© in This Web Service Cambridge University Press Cambridge University Press 978-0-521-14577-0

Cambridge University Press 978-0-521-14577-0 - Probability and Mathematical Genetics Edited by N. H. Bingham and C. M. Goldie Index More information

Index

ABC, approximate Bayesian almost complete, 141 computation almost complete Abecasis, G. R., 220 metrizabilityalmost-complete Abel Memorial Fund, 380 metrizability, see metrizability Abou-Rahme, N., 430 α-stable, see subordinator AB-percolation, see percolation Alsmeyer, G., 114 Abramovitch, F., 86 alternating renewal process, see renewal accumulate essentially, 140, 161, 163, theory 164 alternating word, see word Addario-Berry, L., 120, 121 AMSE, average mean-square error additive combinatorics, 138, 162–164 analytic set, 147 additive function, 139 ancestor gene, see gene adjacency relation, 383 ancestral history, 92, 96, 211, 256, 360 Adler, I., 303 ancestral inference, 110 admixture, 222, 224 Anderson, C. W., 303, 305–306 adsorption, 465–468, 477–478, 481, see Anderson, R. D., 137 also cooperative sequential Andjel, E., 391 adsorption model (CSA) Andrews, G. E., 372 advection-diﬀusion, 398–400, 408–410 anomalous spreading, 125–131 a.e., almost everywhere Antoniak, C. E., 324, 330 Aﬀymetrix gene chip, 326 Applied Probability Trust, 33 age-dependent branching process, see appointments system, 354 branching process approximate Bayesian computation Aldous, D. J., 246, 250–251, 258, 328, (ABC), 214 448 approximate continuity, 142 algorithm, 116, 219, 221, 226–228, 422, Archimedes of Syracuse see also computational complexity, weakly Archimedean property, coupling, Kendall–Møller 144–145 algorithm, Markov chain Monte area-interaction process, see point Carlo (MCMC), phasing algorithm process perfect simulation algorithm, 69–71 arithmeticity, 172 allele, 92–103, 218, 222, 239–260, 360, arithmetic progression, 138, 157, 366, see also minor allele frequency 162–163 (MAF) Arjas, E., 350 allelic partition distribution, 243, 248, Aronson, D. G., 400 250 ARP, alternating renewal process two-allele model, 214 Arratia, R., 102, 106–109, 247, 328 two-allele process, 360, 363 a.s., almost sure(ly) allocation process, 467, 480–481 Askey, R,, 372 Allstat, 303, 317 Asmussen, S., 346

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-14577-0 - Probability and Mathematical Genetics Edited by N. H. Bingham and C. M. Goldie Index More information

510 Index

assay, 205, 326 Bayesian inference, 22, 65, 214, assessment, see computer-based test 222–223 associated random walk, see random Bayesian nonparametric modelling, walk 321–323, 325–326 asymptotic independence, 352, 356 Bayesian partition, 340 Athreya, K. B., 346 Bayesian statistics, 36–38, 40–41, 251, Atiyah, M. F., 20–21, 23, 26 254, 265, 319 Atkinson, F. V., 239 Bayes’s theorem, 81 attraction/repulsion, see point process: BayesThresh method, see wavelet clustering Beagle, see phasing algorithm Austin, T., 59 Beal, M. J., 329 Auton, A., 223 Beaumont, M. A., 214 autonomous equation, 495, 498 Beder, B., 259 auto-Poisson process, see Poisson, S. D. Beerli, P., 214 autoregressive process, 352 Belkhir, K., 221–222 average mean-square error (AMSE), see Bellman, R. E. error Bellman–Harris process, 113, 116, 118 Awadalla, P., 223 Ben Arous, G., 452–453, 458 Axiom, see computer algebra Beneˇs, V. E., 420 Benjamini, I., 392–393 Bachmann, M., 122 Benjamini, Y., 86 Baddeley, A. J., 67, 71–73, 75 Bensoussan, A., 400 Bailleul, I., 452 Bergelson, V., 138, 157, 159, 161, 163 Baire, R.-L. Berkson, J., 189 Baire category theorem, 137 Bertoin, J., 46, 265, 273, 378 Baire function, 138 Bessel, F. W. Baire property, 138, 140 Bessel process, 271 Baire set, 138–164 beta distribution, 270–276, 280–284, Baire space, 142 328–329, 334, 359–360, 364–365, Baire’s theorem, 141 374 balanced incomplete block design, 30 bivariate beta distribution, 368–370 Balding, D. J., 214 Bhamidi, S., 55 Balister, P. N., 384 bias, 193, 196 ballot theorem, 38 Bibby, J. M., 353 balls in cells, 301 binary branching Brownian motion, see Baltr¯unas, A., 173 Brownian motion Banach, S. Bingham, N. H., 194 Banach space, 164 biomass, 137 bandwidth, 189–190, 195–199 birth-and-death model, see Barbour, A. D., 247, 328, 362, 448 mathematical genetics Barry, J., 186 birth-and-death process, 67 Bartlett, M. S., 25, 27, 32–33 bitopology, see topology base, 240 Björnberg, J. E., 391 base measure, 322, 328, 334 Blackwell, D., 322, 330 Batchelor, G. K., 24, 28 Blei, D. M., 329 Bather, J. A., 21, 23, 25, 32 blind transmission, 484–486, 489 Baudoin, F., 461 Blin-Stoyle, R. J., 31 Bayes, T., see also approximate blood group, 239 Bayesian computation (ABC), Bobecka, K., 273 conjugacy, hyperparameter, Bochner, S., 360–361, 368, 370, posterior predictive distribution, 372–373, 378 posterior sampling, posterior Bollobás, B., 384 simulation, predictive density Bolthausen, E., 388 Bayesian curve fitting, 326 Bonald, T., 424–425 Bayesian density estimation, 326 Bond, S., 317 Bayesian hierarchical model, 36, 78, Bonhomme, S., 186 326 Bonnefont, M., 461

Index 511

Borovkov, A. A., 416 Butucea, C., 186 Borwein, D., 137–138, 157 Cambridge, University of, xv–xvi, 19, boundary bias, 190 33, 170 Bousch, T., 466 Mathematical Tripos, 19–24 Bowcock, A. M., 221 Pembroke College, 17, 19, 26 boxcar function, 195 Statistical Laboratory, 22, 24, 29, 34 Bradley, R. C., 356 Campbell, P. J., 92 Bramson, M. D., 118, 120–121, 391, 416 Camps, A., 21 branching process, 137, 260, 346, 493, cancer, 91–110 498, 502, 505–508, see also Cane, V. R., 24, 29 Bellman–Harris process, branching Cannings, C., 242 Brownian motion, branching Cannings model, 248 random walk (BRW), extinction, capacity, 406, see also channel capacity, Galton–Watson process, Markov network capacity, Newtonian branching process, Yule process capacity age-dependent branching process, Carath´eodory, C., 348 114–115 Carpio, K. J. E., 181 continuous-state branching process, Carroll, R. J., 186–187, 191 378 Cartwright, M. L., 20 general branching process, 116, 119 cascade, see multiplicative cascade multitype age-dependent branching Cassels, J. W. S., 29 process, 114 Catalan, E. C. supercritical branching process, 115 Catalan number, 51 weighted branching process, 116 category measure, 147 branching random walk (BRW), 47, Cauchy, A. L. 114–131, see also anomalous Cauchy problem, 398, 401, 408–409 spreading, extinction, spreading Cayley, A. out, spreading speed Cayley’s formula, 51 irreducible BRW, 124, 130 CDP, coloured Dirichlet process multitype BRW, 117, 124–131 cell, 91–97, 100, 110, 206 reducible BRW, 125–131 CFTP, coupling from the past two-type BRW, 125–131 Chafa¨ı, D., 461 Breiman, L., 347 channel, see also Gaussian channel Bressler, G., 489 capacity, 485, 488–490 Briggs, A., 31–32 coding, 484 Bristol, University of, xvi, 19, 381 memoryless channel, 485 British Columbia, University of, 394 characteristic function, 188, 191, Broadbent, S. R., 381 193–194, 197 Brookmeyer, R., 186 characteristic polynomial, 386 Brown, R. charge-state model, see mathematical branching Brownian motion, 118, 120, genetics 123, 128 Chen, H., 422 Brownian diﬀerential, 454 Chen, L., see Stein–Chen approximation Brownian excursion, 53 Chen, M. F., 451 Brownian motion, 51, 60, 271, 361, Chinese restaurant process (CRP), 46, 374, 398–411, 417–443, 447–461, 97–99, 276 467 Christ’s College, Finchley, xv, 18 Brownian network, 418, 426–442 chromosome, 93, 206–212, 214–231 Brownian snake, 53 Chung, K. L., 25, 28, 31 Browning, B. L., 216 clairvoyancy, see compatibility, demon, Browning, S. R., 216 embedding, scheduling BRW, branching random walk clopen, 147 Brydges, D., 502 close-packed lattice, see lattice bulk deletion, see deletion CLT, central limit theorem Buonaccorsi, J. P., 187 cluster(ing), 328, 333–340, see also Burdzy, K., 449 point process burn-in, 230–233, 340 background cluster model, 335–337

