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









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: Stopped Random Walks: Limit Theorems and Applications

: Gut A.

:

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.


: en

: /

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ed2k: ed2k stats

: 1988

: 198

: 19.05.2006

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$(C, \mathcal{C})$      148 157 171176 177
$(D, \mathcal{D})$      149 154157 164 172176 178
$(D, \mathcal{D})$, $D_0$      148 155 164 176
$L^r$-convergence theorem      166
Ahlberg      6 119
Aleskeviciene      87
Anderson      61
Anscombe      3 8 15 42
Anscombe Donsker invariance principle      147149
Anscombe, condition      15 44
Anscombes theorem      3 4 1516 36 44 56 85 137 147 151
Anscombes theorem, $L^r$-analogue of      33
Anscombes theorem, multidimensional      150
Arjas      53
Asmussen      4 48 62 109 125
Athreya      53 61
Barlow      122124
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 9296 102 108 111114 130 137 147 168169
Central limit theorem, moment convergence in      18 168169
Central limit theorem, multidimensional      117
Chang      33 163
Choquet      53
Chow      7 17 23 24 33 34 67 9295 103 106 107 132 137 142144 146
Chromatography      6 118121
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 174175
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 174175
Convergence, almost sure (a.s.)      4 9 1014 17 41 44 54 67 70 8385 97 130 135 145 166 175
Convergence, complete      9 4243 103
Convergence, in $L^r$      4 9 17 37 39 41 44 166
Convergence, in distribution      4 9 10 1517 36 55 58 72 8587 98 102 111112 115117 130131 137138 166 179
Convergence, in probability      4 9 1013 39 166 176
Convergence, in r-mean      4 (see also Convergence in
Convergence, moment      4 9 1720 3639 54 56 57 59 87 89 9297 98102 106 108 109114 130132 143145 165167
Convergence, of finite-dimensional distributions      150 156 171172
Convergence, of probability measures      171176 (see also Weak convergence)
Convergence, rate      4244 103104 131
Convergence, weak      see Weak convergence
Counters      1 2 124125
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
Donskers theorem      147 149 158 164 172 177
Donskers 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 59 22 24 39 50 74107 108 109118 132 133 147 151159 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      103104
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 8384 93 105
First passage time(s), law of the iterated logarithm      102103 163
First passage time(s), momentgenerating function      81
First passage time(s), moments, convergence      55 9297 106
First passage time(s), moments, finiteness      50 7881
First passage time(s), overshoot      76 97102 106
First passage time(s), process      3 5 6 50 55 57 75107 119
First passage time(s), subadditivity      55 83
First passage time(s), uniform integrability      55 57 9092 9495
First passage time(s), weak convergence      151157
First passage times across general boundaries      7 75 109 133146 159161 163
First passage times across general boundaries, central limit theorem      137 144
First passage times across general boundaries, law of large numbers      135137 143
First passage times across general boundaries, law of the iterated logarithm      145146 163
First passage times across general boundaries, momentgenerating function      134
First passage times across general boundaries, moments, convergence      143145
First passage times across general boundaries, moments, finiteness      133134
First passage times across general boundaries, overshoot      145
First passage times across general boundaries, weak convergence      159161
Fluctuation theory      4 46
Fuchs      4 65
Functional central limit theorem      147 172 weak)
Functional limit theorem      7 132 147164
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 8486 9398 100 102 103 107 109 111 113 115 117 119 122 126 127 133 134 136140 144146 150 152 153 156 158 161 182
Hall      163
Hartman      41 177178
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 142144 146 163
Hsu      42 43
Huggins      163
Hunter      62
Iglehart      156 164
Insurance risk theory      1 2 125126
Invariance principle      147164 (see also Functional limit
Invariance principle, almost sure      see Invariance principle strong
