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, 11 parameter estimation, 511–512 delay, 351, 352 Complete Probability Space, 217 difference equations, 528 conditional distribution Dirac¸ delta function, 64, 467 continuous case, 162 directly Riemann Integrable function, 349, 350, discrete case, 162 365 conditional expectation, 163 Dirichlet distribution, 170 general definition, 166 discrete distribution properties, 167 Bernoulli, 53 confidence interval, 284, 285 beta-binomial, 86 consistent estimator, 139, 262 binomial, 53 continuous distribution geometric, 53 beta, 60 hypergeometric, 55 chi square, 59 negative binomial, 54 exponential, 56 Poisson, 55 F distribution, 265 relations, 54 gamma, 58 discrete renewal equation, 347 inverse gamma, 59 distributions Irwin–Hall, 92 transformed random variables, 72 Levy–Smirnov´ distribution, 206 dominated convergence theorem, 130, 131, 222, Levy´ distribution, 205 243, 251, 390, 419, 449 log normal, 62 Doob’s inequalities, 462 Lorentz distribution, 206 Egorov Theorem, 216 mixture of normals, 63 Ehrenfest chain, 373, 374 normal or gaussian, 57 eigenvalues, 175 t, 61, 263 eigenvalues, eigenvectors, 533, 535–537 Truncated Levy´ flight, 206 eigenvector, 441 uniform, 55 embedded Markov chain, 412 continuous mapping theorem, 230 equilibrium renewal process, 362 convergence relations estimating expectations, 111 probability and Lp, 225 adaptive importance resampling, 119 probability and a.s., 224 importance sampling, 110 probability and weak convergence, importance sampling example, 112–115 228 practical considerations, 112 weak convergence and a.s., 230 Euler’s method of approximating SDE’s, 517 p a.s. and L , 221, 223, 226 Euler-Milstein scheme for SDE’s, 520 convergence types Euler-Milstein scheme for SDEs, 519 p L convergence, 215 eventually, 34 almost uniform convergence, 216 expectation, 130 almost sure convergence, 216 properties, 144 as au equivalence, 216 expected time until pattern, 449 convergence in distribution, 217 exponential martingale, 442, 524 convergence in probability, 217 exponential matrix, 513, 534, 535 pointwise convergence, 214 uniform convergence, 214 Fatou’s lemma, 36, 129, 131, 252, 389, 458 convex functions, 133 for sets, 36 convolution, 181, 182, 185, 204, 205, 210, 232, filtration, 295 359, 360 standard filtration, 296 sequences, 183 Fisher information, 272 correlation, 144 Fong and Vasicekˇ model, 508 counting process, 308 forward rate, 140 covariance, 143 Fourier transform, 192 matrix, 169 Fubini’s theorem, 158 INDEX 549 Galton-Watson process, 363 law of large numbers, 149, 238, 261, 262, 274, gambler’s ruin problem, 29, 374 276, 287, 335 gamma function, 263 Lebesque measure, 37 Gaussian process, 469 likelihood ratio, 280, 281 Gaussian vector, 469–471, 482 likelihood ratio test distribution, 283 generalized inverse, 529 linear growth condition, 500 generalized random walk, 373 Lipschitz condition, 500 generating function Longstaff and Schwartz model, 508 discrete probability distribution, 184 examples, 184–188 M/M/1 queue, 429 sequences, 183 marginal distribution, 70 generating gaussian vectors, 173–177 Markov chain, 371, 372, 440 geometric Brownian motion, 497 communicating classes, 379 greatest common divisor, 346 continuous time, 412 ergodic, 390 Holder¨ inequality, 136 finite states general treatment, 396–399 Heston model, 521 first step analysis, 394 hitting time, 476 irreducible, 387, 412 hypothesis test, 277 limit theorems, 387 comparing proportions, 281, 283 periodicity, 380 example, 278 recurrence, 380–383 likelihood ratio, 280, 281 states characterization, 388 testing a single sample proportion, 280 stationary distribution, 388–392, 395 Markov inequality, 132 i.i.d., 71 example, 145 independence Markov process, 417 of sets, 30 Balance equations, 427 random variables, 70 Birth-death process, 423, 428, 431 indicator function, 22 Chapman-Kolmogorov equation, 420 infinitely often, 34 ergodic, 428 integration Kolmogorov differential equations, 419– measurable functions, 129 421 notation, 124 limit