512 Index

cluster size, 324 convex optimization, 423, 425, 433, 437, co-adapted coupling, see coupling 439 coagulation, 273, see also coalescence, cooperative sequential adsorption model Smoluchowski coagulation equation (CSA), 465 coalescence, 39, 46–50, 54, 83, 206–223, Coppersmith, D., 384 265, 361, 364, 367, 375, see also copula, 41 Kingman coalescent, Li and copying, 212, 216–219, 225–227 Stephens model Corander, J., 221–222 inference, 205, 213–215, 220 correlation, 352 n-coalescents, 211 sequence, 368–372 coalescent tree, 210, 214, 217, 361, 366, coupling, 384n, 385, 447–461 376 co-adapted coupling, 447–454, coding, 485–488, see also channel coding 460–461 cofinality, 139 control, 454–458 co-immersed coupling, see coupling: coupling from the past (CFTP), 65, co-adapted 78, 82, 88 coin tossing, 136, 381 dominated CFTP algorithm, 65, 74n collision, 384–385 dominated coupling from the past, colour, 139, 330, 333–336 67–69 coloured Dirichlet process (CDP), 334 γ-coupling, 451 colouring, 268, 272, 275–276, 278, 334, maximal coupling, 447–450, 460–461 360, 368 mirror coupling, 451 combinatorics, 59, see also additive reflection coupling, 447, 450–453, combinatorics 458–460 compatibility rotation coupling, 447, 453, 460 clairvoyant compatibility, 384–385 shift-coupling, 450 complementary slackness, 423, 425 synchronous coupling, 447, 452–453, complete monotonicity, 375 458–460 complete neutrality, 273 time, 447, 452, 454, 458–461 completion, 140 coupon collecting, 271, 282, 300–303 composition, 266–267, 274, 278–280, 287 covariate, 320, 326, 329, 336–337, 339 compound Poisson process, see Poisson, Cox, D. R., 27, 33 S. D. Cox process, 71, 170–173, 181 computational complexity, 69–70, 76, Cramér,C. H., 346 85, 205 Cranston, M., 451–453, 458, 461 computational statistics, see statistics Crick, F. H. C., 240 computer algebra, 453 critical parameter, 493, 498–499 Axiom, 454 critical probability, 382, 389–391 computer-based test, 299–317 Croft, H. T., 137 computer science, 380 Cromwell, V., xvii, 31 Comte, F., 198–199 crossover, 218 concentration, 322, 328 cross-validation method, see wavelet conditional independence, 283–285, 320 Crow, J. F., 245 conjugacy, 323, 327, 329, 331–332 CRP, Chinese restaurant process connection, see network connection CRT, continuum random tree Connor, S. B., 450 Crump, K. S., 116 conservation biology, 222 curvature, 451 contact model, 391 curve, see also Bayesian curve fitting contaminated data, see data integral curve, 495, 498, 504 contiguity, 138–139 probabilistic curve, 495, 498, 500–501, continuum random tree (CRT), see 503–507 diffusion, random tree cycle time, 173–174 convergence rate, see rate of Dai, J. G., 417, 427 convergence Daiches, D., 31 convexity, 117, 120, 122, 127–129, 131, Daley, D. J., 30 139 Damien, P., 328–329 convex minorant, 130 Daniels, H. E., 27, 123

Index 513

Darwin, C. R., 238 Diaconis, P., 39, 58, 60, 97, 105 data difference equation, 386 availability, 205, 213, 238, 326 differential geometry, 24, 60 colorectal cancer data, 92–100 differential operator, see operator contaminated data, 196 diffusion, see also advection-diffusion, genotype data, 216, 220 Fleming–Viot diffusion, generator, HapMap, 229–232 heavy traffic diffusion nutrition data, 186–187 approximation, Jacobi diffusion, rat time-course experiment, 338–340 Kolmogorov diffusion, Redwood seedlings, 65, 74–77 reaction-diffusion equation, standard test functions, 86 transition density, Wright–Fisher Daubechies, I., 87 diffusion Davidson, R., 32 continuum-tree-valued diffusion, 53 Davies, I. M., 508 diffusion process, 362, 364, 366, 399, Davies, R. O., 156, 165 427 Davis, K. B., 196 diffusion sausage, 399–400, 402–410 Dawson, K. J., 221–222 diffusion scaled process, 426 death process, 361, 362, 376 elliptic diffusion, 451 deconvolution, 186–188, 191–201 hypoelliptic diffusion, 451, 453 decrement matrix, 289, 292 jump diffusion, 360, 374 de Finetti’s theorem, see Finetti, B. de L-diffusion, 400–401, 403, 409–410 degrees of freedom, 320 measure-valued diffusion, 50, 206, Dekking, F. M., 120–121 213, 363 deleterious effect, 122 nilpotent diffusion, 452–453 deletion, 265–266, 269, 272–274, 276, relativistic diffusion, 452 278, 283, 285–286 diffusivity, 398, 400–401, 405–406, bulk deletion, 290, 292 409–411 symmetric deletion kernel, 287–288 Diggle, P. J., 74, 186 τ-deletion, 293 Ding, J., 220 demographic model, 96 diploid, 206, 212, 216 demography, 100, 207–213 Dirac, P. A. M., 24 demon, 381–387 Dirichlet, J. P. G. L., see also coloured clairvoyant demon, 381 Dirichlet process (CDP), Denjoy, A., 142 Ferguson–Dirichlet random density deconvolution, see measure, Poisson–Dirichlet deconvolution distribution/process/point density estimation, 189, 192, 325–326, process/random measure 333, see also deconvolution, kernel Dirichlet boundary condition, 398, methods, minimum contrast 408 methods, penalised contrast Dirichlet distribution, 240, 247, 270, methods 273, 322–323, 334–335, 364–365, local constant estimator, 189, 191, 367 192 Dirichlet mixture, 331, 334 local linear estimator, 190–191 Dirichlet–multinomial finite mixture local polynomial estimator, 190–192, model, 332 195 Dirichlet process (DP), 320–336 density point, 141–142, 157–161 Dirichlet process generalisations, 329 density topology, see topology Dirichlet process mixture model, 325, deoxyribonucleic acid, see DNA 332 dependent percolation, see percolation generalized Dirichlet distribution, 367 design, see balanced incomplete block discrete mathematics, 380 design disease mapping, 222 deterministic modelling, 122, 125 distributional bounds, 304 deterministic simple epidemic, see Ditor, S. Z., 137–138, 157 epidemic DNA (deoxyribonucleic acid), 92, 206, De Veciana, G., 424 209, 212, 216, 223, 240, 243, 260 d.f., distribution function mitochondrial DNA, 239

