Wise G.L., Hall E.B. — Counterexamples in Probability and Real Analysis :: Электронная библиотека попечительского совета мехмата МГУ

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Wise G.L., Hall E.B. — Counterexamples in Probability and Real Analysis

Wise G.L., Hall E.B. — Counterexamples in Probability and Real Analysis

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Название: Counterexamples in Probability and Real Analysis

Авторы: Wise G.L., Hall E.B.

Аннотация:

A counterexample is any example or result that is the opposite of one's intuition or to commonly held beliefs. Counterexamples can have great educational value in illuminating complex topics that are difficult to explain in a rigidly logical, written presentation. For example, ideas in mathematical sciences that might seem intuitively obvious may be proved incorrect with the use of a counterexample. This monograph concentrates on counterexamples for use at the intersection of probability and real analysis, which makes it unique among such treatments. The authors argue convincingly that probability theory cannot be separated from real analysis, and this book contains over 300 examples related to both the theory and application of mathematics. Many of the examples in this collection are new, and many old ones, previously buried in the literature, are now accessible for the first time. In contrast to several other collections, all of the examples in this book are completely self-contained — no details are passed off to obscure outside references. Students and theorists across fields as diverse as real analysis, probability, statistics, and engineering will want a copy of this book.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 211

Добавлена в каталог: 12.05.2008

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Предметный указатель

Absolute continuity      xiv 9 10 12 13 18 20 26 53 63 72 81 89 90 115 138 159 175
Admissible estimator      29 201
Analytic set      7 33 47 102
Ancillary statistic      29 197 198
arc-length measure      100 142
Atomic measure      13 26 81 90 94 95 175
Autocorrelation function      23 28 151 152 154 193
Axiom of Choice      vii 33
Bernstein set      6 14 33 41—43 62 101
Binomial distribution      28 195
Borel — Cantelli lemma      204
Brownian motion      24 156—158
Cantor ternary set      5 8 21 33 36—38 41 54 57 60 61 73 137 138 148
Cantor — Lebesgue function      38 54 55 62 76 106
Cantor-like set      5 33 35 36 39 42 62 72
Cardinal      vii 7 33 50 51
Cartesian product      100
Cauchy distribution      163 203
Central limit theorem      171 173
Chapman — Kolmogorov equation      22 147 148
Characteristic function      21 25 141 142 152 171—173
Chebyshev’s inequality      179 199
Co-countable set      57 82 83 89 90 114 130 134 150 160 162
Completely normal space      13 14 17 19 82 92 95—97 113 131
Complex-valued random variable      27 183
Composite hypothesis      30 206 207
Condensation point      40
Conditional expectation      25 103 159 162—164 166 169 187
Conditional independence      24 25 161 162
Conditional probability      24 103 159—161 163
Connected set      10 65
Continuous function      xiii 5 7—11 14 15 17 18 20 33 38 41 47 53—57 59 61 62 66 68 73—77 99 104 106-108 111 113 114 116 127 132—134 142 146 162 173 174 197
Continuum Hypothesis      vii xiii 6 7 10 11 19 44 45 49 68 84 129 130
Convergence      10 25 26 58 61 66 67 169 171 175 179
Convex function      21 53 59 60 127 142
Convex set      18 21 126 127 142
Convolution      18 27 116 118 132 173 183
Correlation      21 23 27 140—143 183 186 187
Cost function      197
Counting measure      57 87 88 90 95 112 116 117 134
Covariance      28 159 183—185
Cramer — Rao lower bound      29 199 200
Critical point      73
Critical value      10 73
Darboux function      9—11 53 64 65 74 75
Derivative      10 11 23 71—75 106 107 152
Detection theory      29 205 206
Diffuse measure      11 13 22 81 83 84 94 146 147
Dini derivative      11 71 75
Dirac measure      98 99 114
Directed set      27 109 180
Dominated Convergence Theorem      104 151 199
Dynkin system theorem      161
entropy      28 138 186
Equipotence      vii 130
Equivalence relation      34 39 67 76 133
Ergodicity      192
extrema      203
Fatou’s Lemma      25 169
Filter      27 164 183 187
Filtration      22 23 25 149 150 169 180
First category      5 7 8 33 37 42 43 49 50 56 78 79
Fisher information      29 198 199
Fourier series      117
Fourier transform      18 116
Fubini’s Theorem      55 112 117 118 129—132 135 149
Fusion      28 170 190 191 207
Gaussian distribution      21—24 26—29 139—146 149 150 152 154—157 163 164 172—174 183—188 190—193 198—202 204—206
Gram — Schmidt procedure      188
Graph      58
Group      67 121
Hahn — Jordan decomposition theorem      88
Hamel basis      6 7 33 43—49 51 57 59 63 65 67
Hausdorff space      12—14 17 19 20 82 85 86 92 95—97 99 101 113 114 131 132 134
Hodges — Le Cam estimator      199
Importance sampling      28 188—190 208
Increments      24 151 156
Independence      24 25 158 161 162
Indiscrete topology      82
Inner measure      xiii 6 33 41 43—45 96 160 164 185
Inner product      117 157
Inner regular      13 14 82 95—97 101
Isolated point      93