theorems, 413 positive measurable functions, 125 summation as an integral, 124 non-regular, 417 Itoˆ formula, 496, 498 pure birth process, 424 Itointegralˆ regular, 417, 418 continuity, 494 transition probability, 418, 419, 425 product rule, 497, 504 transition rate, 418 properties, 494 truncated process, 431 Itoˆ isometry, 490, 492, 494, 495 two state example, 422–423 Yule process, 425, 433 Jacobian, 74 martingale Jensen’s inequality, 133, 137–139 sub super martingales, 438 joint distribution definition, 438 calculation, 73–74 Doob martingale, 439 definition, marginal, 70 martingale reasoning unbounded stopping times, 454–456 key renewal theorem, 350, 352, 355, 356, 359, relationship with Markov processes, 440 365 sub super martingales, 438 Kolmogorov continuity theorem, 472 martingale transform, 443 matrix exponential function, 534, 535 Lp space, 134, 136, 215 measurable function, 45 Laplace transform, 189 memoriless property, 315 lattice random variable, 346–348, 361 mendelian genotype evolution, 401 550 INDEX method of moments, 276 queuing process, 376, 377 Gamma distribution, 277 queuing theory, 376 Minkowski Inequality, 136 MLE consistency, 272 Radon-Nikodym derivative, 50 MLE invariance, 272 random number generation moment generating function, 189 example mixture distribution, 104 example, 190 die rolls, 90 properties, 189 discrete random variables, 90 theorems on, 191 exponential numbers, 89 moments, 143 general, 89 Monotone Convergence Theorem, 126, 131 normal: Box–Muller method, 93 monotone convergence theorem, 251 normal: CLT method, 92 Multinomial distribution, 170 normal: polar rejection method, 93 Multinomial test, 172 normal: Wichura method, 92 multivariate normal density, 469 normal: ziggurat method, 96–101 Multivariate normal distribution, 172 practical considerations, 116 mutation evolution process, 378 rejection sampling, 94 rejection sampling example, 102 Neyman-Scott paradox, 275 sampling importance resampling, 117 Novikov condition, 524 random variables null set, 22 discrete, 48 continuous, 50 definition, 46 order of convergence, 193 random vectors, 168 order statistics, 314 generation, 173 Ørnstein-Uhlenbeck SDE, 503, 507, 508, 516 random walk, 373, 374, 383, 385, 386, 393, 394, 407, 415, 447, 452, 454, 455, 459, p-value, 277 476 percentiles, 67 hitting time distribution, 454 pivotal quantity, 284 renewal equation, 359–361, 364, 365 point processes, 301 renewal measure, 348 Poisson process, 308 Renewal process non-homogeneous, 319 Alternating renewal theorem, 353, 413 approximation with a Binomial, 308–310 strong law of large numbers, 358, 361, compound Poisson process, 320 413 general Poisson process, 317 renewal processes inter-arrival times, 310–315 Central Limit Theorem, 343 merging and splitting events, 316, 317 definition, 332 simulation, 323–325 discrete renewal times, 344 Portfolio Maximization, 140 elementary renewal theorem, 336 positive definite matrix, 169, 174–176, 470 example, 332 power set of a probability space, 10 limiting results, 335 previsible process, 443 renewal function, 333 probability measure strong law of large numbers, 335 conditional probability, 23 renewal reward process, 358 definition, 18 limit theorems, 358 for discrete (countable) spaces, 20 residual life process, 351, 352 monotone convergence property, 32 residual process, 362 properties, 21 reversible stochastic process, 429 probability of extinction, 363 product spaces, 158 sample mean, 139 pseudo-determinant, 529 sample variance, 139 SDE moment calculation, 509 quadratic variation, 476–478, 493, 494 semi Markov process, 412, 413 quantiles, 67 stationary measure, 414 INDEX 551 set event independent components, 298 definition, 10 independent increments, 300 operations, 11 pathwise realization, 296 sigma algebra stationary, 299 Borel sigma algebra, 18 stationary increments, 299 definition, 12 the outcome space, 296 for countable spaces, 17 stochastic volatility model, 508 generated by a class, 12 stoping time, 339 generated by random variables, 69 stopped process, 444