514 Index

Doeblin, W., 448 EPPF, exchangeable partition Dollis Hill, see Post Office Engineering probability function Research Station equilibrium distribution, 350–351, 354 domain of attraction, 209, 211, 305 equivalence class/relation, 210, 257–258 Donnelly, P. [Donnelly, P. J.], 50, 254, equivalent metric, 140, 159, 163 259, 320, 322, 328, 360 Erd˝os, P., 30, 55 Donnelly–Kurtz process, 215 ergodicity, 346–356 Donoho, D. L., 86 ergodic theorem, 346 Donsker, M. D. geometric ergodicity, 402 Donsker–Varadhan invariance ergodic theory, 59, see also principle, 475 subadditivity dose, 188 Erkanli, A., 326, 329 double exponential distribution, see error, 484–485, 487–488 Gumbel distribution average mean-square error (AMSE), DP, Dirichlet process 88 DQES, dietary questionnaire for errors in variables, 186–188, 195 epidemiological studies Escobar, M. D., 326, 329 drift, 345, 359 ESF, Ewens sampling formula duality, 48, 346, 348, 351, 355–356, 360, essentially (un)bounded, 140, 143, 153 362–363, 367, 375 estimation, see deconvolution, kernel dual variable, 418, 423, 425, 428, 435 methods, maximum-likelihood Dubins, L., 321 estimator (MLE), Dufresne, D., 459 minimum-contrast methods, Duke University, 333 Nadaraya–Watson estimator, Dunson, D. B., 329 regression estimator, wavelet curve Durbin, R., 219–220, 226 estimation Durrett, R. T., 391, 475 Etheridge, A. M., 362 dynamical system, 491–507 Ethier, S. N., 209, 360–362, 364 Euclidean Brownian motion, see economics, 422 Brownian motion Eddy, S. R., 219–220, 226 Euler, L. edge, 382 Euler formula, 458 open edge, 392 Euler–Maruyama solution, 411 Edwards, D. A., 27 Euler–Mascheroni constant, 244, 304 effective population size, 218, 226, 229 European Mathematical Society, xvi eigenfunction, 359–360, 365, 368, 370 Evans, J. W., 465 Eisele, T., 398 Evans, S. N., 53 EIT, exponential intersection tails evolution, 238, see also fitness, electro-magnetic force, 73 heterozygote advantage, elementary renewal theorem, see mathematical genetics renewal theory selective force of, 246 embeddability, 137 EVT, extreme-value theory embedding Ewens, W. J., 30, 39, 92, 96, 100, 110, clairvoyant embedding, 385–390 209, 290, 320 penalised embedding, 387 Ewens partition structures, 265, 290, shift-embedding, 139 292 two-dimensional embedding, 388 Ewens sampling formula (ESF), 46, Emery,´ M., 449 92, 95–99, 265, 271–272, 324 emission probability, 219, 226 Ewens sampling property, 248 empirical measure, 364 exchangeability, 35–60, 208–209, 250, Engen, S., see GEM distribution 320–321, 330, 335, see also mixture entrance boundary, 48, 360–361, 364 model entropy, 485 exchangeable increments, 39 epidemic exchangeable representation, 36, 43, deterministic simple epidemic, 123 50 epidemiology, 186–187 exchangeable trials, 276

Index 515

partial exchangeability, 36, 40–42, fitness, 239 57–58, 333 Fleming, W. H. exchangeable partition, see Fleming–Viot diffusion, 206, 208–209, exchangeable partition probability 211, 215, 360, 362, 363, 366 function (EPPF), random partition flow exchangeable partition probability ODE flow, 496 function (EPPF), 267–268, probabilistic flow, 495 270–272, 279–280, 282, 287–288 fluid dynamics, 24 excursion, 271 fluid scaled process, 424–425 explosion, 496 food, 186–187 exponential decay, 346 forensic science, 222 exponential functional, 459 forest, 360, 364, 376 exponential integral, 305 Foss, S. G., 466 exponential intersection tails (EIT), 392 Fourier, J. B. J. extinction, 53, 124, 378, 387 Fourier inverse, 192 extreme-value distribution, 305 Fourier series, 22 extreme-value theory (EVT), 303 Fourier transform, 191–192, 195, 198 local EVT, 304, 306 Fourier integral kernel, see kernel moment convergence in, 304 methods: sinc kernel Fabius, J., 321 fractal, 60, 116 fairness, 417–418, 422, 430, 432–439, 442 fragmentation, 46–50, 54, 265, 273 false discovery rate method, see wavelet Freedman, D. A., 39, 105, 321 Falush, D., 221–223, 231 Freedman, L. S., 187 Fan, J., 190, 192, 195 Frenk, J. B. G., 170 Fannjiang, A., 401 Frobenius, F. G., see Perron–Frobenius Farrington, B., 508 eigenvalue FastPhase, see phasing algorithm full-likelihood inference, 214–215 Fatou, P. J. L. Fulton, J., 31, 32 Fatou’s lemma, 309–312 functional central limit theorem, 419 Fayolle, G., 466 functional law of large numbers, 425 Fearnhead, P., 214, 223 Furstenberg, H., 59 Fearnside, A. T., 333 Gács, P., 384–385 Feller, W., xv, 25, 27, 135–136, 172, Gail, M. H., 186 301, 346, 373, 378 Galton, F. Felsenstein, J., 214 Galton–Watson process, 53, 119, 120, Fenchel, M. W. 378 Fenchel dual, 117 game theory, 422 Feng, S., 240 γ-coupling, see coupling Ferguson, T. S., 321, 329 gamma distribution, 271, 337, 459–460, Ferguson–Dirichlet random measure, see also normal-inverse gamma 363 model Feynman, R. P. gamma process, 271, 322 Feynman–Kac formula, 501 Gani, J. M., 30, 33 FFQ, food frequency questionnaire Gasper, G., 360, 368–369 filtration, 427, 449, 453 Gauss, J. C. F. financial analysis, 186 Gaussian channel, 487–488 Finch, S. R., 246, 253 Gaussian process, 352 Fine, N. J., 137 Gaussian tail estimate, 400 Finetti, B. de, 38, 209, 250, 320 inverse Gaussian distribution, 373, de Finetti’s theorem, 36–39, 43, 46, 378 49, 59 inverse Gaussian process, 361 first category, 149 Gδ first passage time, 373 inner regularity, 157, 159 Fisher, R. A., 24, 241, see also natural set, 140, 147 selection, Wright–Fisher gel electrophoresis, 240, 243 diffusion/model Gelfand, A., 329 Fisher/KPP equation, 118 Gelman, A., 320

516 Index

GEM, generalised Engen–McCloskey or Griffeath, D., 448 Griffiths–Engen–McCloskey Griffin, J. E., 329 GEM distribution, 251–256, 273, 294, Griffiths, R. C., 205, 213–214, 259, 320, 296–297, 322, see also see also GEM distribution mathematical genetics, Griffiths–Tavaré approach (GT), Poisson–Dirichlet distribution 214–215 gene, 91, 240–260, 324 Griffiths, S., 388 ancestor gene, 241 Gromov, M. L. class, 341 Gromov–Hausdorff distance, 50 expression, 326–327, 335, 338 Gromov–Hausdorff space, 54 flow, 222 group frequency, 122, 359, 364 locally compact group, 137 locus, 239–240 metric group, 141 sequencing, 240 normed group, 139, 164 genealogy, 211, 378 group theory, 116 generator, 359, 362, 375, 399, 400 GT, Griffiths–Tavaré approach backward generator, 364 Guess, H. A., 244–245, 256 forward generator, 360 Gumbel,E.J. generically all, 138 Gumbel distribution, 56, 304, 316 genetic diversity, 30 Gurel-Gurevich, O., 392–393 genetic drift, 222 Haar, A. genetic map, 218, 223 Haar wavelet, 87 genetics, see mathematical genetics Haas, B., 265, 273, 297 genetic type, see type Häggström,O., 392 genetic variation, 92, 205, 213, 240–241, Hahn, H., see Vitali–Hahn–Saks 243 theorem Geneva, University of, 394 Haigh, J., 30, 32 genome, 91, 208, 222, 239 Hairer, M., 451 genotype data, see data Hall, P. G., 109 geometric ergodicity, see ergodicity Halmos, P. R., 159, 253, 348 Gibbens, R. J., 430 Hammersley, J. M., 113–114, 116–117, Gibbs, J. W. 122, 381 Gibbs sampler, 216, 221, 223, 228, haploid, 206, 223 328, 330–331, 335 haplotype, 216–220, 223–233 Gijbels, A., 187 HapMap, see data Gijbels, I., 190 Harbord, B., 430 Glad, I. K., 196 harmonic map, 450 Gnedin, A. V., 251, 328 Harris, S. C., 492 Goldie, C. M., 32, 138–139, 142, Harris, T. E., see Bellman–Harris 144–146, 194 process Goldstein, S., 448 Harrison, J. M., 416, 420–421 Golomb, M., 137 Haulk, C., 251, 328 goodness of fit, 97, 106 Hausdorff, F., see Gromov–Hausdorff Google, 303 distance/space Google Scholar,59 Hausdorff metric/topology, 291, 296 Gould, S. H., 137 Haynor, D. R., 338 Graham, R. L., 139 Hazelton, M., 67 graph, 44, 54–55, 57–59, 381–384, 391 heat flow/loss, 398–400 ancestral recombination graph, 213 Heath, D., 170 complete graph, 384 heat kernel, 400 connected graph, 383, 393 Heaviside, O., 23, 123, 126 directed graph, 382 heavy-tailed, see tail behaviour threshold graph, 59 heavy traffic, xv, xvii, 23, 416–443, see Green, G., see Taylor–Green vector field also queue Green,P.J.,33 diffusion approximation, 421, 429, 435 Greenman, C., 91 Heiberg, C., 146 Grey, D. R., 119 Heidergott, B., 466