Invariance principle, Anscombe Donsker      147149
Invariance principle, strong      7 42 147 161164 177178
Invariance principle, weak      7 147161 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 6566 7778 98 104106 129
Ladder, epoch, ascending      8 65
Ladder, epoch, ascending, strong      5 65 6870 77
Ladder, epoch, ascending, weak      67 70
Ladder, epoch, descending, strong      67
Ladder, epoch, descending, weak      66
Ladder, height      6667 98 101 129
Ladder, height, first passage times for      see First passage times
Ladder, height, strong ascending      66 6869 77
Ladder, method      76 7778 81 83 8789 91 93 9496 97 101 152
Ladder, variable      2 4 26 46 6567 77
Lai      7 17 28 33 34 59 67 85 91 92 94 100 133 142144
Lalley      62 87 102
Lamperti      61
Law of Large Numbers      1 8 1314 17 29 33 37 44 54 70 8384 9294 108 109111 130 135137
Law of large numbers, convergence rate      4243
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      1718 44
Law of the iterated logarithm      1 8 9 4142 43 102103 108 117 131 147 161164 177178
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      4142 162164 177178
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 167171 178
Martingale, convergence      168
Martingale, moment inequalities      see Moments
Martingale, optional sampling theorem      2125 167 170171
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 2528 49 51 78 79 97 132 133134 145
Moments, inequalities for martingales      167 169
Moments, inequalities for stopped random walks      2028
Moments, inequalities for sums of independent random variables      167 168169
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      4849
Number of visits      6 6465
Number of visits, expected      6465 90
Nummelin      53
Optional sampling theorem      see Martingale
Orey      90
Ornstein      4 6 65
Partial maxima      5 7 46 6773 75 108 128132 155 159 164
Partial minima      5 6770 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      122124
Pyke      20 124
Queueing theory      1 2 126
Random change of time      148 151 160 176
Random index      19 17 46 68
Random walk      17 8 46 6273 99100
Random walk, arithmetic      47 65 97 105 107
Random walk, arithmetic with span d      47 63
Random walk, arithmetic, d-arithmetic      47 88 90 9599
Random walk, Bernoulli      1 47
Random walk, Bernoulli, symmetric      1
Random walk, classification      6368
Random walk, coin-tossing      1
Random walk, drifting      4 6
Random walk, drifting to $+\infty$      5 7 6467 70 104 with
Random walk, drifting to $-\infty$      6467 69 70
Random walk, maximum of      see Partial maxima
Random walk, minimum of      see Partial minima
Random walk, nonarithmetic      47 63 65 88 90 95100 102
Random walk, oscillating      4 6 64 6668
Random walk, persistent      4 6465
Random walk, randomly indexed      19 (see also Stopped random walk)
Random walk, simple      1 6365 104106 107
Random walk, simple, symmetric      1 24 35 66
Random walk, stopped      see Stopped random walk
Random walk, transient      4 6 6465
Random walk, two-dimensional      3 6 108 109 stopped)
Random walk, with multidimensional indices      107
Random walk, with positive drift      3 5 73 74107 108 128132 133146 151157 159161 163 164
Recurrent even      4
Regularly varying function      53 57 61 87 116 134 155 180182
Relative compactness      162 164 178
Reliability theory      1 2 8 123124
Renewal counting process      26 8 9 4857 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      5457
Renewal counting process, moments, finiteness      49
Renewal counting process, uniform integrability      5457
Renewal function      5 6 4953 61 90
Renewal function, extended      89
Renewal measure      63 90
Renewal measure, harmonic      90
Renewal process      16 4662 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 5658
Renewal process, coupling proofs      53
Renewal process, delayed      62
Renewal process, integral equation      50 52
Renewal process, lifetime      58 60
Renewal process, lifetime, residual      5860 61 97
Renewal process, nonarithmetic      47 48 5254 5660
Renewal process, pure      62
Renewal process, terminating      62 66
Renewal Theorem      6 8 5153 61 8790
Renewal Theorem, Blackwells      52 61 90
Renewal theorem, Blackwells, random walk analogue      88
Renewal theorem, elementary      5 51 61 89 90
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