Jensen’s Inequality      142
Kalman filter      28 186 187 207
Lebesgue integral      103 110 137 138
Lebesgue — Stieltjes measure      12 89
Limit point      33 40 86 92 98
Linear function      17 113 114
Lipschitz property      9 53 62
Locally compact space      12—14 19 20 85 92 96—98 131—133
Lower Bemicontinuous      8 53 55
Maclaurin series      194
Markov property      22 28 147 148 161 179 191 192
Martingale      25 27 169 171 178—181 185
Martingale convergence theorem      28 179 185 208
Maximum Likelihood Estimation      29 199—204 207
Mean-square continuity      23 154
Mean-square derivative      23 152
Mean-square error      25 28 164 184 185 195 201
Median      25 172 203
Method of moments      28 194
Metric space      5 8 12 14 20 21 33 34 58 59 82 86 97—99 104 132 133 139 171
Midpoint convexity      8 53 59 60
Minimization      164
Modification      22 149
Monte Carlo estimation      188
Nesting property      163
Neyman — Pearson test      29 205 206
Noise      29 187 206
Nowhere dense      6 16 20 33 36 40 42 49 50 57 68 73 78 79 98 107 108 138
Ordinal      vii xiii 51 92 93 96 99 109 110 113 130 132 134
Orthogonal      121 151 188
Orthonormal      202
Oscillation      56
Outer measure      xiii 13 14 48 83 90 96 101 144 160 164
Outer regular      13 14 82 95—98
Pairwise independence      147 148
Partial derivative      18 127
Partition      6 7 18 19 21 23 33 34 45 46 48 50 51 65 67 77 85 87 121—126 129 139 142 145 150 155 156 184 186 187 204 205
Perfect set      6 9 24 33 36 40 46 47 51 62 64 98 155
Periodic function      8 23 53 57—59 152
Poisson distribution      194 196 199
Polish space      33 47
Product space      19 20 131—134 145 150 155
Property of Baire      6 33 42 43
Pseudometric      23 153 154
Quantization      27 28 158 183—187 207
Radon — Nikodym      17 18 103 114 115 137 159
Regression function      25 165 170
Regular measure      13 14 82 94 96 97
Riemann integral      10 14 15 20 72 73 103—105 137 138
Riemann — Stieltjes integral      15 106
Rigid motion      121 125 126
Rotation      121 123—125
Sample path      22—24 148—156 183 193
Saturated measure      13 83 90 91
Saturated nonmeasurable set      7 19 21 22 24 33 50 51 61 90 129 139 144 150 151 155 181 205
Second category      6 7 33 43 49
Separability      21 23 33 139 153 154
Signed measure      12 17 81 87 88 114 159
Skorohod imbedding theorem      24 156
Sorgenfrey plane      86
Spectral measure      23 154
Spectral representation      23 151 152
Stationarity      23 24 28 29 150—152 154—156 192 193 204 205
Stieltjes integral      15 105
Stochastic integral      151 182
Stopping time      25 156 157 169 170
Strong Law of Large Numbers      172 199
Sufficiency      25 159 165—168
Superemciency      200
Taylor’s series      10 71
Threshold      206
Tightness      21 139 172
Tonelli’s Theorem      104
Topological space      xiii 33 81 82 85 86 96 97 121
translate      48 98 99 121 126 174
Transnnite induction      43 44 47 49—51 64
Unbiased estimator      28 29 187 188 194—197 201
Uniform distribution      138 144 152 165 175
Upper semicontinuous      53
Vitale set      39 66
Weak law of large numbers      27 176 202
Weak* topology      26 171 174 175
Zermelo — Fraenkel axioms      vii

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