Index 517

Hein, J., 205, 211, 256 hypergraph, 59 Heisenberg,W.K. hyperparameter, 321, 331–332 Heisenberg group, 453 hyperprior, 79 Henschel String Quartet, 508 ice crystals, 398 hereditary, 159, 163 iff, if and only if heredity, 238 Iglehart, D. L., 26–27, 416 Hermite, C. i.i.d., independent and identically Hermitian matrix, 43 distributed Hesse, C., 193 Iksanov, A., 114 heterozygote advantage, 246 IMPUTE 2, see phasing algorithm Heyer, H., 137 inclusion-exclusion, 58, 105, 158, 162, hidden Markov model (HMM), see 301, 406 Markov, A. A. indicator process, 172 hierarchical modelling, 320, see also infection rate, 391 Bayesian hierarchical model inference, see Bayesian inference, high-frequency information, 195 coalescence, full-likelihood Hilbert, D. inference, statistical inference Hilbert–Schmidt norm, 455 infinite divisibility, 137, 378 Hindman, N., 138–139, 157, 159, 161, infinitely-many-alleles model, see 163–165 mathematical genetics hitting probability, 346 infinitely-many-sites model, see hitting time, 400 mathematical genetics Hjort, N. L., 196, 320, 329 information feed, 484, 486–490 HLA, human leukocyte antigen initial rank, 293 HMM, hidden Markov model Institut Henri Poincaré, 394 Hodge, W. V. D., 21, 24, 26, 29, 31 intensity measure, 115–116 Hoffman, C., 392 interacting particle system, 206, 448 Hoffmann-Jørgensen, J., 138, 152 International HapMap Consortium, 216, Hofstad, R. van der, 502 218, 229 Holmes, C., 329 International Mathematical Union, 380 Holmes, S., 58, 102 Internet, 417, 422, 442 Holroyd, A. E., 388n, 394 interrogation, 484 Holst, L., 328, 448 invariant manifold, 425 homeomorphism, 144–145, 147–149 inverse function theorem, 457 homogenization, 398–401, 408–410 inverse gamma distribution, 459 homozygosity, 98 inverse temperature, 391 Hoover,D.N.,40 Isaac Newton Institute for Hopf,E.F.F.,see Wiener–Hopf theory Mathematical Sciences, 381 Hoppe, F., 253, 255 Ishwaran, H., 323, 328, 330 Hörmander, L. Ising, E. Hörmander regularity theorem, 451 Host, B., 120–121 quantum Ising model, 391 Howie, B. N., 216, 220, 228 isometry, 42, 44, 50 Hoyle, F., 20, 24 quasi-isometric, 390 Hsu, E. P., 450 rough isometry, 390 Hudson, R. R., 205 Itô, K., see also multiple Wiener–Itô Hughes, I. M., 239 integral human anthropology, 222 Itô calculus, 33, 454, 461 human disease genetics, 222 Jacka, S. D., 450 Human Genome Project, xvii Jacobi, C. G. J. human leukocyte antigen (HLA), 220 Jacobi diffusion, 359–361, 368, Hurst, H. E. 370–372, 374–375 Hurst index, 171–174, 180 Jacobi–Poisson kernel, 372–373 Hu, Y., 120 Jacobi polynomial, 359–360, 365, 368, hydrodyamical limit, 48 370–373 hypercube, 450 Jagers, P., 116 hypergeometric function, 359 James, L. F., 328, 330

518 Index

Janson, S., 58, 448 influence, xvii, 39, 48, 110, 113–114, Jansons, K. M., 452 164, 170, 205–206, 239–260, 341, Jeffreys, H., 22 380–381, 416–418, 446, 491 Jensen, A., 23 Kingman coalescent, xv, xvii, 30, Jikov, V. V., 400 48–49, 96, 205–206, 239, 241, Johnstone, I. M., 86 250, 252, 256–260, 320, 360, 376, Jonasson, J., 391–392 381, 448 Jones, M. C., 190 Kingman coalescent tree, 361, 364 Jordan, M. I., 329 Kingman distribution, 240, 244, Josephs, H. J., 23 247–248, 250–251, 253 Joyce, P., 211, 256, 322, 328 Kingman paintbox, 39, 44–46, jump process, 372 268–269, 274, 280, 283, 290–291 Kac, M., 164, 398, see also Kingman subadditive ergodic Feynman–Kac formula theorem, 381, 400, 402 K-allele model, see mathematical Kingman, M. E., née Harley, 18 genetics Kingman, Robert, 18 Kallenberg, O., 41, 320 Kingman, William, 17 Kang, W. N., 418, 422, 427–429, 438n Kipnis, V., 187 Karlin, S., 30, 137, 243, 346 Klein, A., 391 Karpelevich, F. I., 469 Klüppelberg, C., 173 KBD, Kestelman–Borwein–Ditor König, W., 502 Kelly, F. P., 33, 254–256, 259 Kolmogorov, A. N., 23, 27, 33, see also Kemperman, J. H. B., 138, 157, 163 Fisher/KPP equation Kendall, D. G., xv, 23, 25–34, 136–137 Kolmogorov diffusion, 452 Kendall, M. G., 28 Kolmogorov extension theorem, 50 Kendall, W. S., 67–71 Konstantopoulos, T., 424 Kendall–Møller algorithm, 68–71 Kotsialis, A., 430 Kent, J. T., 353 Kottas, A., 329 kernel methods, 186–201 Kozlov, S. M., 400 deconvolution-kernel estimator, 192, KPP, Kolmogorov–Petrovskii–Piskunov 198 Kragh, E., 186 infinite order kernel, 195 Kramer, P. R., 413 kernel density estimator, 189 Krogh, A., 219–220, 226 second-order kernel, 196–197, 200 Kuhner, M. K., 214 sinc kernel, 186, 192, 195–199 Kuratowski, K., 147 kernel polynomial, 365 Kurtz, T. G., 50, 209, 211, 360, see also kernel stick-breaking process, 329 Donnelly–Kurtz process Kerov, S., 271 Kutta, M. W., see Runge–Kutta Kestelman, H., 137–138, 157 method Kesten, H., 242, 393–394, 475 Kuwada, K., 450 k K-function, see point process -wise independence, 384, 392 Khinchin, A. Ya. [Khintchine, A. Y.], Lagrange, J.-L. 23, 162, see also Lévy–Khintchine Lagrange multiplier, 423, 433 formula Lamperti, J. killing, 374 Lamperti-stable process, 293 Kimura, M., 242, 243, 246 Lang, R., 398 Kingman, Charles, 17 Laplace, P.-S. Kingman, F. E. T., 17–19 Laplace distribution, 199 Kingman, J. F. C., 35, 41, 89, 115, 117, Laplace transform, 116, 244, 350, 352, 136–140, 143–154, 181, 211, 296, 354, 371, 373 320, 322–324, 328, 332, 354, 363, large deviations, 492, 501–502 419, see also partition structure, latent variables, 336 p-function, regenerative lattice phenomenon close-packed lattice, 394 career, xv–xvii, 19–32, 381 dual lattice, 382 early life, xv, 17–19 square lattice, 382, 391–392, 394 honours, xvi stretched lattice, 392

