Continuity of function(al)s, composition      149 151
Continuity of function(al)s, first passage time      156
Continuity of function(al)s, inversion      155
Continuity of function(al)s, largest jump      154
Continuity of function(al)s, projections      150 156 178
Continuity of function(al)s, supremum      155
Continuous mapping theorem      149 150 154 163 174—175
Convergence, almost sure (a.s.)      4 9 10—14 17 41 44 54 67 70 83—85 97 130 135 145 166 175
Convergence, complete      9 42—43 103
Convergence, in       4 9 17 37 39 41 44 166
Convergence, in distribution      4 9 10 15—17 36 55 58 72 85—87 98 102 111—112 115—117 130—131 137—138 166 179
Convergence, in probability      4 9 10—13 39 166 176
Convergence, in r-mean      4 (see also “Convergence in
Convergence, moment      4 9 17—20 36—39 54 56 57 59 87 89 92—97 98—102 106 108 109—114 130—132 143—145 165—167
Convergence, of finite-dimensional distributions      150 156 171—172
Convergence, of probability measures      171—176 (see also “Weak convergence”)
Convergence, rate      42—44 103—104 131
Convergence, weak      see “Weak convergence”
Counters      1 2 124—125
Coupling      53
Cox      106 109 122
Cramer      125
Daley      89
Davis      20 169
De Acosta      177 178
de Groot      23
de Haan      134 180
DENY      53
Doeblin      53
Domain of attraction      see “Stable law”
Doney      62 67
Donsker      7 147
Donsker’s theorem      147 149 158 164 172 177
Donsker’s theorem, Anscombe version of      147 162
Doob      21 48 53 54 87 167 168 170
Dynkin      61
Englund      61
Erdos      6 7 42 52 73 85
Erickson      53 61
Esseen      61
Essen      89
Farrell      62
Feller      4 6 24 36 48 52 54 60 61 65 90 134 180
First passage time(s)      2 5—9 22 24 39 50 74—107 108 109—118 132 133 147 151—159 163
First passage time(s), auxiliary      138 139 144
First passage time(s), central limit theorem      56 85 93
First passage time(s), complete convergence      103
First passage time(s), convergence rate      103—104
First passage time(s), excess over the boundary      76 97 overshoot”)
First passage time(s), for the ladder height process      77 81
First passage time(s), law of large numbers      55 83—84 93 105
First passage time(s), law of the iterated logarithm      102—103 163
First passage time(s), momentgenerating function      81
First passage time(s), moments, convergence      55 92—97 106
First passage time(s), moments, finiteness      50 78—81
First passage time(s), overshoot      76 97—102 106
First passage time(s), process      3 5 6 50 55 57 75—107 119
First passage time(s), subadditivity      55 83
First passage time(s), uniform integrability      55 57 90—92 94—95
First passage time(s), weak convergence      151—157
First passage times across general boundaries      7 75 109 133—146 159—161 163
First passage times across general boundaries, central limit theorem      137 144
First passage times across general boundaries, law of large numbers      135—137 143
First passage times across general boundaries, law of the iterated logarithm      145—146 163
First passage times across general boundaries, momentgenerating function      134
First passage times across general boundaries, moments, convergence      143—145
First passage times across general boundaries, moments, finiteness      133—134
First passage times across general boundaries, overshoot      145
First passage times across general boundaries, weak convergence      159—161
Fluctuation theory      4 46
Fuchs      4 65
Functional central limit theorem      147 172 weak”)
Functional limit theorem      7 132 147—164
Gafurov      62
Garsia      61 167
Generalized arc sine distribution      61
Gikhman      151 173
Gnedenko      122
Goldie      103
Grubel      90
Gundy      7
Gut      6 7 22 26 28 34 35 40 43 44 75 76 78 84—86 93—98 100 102 103 107 109 111 113 115 117 119 122 126 127 133 134 136—140 144—146 150 152 153 156 158 161 182
Hall      163
Hartman      41 177—178
Hatori      54
Heath      128
Heyde      73 78 80 81 83 85 86 89 95 132 163
Hogfeldt      158
Hoist      126 127
Horvath      163
Hsiung      17 33 34 92 94 103 132 137 142—144 146 163
Hsu      42 43
Huggins      163
Hunter      62
Iglehart      156 164
Insurance risk theory      1 2 125—126
Invariance principle      147—164 (see also “Functional limit
Invariance principle, almost sure      see “Invariance principle strong”
Invariance principle, Anscombe — Donsker      147—149
Invariance principle, strong      7 42 147 161—164 177—178
Invariance principle, weak      7 147—161 175
Inverse relationship      138 146 155
Inverse relationship between partial maxima and first passage times      75 80 85 103 132
Inverse relationship between renewal and counting processes      49 56 75
Jagers      4 48 53

Janson      6 11 12 28 34 35 69 73 78 82 97 101 107 109 111 113 115 118 122 132 139 140 158
Kac      6 7 73
Kaijser      121
Karamata      180
Katz      42 43
Kemperman      62
Kesten      62
Kiefer      132
Kingman      83
Kolmogorov      14 27 52 84
Ladder, epoch      5 65—66 77—78 98 104—106 129
Ladder, epoch, ascending      8 65
Ladder, epoch, ascending, strong      5 65 68—70 77
Ladder, epoch, ascending, weak      67 70
Ladder, epoch, descending, strong      67
Ladder, epoch, descending, weak      66
Ladder, height      66—67 98 101 129
Ladder, height, first passage times for      see “First passage times”
Ladder, height, strong ascending      66 68—69 77
Ladder, method      76 77—78 81 83 87—89 91 93 94—96 97 101 152
