A User's Guide to Measure Theoretic ProbabilityThis book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not just a presentation of mathematical theory, but is also a discussion of why that theory takes its current form. It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguments and understand what they mean. |
Contents
II | 1 |
V | 3 |
VI | 5 |
VII | 7 |
VIII | 11 |
IX | 13 |
X | 14 |
XI | 17 |
LXXIII | 203 |
LXXIV | 205 |
LXXV | 206 |
LXXVI | 208 |
LXXVII | 211 |
LXXVIII | 213 |
LXXIX | 215 |
LXXX | 217 |
XII | 22 |
XIII | 26 |
XIV | 29 |
XV | 31 |
XVI | 33 |
XVII | 36 |
XVIII | 37 |
XIX | 39 |
XX | 41 |
XXI | 43 |
XXII | 45 |
XXIII | 51 |
XXIV | 53 |
XXV | 58 |
XXVI | 59 |
XXVII | 65 |
XXVIII | 68 |
XXIX | 70 |
XXX | 71 |
XXXI | 75 |
XXXII | 77 |
XXXIII | 80 |
XXXIV | 83 |
XXXV | 88 |
XXXVI | 93 |
XXXVII | 95 |
XXXVIII | 97 |
XXXIX | 99 |
XL | 102 |
XLI | 108 |
XLII | 111 |
XLIII | 113 |
XLIV | 116 |
XLV | 118 |
XLVI | 121 |
XLVII | 123 |
XLVIII | 128 |
XLIX | 131 |
L | 135 |
LI | 138 |
LII | 142 |
LIII | 147 |
LIV | 151 |
LV | 152 |
LVI | 153 |
LVII | 155 |
LVIII | 159 |
LIX | 162 |
LX | 166 |
LXI | 169 |
LXII | 176 |
LXIII | 182 |
LXIV | 184 |
LXV | 186 |
LXVI | 190 |
LXVII | 193 |
LXVIII | 195 |
LXIX | 198 |
LXXI | 200 |
LXXII | 202 |
LXXXI | 219 |
LXXXII | 222 |
LXXXIII | 226 |
LXXXIV | 228 |
LXXXV | 230 |
LXXXVI | 234 |
LXXXVII | 237 |
LXXXVIII | 239 |
LXXXIX | 242 |
XC | 244 |
XCI | 248 |
XCII | 249 |
XCIII | 256 |
XCIV | 258 |
XCV | 261 |
XCVI | 264 |
XCVII | 266 |
XCVIII | 268 |
XCIX | 271 |
C | 272 |
CI | 274 |
CII | 275 |
CIII | 276 |
CIV | 278 |
CV | 280 |
CVI | 285 |
CVII | 287 |
CVIII | 289 |
CIX | 291 |
CX | 292 |
CXI | 294 |
CXII | 295 |
CXIII | 296 |
CXIV | 300 |
CXVI | 301 |
CXVII | 302 |
CXVIII | 303 |
CXIX | 305 |
CXX | 306 |
CXXII | 307 |
CXXIII | 308 |
CXXIV | 310 |
CXXV | 312 |
CXXVI | 313 |
CXXVII | 315 |
CXXVIII | 316 |
CXXIX | 317 |
CXXX | 320 |
CXXXI | 324 |
CXXXII | 329 |
CXXXIII | 332 |
CXXXIV | 334 |
CXXXV | 336 |
CXXXVI | 338 |
CXXXVIII | 339 |
CXXXIX | 342 |
CXL | 343 |
CXLI | 345 |
| 347 | |
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Common terms and phrases
A₁ absolutely continuous approximation argument assertion B₁ Borel sigma-field bounded Brownian motion Chapter compact conditional distribution conditional expectation constant convergence in distribution convex countably additive Deduce defined definition denote density derivative disjoint Dominated Convergence equals equivalent Example exists F₁ filtration finite measure fixed follows Fourier transform function f h₁ Hint image measure independent random variables indicator function inequality interval kernel Kolmogorov Lebesgue measure Lemma Lévy lim sup limit theorem linear functional martingale Mathematical measurable functions measure theoretic metric space Monotone Convergence multivariate negligible sets nonnegative normal distribution numbers pointwise probability measure probability space Problem proof random vectors real line REMARK sample paths Section sequence Show sigma-finite measure stochastic submartingale subset supermartingale Suppose surely Write X₁ y₁ zero σ²