Index 519

triangular lattice, 382, 393 load, see workload Laud, P. W., 328–329 Lo, A. Y., 325 Lau, J. W., 326–327, 340 local constant estimator, see density Lauritzen, S. L., 284 estimation Lavine, M., 329 local least squares, 190 Lawler, G. F., 382 local linear estimator, see density Laws, R., 186 estimation LDhat program, 223 locally compact group, see group L-diffusion, see diffusion locally stable point process, see point Lebesgue,H.L. process Lebesgue density theorem, 141, 142, local polynomial estimator, see density 148 estimation Ledermann, W., 31 Loewner, C. [Löwner, K.], see Lee, N. H., 418, 422, 427–429, 438n Schramm–Loewner evolutions Lee, P. M., 30 (SLE) Lee, T. J., 424 logistic distribution, 56 Le Gall, J.-F., 53, 403 London Mathematical Society, xvi, 27 Levinson, D., 430 long-range dependence, see point Lévy, P. process Lévy–Khintchine formula, 137, 292 look-down, 50, 360 Lévy measure, 292–294 loss function, 340 Lévy process, 33, 286, 360, 371, 373 Lovász, L., 58 Lévy stochastic area, 447, 453–461 lower semi-continuous, 127 Lewis, M. A., 126–128 Loynes,R.M.,25 p Lewontin, R. C., 92, 100, 110 L -bounded, 304 p L-function, see point process L norm, 403 Li, B., 126–128 LRD, long-range dependent/dependence Lickorish, W. B. R., 21 Lui, R., 123, 125 Lieshout, M. N. M. van, 71–73 Lu, Z., 346, 350, 352 lifetime, 173, 182 Luzin, N. N., 142 lifetime distribution, 170, 172 Lyapunov, A. M. lifting, 147 Lyapunov function, 425 Liggett, T. M., 113n, 386–387, 391, 394, MacEachern, S. N., 329 448 machine learning, 265 light-tailed, see tail behaviour MacKay, D. J. C., 486 likelihood, 221, 331–332, see also MacQueen, J. B., 330 full-likelihood inference MAF, minor allele frequency marginal likelihood, 327 Mairesse, J., 466 pseudo-likelihood, 77, 88 Majda, A. J., 413 LIL, law of the iterated logarithm Malliavin, P. Lima, B. N. B. de, 388–389 Malliavin calculus, 43 Limic, V., 475 Malyshev, V. A., 466, 469 Li, N. Mandel, S. P. H., 239 Li and Stephens model, 205, 215–226 Mandel–Scheuer inequality, 24, 30 Lindelöf,E. L., 151 manifold, 450–451 Lindley, D. V., 21–24, 29, 354 Mannion, D., 30 Lindvall, T., 447, 450–451 Manolescu, I., 384n lineage, 217, 360–361, 376 Marchini, J., 216, 220, 228 linearization, 499, 505–506 Mardia, K. V., 353 line of descent, 48, 360–361, 367 Marjoram, P., 92–95, 100, 213, 256 linkage disequilibrium, 222–223, 229 Markov, A. A., 213, see also reverse Lions, J.-L., 400 Markov chain Lipschitz, R. O. S., 398, 400–401, 408 hidden Markov model (HMM), 219 Li, S. F., 451 Markov branching process, 118 Littlewood, J. E., 142, 303 Markov chain, xv, xvii, 23, 25, 27, 31, Liu, Q., 116, 119 40, 48, 53, 136, 181, 206, 214, Li, Y., 220 218, 350, 371, 383, 448, 464–481

520 Index

Markov chain Monte Carlo (MCMC), mean squared error, 194 65, 214, 329–330, 400, 411–414 measurability, 383 Markov inequality, 311 measurement error, 186–191 Markov point process, 71 measure-preserving map, 41 Markov process, 27, 47–50, 67, 208, measure theory, 138–164, 380 211, 370–371, 373, 378, 419–420, measure-valued diffusion, see diffusion 423–424, 449 measure-valued process, 368 Markov property, 350, 401–404 median, 78, 84, 86, 118, 122 Markov renewal process (MRP), 182 meiosis, 218 martingale, 33, 114, 346, 349, 351–352, Meister, A., 186 354, 427, 466, see also Wald Meixner, J., 361 martingale Melville, H. W., 18 martingale differential identity, 455 Mendel, G. J., 238 Maruyama, G., see Euler–Maruyama Menshiikov, M. V., 466 solution Merl, D., 333 Mascheroni, L., see Euler–Mascheroni Merlo, L. M. F., 92 constant metastasis, 98 Massoulié, L., 423–425, 429 Metcalfe, P. D., 452 Mathematica, 499–500, 503 methylation, 92–96 mathematical finance, 33 metric measure space, 60 mathematical genetics, xv, xvii, 24, 30, metrizability 39, 46, 48, 92, 110, 205–233, almost-complete metrizability, 138 238–260, 265, 272, 320, 324, 360, complete metrizability, 138, 140, 159, 381, see also allelic partition 163 distribution, Cannings model, microbiology, 326 evolution, Mandel–Scheuer Midthune, D., 187 inequality, mixture model, migration, 224 monomorphism, Moran model birth-and-death model, 247 Miles, R. E., 25 charge-state model, 241–242 Miller, H. D., 25 infinitely-many-alleles model, 96, 242, minimax, 195 244, 250 minimum contrast methods, 186, infinitely-many-sites model, 260 195–201 K-allele model, 247–248 Minkowski, H. neutrality, 207–208 Minkowski addition, 72 neutral model, 96, 100 minor allele frequency (MAF), 232 neutral theory, 243, 246, 259 Mitchell, J. M. O., née Speakman, 30 random-sampling model, 243, Mitchell, J. S., 18 247–250, 257–258 Mitchison, G., 219–220, 226 standard neutral model, 208 mitochondrial DNA, see DNA mathematical physics, 380 mitochondrial Eve, 239, 259 Mathematical Reviews,59 mixing, 356 Mathematical Tripos, see Cambridge, mixture model, 222–223, 320–321, 323, University of 333, see also Dirichlet process Matlab, 314 mixture model Maulloo, A., 417, 422 exchangeability, 320 maximum likelihood, 220 MLE, maximum-likelihood estimator maximum-likelihood estimator Mode, C. J., 116 (MLE), 97, 106 molecular variation, 92 max-plus algebra, 466 Møller, J., 67–71, see also McCloskey, J. W., 322, see also GEM Kendall–Møller algorithm distribution Mollison, D., 123 McDiarmid, C., 120–121 moment McGregor, J. L., 137, 243 bound, 304 MCMC, Markov chain Monte Carlo index, 173 McVean, G. A. T., 220, 221, 223, 228 monomorphism, 245, 255 meagre, 138–163 monotone-convergence theorem, 505