Ladder, variable      2 4 26 46 65—67 77
Lai      7 17 28 33 34 59 67 85 91 92 94 100 133 142—144
Lalley      62 87 102
Lamperti      61
Law of Large Numbers      1 8 13—14 17 29 33 37 44 54 70 83—84 92—94 108 109—111 130 135—137
Law of large numbers, convergence rate      42—43
Law of large numbers, converse      27
Law of large numbers, Erdos — Renyi      85
Law of large numbers, martingale proof of      44
Law of large numbers, moment convergence      17—18 44
Law of the iterated logarithm      1 8 9 41—42 43 102—103 108 117 131 147 161—164 177—178
Law of the iterated logarithm, Anscombe version of      42 162
Law of the iterated logarithm, converse      41
Limit points      41 177
Limit points, set of      41—42 162—164 177—178
Lindberger      161 164
Lindvall      53 175
Local limit theorem      87 106
Loeve      11 14 166
Lorden      100 101
Maejima      106 107
Marcinkiewicz      14 19 20 29 44 61 84 110 136 167 169
Martingale      44 53 167—171 178
Martingale, convergence      168
Martingale, moment inequalities      see “Moments”
Martingale, optional sampling theorem      21—25 167 170—171
Martingale, reversed      44 168 178
McDonald      53
Meyer      53
Miller      122
Mohan      53 57 61
Moments, boundedness      39 40
Moments, convergence      see “Convergence”
Moments, finiteness      24 25—28 49 51 78 79 97 132 133—134 145
Moments, inequalities for martingales      167 169
Moments, inequalities for stopped random walks      20—28
Moments, inequalities for sums of independent random variables      167 168—169
Moments, Marcinkiewicz — Zygmund      19 20 167 169
Mori      107
Nagaev      62
Negative binomial process      47 49 57 59 106
Nerman      100
Neveu      24 168
Ney      53 107
Niculescu      53
Number of renewals      48—49
Number of visits      6 64—65
Number of visits, expected      64—65 90
Nummelin      53
Optional sampling theorem      see “Martingale”
Orey      90
Ornstein      4 6 65
Partial maxima      5 7 46 67—73 75 108 128—132 155 159 164
Partial minima      5 67—70 128 132
Plucihska      122
Point, persistent      4 65
Point, possible      4
Point, transient      65
Poisson process      47 48 53 57 59 125 126
Pollard      52 90
Port      6
Prabhu      4 48 65 68 69 73 75 90 109 125 126
Projections      150 171
Projections, continuity of      see “Continuity of function(al)s”
Proschan      122—124
Pyke      20 124
Queueing theory      1 2 126
Random change of time      148 151 160 176
Random index      1—9 17 46 68
Random walk      1—7 8 46 62—73 99—100
Random walk, arithmetic      47 65 97 105 107
Random walk, arithmetic with span d      47 63
Random walk, arithmetic, d-arithmetic      47 88 90 95—99
Random walk, Bernoulli      1 47
Random walk, Bernoulli, symmetric      1
Random walk, classification      63—68
Random walk, coin-tossing      1
Random walk, drifting      4 6
Random walk, drifting to $+\infty$       5 7 64—67 70 104 with
Random walk, drifting to $-\infty$       64—67 69 70
Random walk, maximum of      see “Partial maxima”
Random walk, minimum of      see “Partial minima”
Random walk, nonarithmetic      47 63 65 88 90 95—100 102
Random walk, oscillating      4 6 64 66—68
Random walk, persistent      4 64—65
Random walk, randomly indexed      1—9 (see also “Stopped random walk”)
Random walk, simple      1 63—65 104—106 107
Random walk, simple, symmetric      1 24 35 66
Random walk, stopped      see “Stopped random walk”
Random walk, transient      4 6 64—65
Random walk, two-dimensional      3 6 108 109 stopped”)
Random walk, with multidimensional indices      107
Random walk, with positive drift      3 5 73 74—107 108 128—132 133—146 151—157 159—161 163 164
Recurrent even      4
Regularly varying function      53 57 61 87 116 134 155 180—182
Relative compactness      162 —164 178
Reliability theory      1 2 8 123—124
Renewal counting process      2—6 8 9 48—57 68 82 118
Renewal counting process, Berry — Esseen theorem      61
Renewal counting process, central limit theorem      55
Renewal counting process, for random walks with positive drift      82
Renewal counting process, large deviation      62
Renewal counting process, law of large numbers      54
Renewal counting process, law of the iterated logarithm      61
Renewal counting process, momentgenerating function      49
Renewal counting process, moments, convergence      54—57
Renewal counting process, moments, finiteness      49
Renewal counting process, uniform integrability      54—57
Renewal function      5 6 49—53 61 90
Renewal function, extended      89
Renewal measure      63 90
Renewal measure, harmonic      90
Renewal process      1—6 46—62 74 76 82 105 108 113 118 125 152 164
Renewal process, age      60 61
Renewal process, alternating      6 122
Renewal process, arithmetic      47 48 54 66
Renewal process, arithmetic with span d      47
Renewal process, arithmetic, d-arithmetic      47 50 52 53 56—58
Renewal process, coupling proofs      53
Renewal process, delayed      62
Renewal process, integral equation      50 52
Renewal process, lifetime      58 60
Renewal process, lifetime, residual      58—60 61 97
Renewal process, nonarithmetic      47 48 52—54 56—60
Renewal process, pure      62
Renewal process, terminating      62 66
Renewal Theorem      6 8 51—53 61 87—90
Renewal Theorem, Blackwell’s      52 61 90
Renewal theorem, Blackwell’s, random walk analogue      88
Renewal theorem, elementary      5 51 61 89 90

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