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Index alternating renewal process, 353, 354 reflection principle, 474 scaling invariance, 471 Bayes rule, 23, 28 simulation, 480 Bernoulli process, 228, 293, 301–305, 309, 316 time invariance, 472 beta function, 60, 170, 265 Buffon’s needle problem, 83, 94 birth and catastrophe chain, 377 birth and death chain, 375, 392 Cauchy–Bunyakovsky–Schwarz inequality, 135, birthday problem, 30–31 137, 143, 149 Black–Karasinski model, 509 Central Limit Theorem, 2, 87, 92, 241, Blackwell renewal theorem, 346, 348 244, 245, 273, 282, 302, 342, 343, block replacement policy, 344 482, 523 Borel–Cantelli lemmas, 34–142, 224, 304 DeMoivre-Laplace version, 151 bounded convergence theorem, 131, 251, 422, changing the order of derivation and integra- 448, 455 tion, 243 branching process, 363, 366, 368, 441, 461 Chapman-Kolmogorov equation, 378, 379, 381, Brownian bridge, 483 389, 420 Brownian motion conditional distribution, 468 characteristic function, 192–194 covariance structure,COPYRIGHTED 471 MATERIALcontinuity theorem, 203 definition, 467 examples, 194–199 drift, 471 multivariate random vector, 194 examples, 468, 471 properties, 193 generation, 176 uniqueness, 200 hitting time, 475, 476 characterization theorem for convergence in dis- multidimensional, 473 tribution, 218 quadratic variation, 478 Cholesky decomposition, 174–176, 514 547 Probability and Stochastic Processes, First Edition. Ionut¸Florescu C 2015 John Wiley & Sons, Inc. Published 2015 by John Wiley & Sons, Inc. 548 INDEX CIR model, 507 Darboux sums, 349 moments, 510, 511 De Morgan laws,
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