Index 521

Monte Carlo, see Markov chain Monte neural network, 469 Carlo (MCMC) neutrality/neutral theory, see complete Moran, P. A. P., 30, 239, 242, 247, 250, neutrality, mathematical genetics, 255, 362 selection Morris, W., 33 neutral to the right, 292 Morse, H. C. Newell, G. F., 417 Morse–Thue binary sequence, 452, Newman, C. M., 391 461 Newton, I., see also Isaac Newton Mossel, E., 391–392 Institute for Mathematical Sciences motorway, see road traffic Newtonian capacity, 398, 406 moving average process, 352 Nikodým, O. M., see Radon–Nikodým MRCAS, most recent common ancestor derivative of the sample Nobile, A., 333 mRNA, 338 non-arithmetic distribution, 172, 176, MRP, Markov renewal process 180 Müller, P., 326, 329 non-interference, 248, 250 Mulholland, H. P., 239 non-standard analysis, 42 Muliere, P., 323 Nordborg, M., 256 multiple observers, 92, 97–104 normal–inverse gamma model, 327, 331 multiple Wiener–Itô integral, 43 nowhere dense, 141 multiplicative cascade, 116 nowhere differentiable, 60 multiplicatively regenerative, see number theory, 116, 162, 380 random permutation nutrition, see data multiscale analysis, 391 multiscale process, see point process observer, see multiple observers multitype age-dependent process, see ODE, ordinary differential equation branching process ODE flow, see flow multivariate t distribution, 327, 331, Office of National Statistics, xvi 337–338 Ohta, T., 242 mutation, 91–96, 99–100, 207–211, Oldser, G. J., 466 213–215, 217–220, 227, 241–245, Ole˘ınik,O. A., 400 252, 256, 258–259, 359–361, ON/OFF periods, 170, 172, 178, 181 364–368, 376–378 operator, 398, 410, see also mutation operator, 363 operator parent-independent mutation, linear operator, 125, 130 214–215 updating operator, 122 passenger mutation, 92–94 winding operator, 506 rate, 100, 101, 218, 226, 359 optimization, see convex optimization Myers, S. R., 220, 223, 228 orderly, see point process myopically continuous function, 71 order-preserving, 122 order statistics, 240, 244–245, 248 Nadaraya,E.A. ordinary differential equation (ODE), Nadaraya–Watson estimator, 189 492, 496–499 Nakry, D., 461 Orey, S., 26–27 Nash, J. F., 422 Ornstein, L. S. Nason, G. P., 87 Ornstein–Uhlenbeck process, 450 National Academy of Sciences, xvi orthogonal-function expansion, 368 natural selection, 222 orthonormality, 365–366, 370 fundamental theorem of, 24 Neal, R. M., 329 Osborne, T. J., 391 netflow, 428 Ostrowski, A. M., 138 network, 416–418, 422–442, see also Owen, A. R. G., 24 Brownian network Oxford, University of, xv–xvii, 26, 33, capacity, 417–418, 422–424, 427, 205 430–433, 438–439 PAC, product of approximate capacity allocation policy, 422, 433, conditionals 437, 439 paintbox, see Kingman paintbox connection, 422–424, 427–429 theorem

522 Index

Palm, C., 23 cluster percolation, 392 Papageorgiou, M., 430 dependent percolation, 390–393 Papangelou, F., 67 directed percolation, 382, 389–390, Papangelou conditional intensity, 67, 392 69, 73 of words, 385, 389, 393–394 Papanicolau, G. C., 398, 400–401 oriented percolation, 382 parallel transport, 451 site percolation, 381, 389–391, 393 Parekh, A., 489 vertically dependent percolation, 391 Park, J.-H., 329 vertically directed percolation, 391 partial diﬀerential equation, 380, see Peres, Y., 391–392 also Fisher/KPP equation perforated domain, 398–400, 408 partially exchangeable partition, see Perman, M., 271 random partition Perron, O. particle process, 50 Perron–Frobenius eigenvalue, 124, partition, 266, see also Bayesian 131, 350 partition, exchangeable partition Petrone, S., 340 probability function (EPPF), Petrovskii, I. G., see Fisher/KPP partition structure, random equation partition Petruska, Gy., 147 colour partition, 268 p-function, xv, 27, 136–137, 170, degree, 325 181–182 distribution, 335 pgf, probability generating function model, 321–332, 334 PHASE, see phasing algorithm ordered partition, 267 phase, see renewal theory partition structure, 247, 249–251, phase transition, 392, 491, 496, 504 265–274, see also Ewens partition phasing, 223 structure phasing algorithm extended two-parameter family, Beagle, 216 271–273, 282, 289 FastPhase, 216 regenerative partition structure, 286, IMPUTE 2, 216 292 PHASE, 216, 221, 228 Pascal, B. physical chemistry, 48 Pascal programming language, 314 physics, 391 path, 384, 388, 393 Pidcock, D., 317 functional, 451 Piskunov, N. S., see Fisher/KPP integral, 447, 461 equation open path, 391–392 Pitman, J. [Pitman, J. W.], 36, 44, 46, undirected path, 384 54, 60, 97, 251, 258, 328, 330, 332, Patil, C. P., 322 448 Patterson, N., 221 Plancherel, M. Pavliotis, G. A., 413 Plancherel’s identity, 193 Pearson, E. S., 28 p.m., probability measure Pee, D., 187 point process, 171, 173, see also Markov Pegrum, H. B., 18 point process, Strauss point process Peled, R., 387–388, 390, 392–394 area-interaction process, 65, 67, Pemantle, R., 508 70–77, 79, 81 Pembroke College, see Cambridge, clustering, 70–77, 79, 81 University of K-function, 74–75 penalised contrast methods, 186, 198, L-function, 74–77 200 locally stable point process, 65, 67–70 Penrose, M. D., 465 long-range dependent point process, Pepper, J. W., 92 171, 173, 181 percolation, 55, 381–382, 384, 388, 392, multiscale area-interaction process, see also critical probability, phase 65, 71–74 transition orderly point process, 173, 182 AB-percolation, 393 spatial point process, 65, 67, 72 bond percolation, 382, 391 stationary point process, 180

Index 523

T-function, 75–76 pseudo-likelihood, see likelihood Poisson, S. D., see also Jacobi–Poisson quantum kernel, two-parameter Ising model, see Ising, E. Poisson–Dirichlet process mechanics, 24, 43 auto-Poisson process, 80 quasi all, 146 compound Poisson process, 293, 371 quasi-isometric, see isometry Poisson approximation, 100–110 question bank, 300, 302, 314 Poisson–Dirichlet distribution, 39, 46, questionnaire, 187 240, 248, 265, 273, 297, 328 queue, xv–xvi, 23, 25–27, 33, 136, Poisson–Dirichlet point process, 363 354–355, 466, 469, see also heavy Poisson–Dirichlet random measure, traffic, workload 363–364 discipline, 419–420, 439 Poisson process, xv, 53–54, 67–68, 70–72, 75, 257, 293, 355, 364, Radon, J. 371, 390, 418, 420, 424, 434 Radon–Nikodým derivative, 349, 355 Poisson random measure, 398, 408 ramp metering, 417–418, 430–442 Polish space, 147 Ramsey, F. P., 139, 162 political philosophy, 422 random colouring, see colouring Politis, D. N., 196 random curve, 382 Pollaczek, F., 23 random discrete distribution, 265, 328 Pólya, G. random drift, see drift Pólya sequence, 330 random environment, 346, 390–391 Pólya tree, 329 random mapping, 46, 54, 246 Pólya urn, 276, 326, 330 random mating, 239 Pólya urn sampler, 332, 335–336, 339 random matrix, 43, 57 population density, 125 random measure, 37, 171, 173, 364, see population genetics, see mathematical also Poisson random measure genetics stationary random measure, 174, 178 population structure, 221–233 random medium, 382 geographical population structure, random metric space, 390 205, 208, 211, 221 random mutation, see mutation portmanteau theorem, 52, 54 random network, 55–56 positive definite, 368–369 random order, 293–296 posterior predictive distribution, 331 random partition, 39, 44–46, 48, 50, Post Office Engineering Research 243–251, 265–297, 324–325, 329, Station, 22–23, 25 see also exchangeable partition potential, 465–470, 474–475, 479–481 probability function (EPPF) predictive density, 333 exchangeable random partition, 265, Price, A. L., 221 267, 270, 273, 276, 278–291 Price, C. J., 452, 461 partially exchangeable random prime divisor, 145 partition, 274–278 prime factorization, 260 regenerative random partition, 265, Pritchard, J. K., 221–223, 228, 231 287–292 Privman, V., 465 τ-regenerative random partition, 289 probability, 17, 21–24, 32, 36, 42, 88, random permutation, 46, 246 113–114, 213, 320, 380, 382, 391 size-biased random permutation, probability generating function (pgf), 269–270, 293–294 128, 136, 377–378, 498 τ-biased random permutation, 294, processor sharing, 420, 422 296 product of approximate conditionals random-sampling model, see (PAC), 221 mathematical genetics, sampling product topology, 57 random set, 390, 399 profile plot, 339 multiplicatively regenerative random Prohorov, Y. V., 416 set, 266, 289–297 projection, 402, 412 regenerative random set, 265 proposal distribution, 215–216 random shuffle, 297 Propp, J. G., 66 random tree, 51–55, 206, 210, 265, 273

524 Index

random walk (RW), 51, 136, 345, 354, phase, 173–174, 178, 181 377–378, 383–384, 392–393, 450, renewal density function, 170 469, 479, see also branching renewal function, 172–173, 182 random walk (BRW), ergodicity, renewal process, 173 random environment, renewal sequence, 136 self-avoidance, stationary stationary renewal process, 172–173, increments 182 associated random walk, 346, 348–354 Rényi, A., 30, 55 ranked empirical frequencies, 45 repulsion, see point process: clustering ranked frequencies, 268, 271, 273 residual allocation model, 253 ranking, 328 residual allocation scheme, 275 rat, 338 residual fraction/proportion, 271, 273, rate distortion theory, 485 275–277, 282, 284, 290 rate of convergence, 119, 346 Resnick, S. I., 170, 304 rationally invariant function, 152–155 resource-route incidence matrix, 422, Rauch, J., 398 431, 436 r.d.f., renewal density function Reuter, G. E. H., 27, 33, 144 reaction-diffusion equation, 122–123, reverse Markov chain, 351 128 reversibility, 245, 359, 364, 370 recombination, 212–213, 217–220, reversible jump method, 329 223–229 Révész, P., 346 record, 294–295 Richardson, S., 320, 323, 329 recovery rate, 391 Richthammer, T., 386–387, 394 recurrence time, 420 Rideal, E. K., 18 recurrent event, xv, 25, 27, 135–136 Riemann, G. F. B. Redwood seedlings, see data Riemann integral, 195 Reed, B., 120–121 Ripley, B. D., 74 reflection principle, 450 rising factorial, 270 regenerative partition, see random road traffic, 417–418, 430–442 partition Roberts, J., 423, 429 regenerative phenomenon, xv, xvii, 27, Roberts, S., 21 136, 170, 182, 265, 286 Robin, J.-M., 186 regenerative set, see random set robustness, 247, 251 regression, 188, 326, 336–337, see also Rogers, L. C. G., 451, 508 errors in variables, deconvolution, Rolski, T., 178, 182 kernel methods, measurement error Romano, J. P., 196 nonparametric regression, 65, 77–88 root signal-to-noise ratio (RSNR), see regression estimator, 192 signal-to-noise ratio regression mean, 193 Ross, S. M., 303 regular variation, 137, 139, 170, 194 rotatable array, 43 Reich, D., 221 rotatable matrix, 41 Reid, B. J., 92 rotatable random functional, 43 Reiman, M.I., 416 Rothschild, B. L., 139 reinforced process, 466 Royal Society, xvi rejection method, 98 Royal Statistical Society (RSS), xvi, 23, relativity, 24, see also diffusion: 28 relativistic diffusion RoyChoudhury, A., 92, 100, 110 Renesse, M. von, 451 Roy, R., 372 renewal process, 420 Rozenholc, Y., 198–199 renewal theory, 346, see also cycle time, RSNR, root signal-to-noise ratio lifetime, lifetime distribution, RSS, Royal Statistical Society Markov renewal process (MRP), R statistical package, 87 ON/OFF periods ruin, 346, 354 alternating renewal process, 170–182 Ruiz-Linares, A., 221 elementary renewal theorem, 172–173, Runge, C. D. T. 175 Runge–Kutta method, 499 generalized renewal theorem, 175 Ruppert, D., 187

Index 525

Ruziewicz, St., 162–163 selectively neutral, 324 Ruzzo, W. L., 338 self-avoidance, 381–382, 393–394 r.v., random variable self-similar, 47 RW, random walk semicircle distribution, 43 Ryan Jr., T., 116 semigroup, 137 Rybko, A. N., 469 Seneta, E., 146 Saatci, Y., 430 Servin, B., 220 Saks, S., see Vitali–Hahm–Saks theorem Sethuraman, J. Samorodnitsky, G., 170 Sethuraman stick-breaking model, 322 sampling, 30, 38–39, 49, 58, 92–93, Sgibnev, M. S., 170, 172–173, 176, 181 217–231, 241, 247–250, 253–258, Shah, S. P., 91 271, 283, 291, 296, 324, see also Shannon, C. E., 483 Ewens sampling formula, Gibbs Shannon information, 485 sampler, Pólya urn sampler, Shibata, D., 92–95, 100 size-biased pick shift-coupling, see coupling hypergeometric sampling, 97, 363 shifted-filter property, 163 importance sampling, 214–216 shift-embedding, see embedding multinomial sampling, 323 Shi, Z., 120 posterior sampling, 330–331 Sidoravicius, V., 393–394 sampling-consistent, 266–268 Siegmund, K. D., 92–95, 100 sequential sampling, 98 Sierpiński, W., 142–143 size-biased sampling, 251, 253, 322, Sierpiński gasket, 60 328 signal-to-noise ratio, 88 spatial sampling, 98 root signal-to-noise ratio (RSNR), 86 species sampling, 265, 322, 330 Silva, R. W. C., 389 Sanchis, R., 389 Silverman, B. W., 33, 189 sausage, see diffusion sausage, Wiener Simonoff, J. S., 190 sausage simulation, 92, 99, 198, 216, 413, see scaling exponent, 47 also algorithm scaling invariance, 288 exact/perfect simulation, 65–88, 211 scheduling health warning, 386 clairvoyant scheduling, 383–384, 392 posterior simulation, 332 Scheet, P., 216, 220, 223 simultaneous re-allocation, 332 Scheuer, P. A. G., 239, see also sinc kernel, see kernel methods Mandel–Scheuer inequality single nucleotide polymorphism (SNP), Schierup, M. H., 205, 211, 256 216–233 Schladitz, K., 75 site, 381 Schmidt, E., see Hilbert–Schmidt norm size-biased pick, 269, 273, 279, 289–290, Schoenberg, I. J., 39 297 Schonmann, R. H., 391 τ-biased pick, 288–290 Schramm, O. size-biased permutation, see random Schramm–Loewner evolutions (SLE), permutation 382 size biasing, 45, 251, 254, 265, 297, 378, Schwartz, L., 28 see also sampling Schwerdt, R., 26 Sjöblom, T., 91 Science and Engineering Research skeleton Council, xvi, 205 discrete skeleton, 118, 124, 128, 136, Science Research Council, xvi, 18 156 Scott, D. B., 31 integer skeleton, 138, 145 Scott, D. W., 189 perturbed skeleton, 154 Scott, J. F., 32 rational skeleton, 138, 145, 151–156 Scudo, P. F., 391 set, 146 Secchi, P., 323 SLE, Schramm–Loewner evolutions second countable, 151 SLLN, strong law of large numbers seismology, 186 Smirnov, S., 382 selection, 362, see also natural selection Smith, A. F. M., 328–329 multi-locus selection, 24 Smith, C. A. B., 239

526 Index

Smithies, F., 29 stick-breaking process, Sethuraman Smith, N. J., 216, 223, 228 stick-breaking model Smith, W. L., 346 Stirling, J. Smoluchowski, M. R. von S. Stirling number, 243 Smoluchowski coagulation equation, stochastically bounded, 241, 304 48 stochastically monotone, 66 smoothing parameter, 193, 200 Stochastic Analysis Group (StAG), 27 snapshot principle, 419, 431 stochastic area, see Lévy stochastic area SNP, single nucleotide polymorphism stochastic differential equation, 399, Solecki, S., 147 405, 411, 449, 454 spanning subtree, 52 stochastic differential system, 456–458, Speakman, J. M. O., see Mitchell, J. M. 460 O. stochastic geometry, 391 spectrum, 370 stochastic integral, 449, 452 Speed, T. P., 350 Stoneley, R., 21, 23 Spencer, J. H., 139 storage strategy, 240 spin glass, 116 Strauss, D., 139, 165 Spitzer, F. L., 381 Strauss, D. J., 74 Spitzer’s identity, xvi Strauss point process, 71 spreading out, 116–131, see branching strong law of large numbers (SLLN), 45, random walk (BRW) 345, 457 spreading speed, 125–131 strong nuclear force, 73 spread of population, 122 structured data, 328 Srikant, R., 417, 422 STRUCTURE program, 222 stable density, 373 Stuart, A. M., 413 stable process, 373, see also Sturm, K.-T., 450 Lamperti-stable process subadditivity, xv, xvii, 113, 139, 170, Stacey, A. M., 384 182, 403, see also Kingman Stanford University, 187 subadditive ergodic theorem started from dust, 48, 54 subordination, 360–361, 371–372, state space collapse, 420, 428, 436 374–376, 378 stationary increments, 346–354 subordinator, 46, 286, 292–294, 376 α stationary measure, 363 -stable subordinator, 293 stationary random measure, see random stable subordinator, 271 measure subsequence principle, 39 statistical genomics, 320 sufficient statistic, 40, 97, 106, 243, 324 statistical inference, 208, 213, 221 supercritical, see branching process Statistical Laboratory, see Cambridge, SureShrink method, see wavelet University of Sussex, University of, xvi, 17, 21, 31–32 statistical physics, 382, 391 symmetric deletion kernel, see deletion statistical smoothing, 190 symmetric distribution, 188, 193, 195 statistics, 21, 24, 29–31, 33, 92, 113, 292, symmetry, 105 328, see also Bayesian statistics Szegedy, B., 58 computational statistics, 88 Szemerédi,E. Statistics Commission, xvi Szemerédi’s theorem, 59 Steel, M. F. J., 329 tail behaviour, 194–195, 197, 304, 460, Stefanski, L., 186–187, 192 475, see also Gaussian tail estimate Stein, C. M. heavy-tailed, 88, 170 Stein’s method, 59 light-tailed, 88 Stein–Chen approximation, 448 Taillie, C., 322 Steinhaus, H. D., 139, 142 tail σ-algebra, 452 Stephens, M., 214, 216, 220–223, 228, Tajima, F., 205 231, see also Li and Stephens Takács, L., 23 model Tan, D., 417, 422 stick breaking, 53, 270, 273, 275–276, Tao, T., 59 293, 328, 334, see also kernel Tarrès, P., 466

Index 527

τ-biased permutation, see random coalescent tree, random tree, permutation spanning subtree τ-biased pick, see size-biased pick process, 364 τ-deletion, see deletion real tree, 52, 60 Taupin, M.-L., 198–199 Truong, Y. K., 192, 195 τ-regenerative, see random partition Tsao, J. L., 92 Tavaré, S., 205–215, 247, 254, 258, 328, Tse, D., 489 340, 361, see also Griffiths–Tavaré Tsybakov, A. B., 186 approach (GT) tumour, 92–100 Taylor, B. Turl, H. W., 19 Taylor–Green vector field, 412 Turner, R., 67 Taylor, M., 398 Turova, T. S., 469 Taylor, S. J., xvi, 28 two-allele model, see allele Teh, Y. W., 265, 329 two-allele process, see allele teletraffic, 22, 23 two-parameter Poisson–Dirichlet Temple, G. F. J., 28, 31 process, 328, 332 tensor field, 400–401 type, 124, 207–217, 219–220, 227 test, see computer-based test two-type, 125 Tetali, P., 384 type space, 124–126, 207, 214 Tetris, 466 Uhlenbeck, G. E., see Teugels, J. L., 138–139, 142, 144–146, Ornstein–Uhlenbeck process 194 ultraspherical polynomial, 370 Teung, K. Y., 338 uniform boundedness theorem, 139 T-function, see point process uniform integrability, 312, 405 Thorisson, H., 450 uniform kernel, see kernel methods Thue, A., see Morse–Thue binary unimodal density, 326 sequence upper record, see record tight, 121–122 urn, 102, see also Pólya urn tile, 291 Ushakov, N., 196 tilt, 373 Value at Risk, 41 time-course experiment, see data Varadhan, S. R. S., 398, 401, see also time-series, 328 Donsker–Varadhan invariance time reversal, 48, 448 principle Titchmarsh, E. C., 26 Varaiya, P., 430 Tiwari, R. C., 322 variance bound, 305 Todd, A. R., 29 vector field, 398, 408, see also Toland, J. F., 491, 508 Taylor–Green vector field Tomfohrde, J., 221 velocity field, 401 topology, see also Hausdorff Vere-Jones, D., 171–172 metric/topology Vershik, A. M., 42 bitopology, 138–164 vertex density topology, 137–143, 146 closed vertex, 381 fine topology, 147 degree, 383 torus, 401 open vertex, 381, 384, 388, 391 total-variation metric, 100–109 Vesilo, R., 171, 178, 182 traffic, see road traffic Viot, M., see Fleming–Viot diffusion Trajstman, A. C., 247 virtual resource, 440–442 transience, 366 Vitali, G. transition density, 359–361, 365–368, Vitali–Hahn–Saks theorem, 348 371, 374–375 Volchan, S., 391 transition function, 360–364, 376, 378 Waerden, B. L. van der, 138, 157, transition matrix, 350, 383 162–163 translation-invariant, 122 waiting time, 419 Trautner, R., 138 Wakeley, J., 205, 211, 256 travelling wave, 124 Wald, A. tree, 294, 360, 378, see also Kingman Wald martingale, 346, 349

528 Index

Waldmann, P., 221–222 best word, 387 Walker, A. M., 24, 29 periodic word, 393 Walker, S. G., 328–329 worst word, 387 Wand, M. P., 190 workload, 418–421, 424–443 Ward,G.N.,31 cone, 426–427, 434–438 Warren, J., 492 Woude, J. van der, 466 water vapour, 398 Wright, S., 241 Watson, G. S., see Nadaraya–Watson Wright–Fisher diﬀusion, 208, 359, estimator 363–364, 367–368 Watson, H. W., see Galton–Watson Wright–Fisher model, 207–211 process w.r.t., with respect to Watson, J. D., 240 Yamato, J., 214 Watterson, G. A., 30, 39, 98, 100, Yao, D. D., 422 102–103 Yatabe, Y., 92–93 wavelet, see also Haar wavelet Yor, M., 46, 271, 273, 328, 332, 459 BayesThresh method, 78, 87 Yule,G.U. cross-validation method, 78, 87 Yule process, 118 curve estimation, 82, 86–88 false discovery rate method, 78, 87 Zarepour, M., 323 regression, 65 zebra, 73 Zeitouni, O., 121 SureShrink method, 78, 87 Zhang, L., 430 thresholding, 65, 71, 77–88 Zhang, W., 214 WaveThresh package, 87 Zhang, Y., 393–394 Weierstrass, K. T. W., 60 weighted branching process, see branching process Weinberger, H. F., 122–123, 126–128 Weiss, B., 138, 157, 159, 161, 163 Weissman, I., 139 Welsh, D. J. A., 206 Wen, X., 338–340 Werft, W., 430 Werner, W., 382 Wesolowski, J., 273 West, M., 326, 329, 333 White, J., 430 Whittle, P., xv, 24–25, 27, 29, 32, 34 Whitt, W., 417 Wiener, N., see also multiple Wiener–Itˆo integral Wiener–Hopf theory, 492–496, 506, 508 Wiener sausage, 399 Wilkinson, W. E., 346 Willer, C. J., 220 Williams, D., 27, 32 Wilson, D. B., 66 Winkel, M., 265–266, 273, 293–294, 297 Winkler, P., 384 Wirth, T., 223 Witsenhausen, H., 489 Wiuf, C., 92, 100, 110, 205, 211, 256 wlog, without loss of generality Wood, F., 265 Woo, Y.-J., 92–95, 100 word, 385, see also percolation of words alternating word, 386, 393