References

[1] Aigner M, Ziegler GM (1999) Proofs from the Book. Springer, NewYork

[2] Aldous D (1989) Probability Approximations via the Poisson ClumpingHeuristic. Springer, New York

[3] Aldous D, Diaconis P (1986) Shuffling cards and stopping times. AmerMath Monthly 93:333–348

[4] Alon N, Spencer JH, Erdos P (1992) The Probabilistic Method. Wiley,New York

[5] Andel J (2001) Mathematics of Chance. Wiley, New York

[6] Apostol T (1974) Mathematical Analysis, 2nd ed. Addison-Wesley,Reading, MA

[7] Angus J (1994) The probability integral transform and related results.SIAM Review 36:652–654

[8] Apostol T (1976) Introduction to Analytic Number Theory. Springer,New York

[9] Arnold BC, Balakrishnan N, Nagaraja HN (1992) A First Course inOrder Statistics. Wiley, New York

[10] Arratia R, Goldstein L, Gordon L (1989) Two moments suffice forPoisson approximations: the Chen-Stein method. Ann Prob 17:9–25

K. Lange, Applied Probability,DOI 10.1007/978-1-4419-7165-4, © Springer Science+Business Media, LLC 2010

416 References

[11] Arratia R, Goldstein L, Gordon L (1990) Poisson approximation andthe Chen-Stein method. Stat Sci 5:403–434

[12] Asmussen S, Glynn PW (2007) Stochastic Simulation: Algorithms andAnalysis. Springer, New York

[13] Asmussen S, Hering H (1983) Branching Processes. Birkhauser, Boston

[14] Athreya KB, Ney PE (1972) Branching Processes. Springer, New York

[15] Baclawski K, Rota G-C, Billey S (1989) An Introduction to the Theoryof Probability. Massachusetts Institute of Technology, Cambridge, MA

[16] Baker JA (1997) Integration over spheres and the divergence theoremfor balls. Amer Math Monthly 64:36–47

[17] Balasubramanian K, Balakrishnan N (1993) Duality principle in orderstatistics. J Roy Stat Soc B 55:687–691

[18] Barbour AD, Holst L, Janson S (1992) Poisson Approximation. OxfordUniversity Press, Oxford

[19] Barndorff-Nielsen OE, Cox DR (1989) Asymptotic Techniques for Usein Statistics. Chapman & Hall, London

[20] Bender CM, Orszag SA (1978) Advanced Mathematical Methods forScientists and Engineers. McGraw-Hill, New York

[21] Benjamin AT, Quinn JJ (2003) Proofs That Really Count: The Art ofCombinatorial Proof. Mathematical Association of America, Washing-ton, DC

[22] Berge C (1971) Principles of Combinatorics. Academic Press, NewYork

[23] Bhattacharya RN, Waymire EC (1990) Stochastic Processes with Ap-plications. Wiley, New York

[24] Billingsley P (1986) Probability and Measure, 2nd ed. Wiley, New York

[25] Blom G, Holst L (1991) Embedding procedures for discrete problemsin probability. Math Scientist 16:29–40

[26] Blom G, Holst L, Sandell D (1994) Problems and Snapshots from theWorld of Probability. Springer, New York

[27] Boas RP Jr (1977) Partial sums of infinite series, and how they grow.Amer Math Monthly 84:237–258

[28] Bradley RA, Terry ME (1952), Rank analysis of incomplete block de-signs, Biometrika 39:324–345

References 417

[29] Bragg LR (1991) Parametric differentiation revisited. Amer MathMonthly 98:259–262

[30] Brezezniak Z, Zastawniak T (1999) Basic Stochastic Processes.Springer, New York

[31] Brigham EO (1974) The Fast Fourier Transform. Prentice-Hall, En-glewood Cliffs, NJ

[32] Brualdi RA (1977) Introductory Combinatorics. North-Holland, NewYork

[33] Cao Y, Gillespie DT, Petzold LR (2006). Efficient leap-size selectionfor accelerated stochastic simulation. J Phys Chem 124:1–11

[34] Casella G, Berger RL (1990) Statistical Inference. Wadsworth andBrooks/Cole, Pacific Grove, CA

[35] Casella G, George EI (1992) Explaining the Gibbs sampler. AmerStatistician 46:167–174

[36] Chen LHY (1975) Poisson approximation for dependent trials. AnnProb 3:534–545

[37] Chib S, Greenberg E (1995) Understanding the Metropolis-Hastingsalgorithm. Amer Statistician 49:327–335

[38] Chung KL, Williams RJ (1990) Introduction to Stochastic Integration,2nd ed. Birkhauser, Boston

[39] Ciarlet PG (1989) Introduction to Numerical Linear Algebra and Op-timization. Cambridge University Press, Cambridge

[40] Cochran WG (1977) Sampling Techniques, 3rd ed. Wiley, New York

[41] Coldman AJ, Goldie JH (1986) A stochastic model for the origin andtreatment of tumors containing drug resistant cells. Bull Math Biol48:279–292

[42] Comet L (1974) Advanced Combinatorics. Reidel, Dordrecht

[43] Cressie N, Davis AS, Folks JL, Polocelli GE Jr (1981) The moment-generating function and negative integer moments. Amer Statistician35:148–150

[44] Crow, JF, Kimura M (1970) An Introduction to Population GeneticsTheory. Harper & Row, New York

[45] David HA (1993) A note on order statistics for dependent variates.Amer Statistician 47:198–199

418 References

[46] de Bruijn NG (1981) Asymptotic Methods in Analysis. Dover, NewYork

[47] De Pierro AR (1993) On the relation between the ISRA and EM al-gorithm for positron emission tomography. IEEE Trans Med Imaging12:328–333

[48] D’Eustachio P, Ruddle FH (1983) Somatic cell genetics and gene fam-ilies. Science 220:919–924

[49] Diaconis P (1988) Group Representations in Probability and Statistics.Institute of Mathematical Statistics, Hayward, CA

[50] Dieudonne J (1971) Infinitesimal Calculus. Houghton-Mifflin, Boston

[51] Dorman K, Sinsheimer JS, Lange K (2004) In the garden of branchingprocesses. SIAM Review 46:202–229

[52] Doyle PG, Snell JL (1984) Random Walks and Electrical Networks.The Mathematical Association of America, Washington, DC

[53] Durrett R (1991) Probability: Theory and Examples. Wadsworth &Brooks/Cole, Pacific Grove, CA

[54] Dym H, McKean HP (1972) Fourier Series and Integrals. AcademicPress, New York

[55] Erdos P, Furedi Z (1981) The greatest angle among n points in thed-dimensional Euclidean space. Ann Discrete Math 17:275–283

[56] Ewens, W.J. (1979). Mathematical Population Genetics. Springer, NewYork

[57] Fan R, Lange K (1999) Diffusion process calculations for mutant genesin nonstationary populations. In Statistics in Molecular Biology andGenetics. edited by Seillier-Moiseiwitsch F, The Institute of Math-ematical Statistics and the American Mathematical Society, Provi-dence, RI, pp 38-55

[58] Fan R, Lange K, Pena EA (1999) Applications of a formula for thevariance function of a stochastic process. Stat Prob Letters 43:123–130

[59] Feller W (1968) An Introduction to Probability Theory and Its Appli-cations, Vol 1, 3rd ed. Wiley, New York

[60] Feller W (1971) An Introduction to Probability Theory and Its Appli-cations, Vol 2, 2nd ed. Wiley, New York

[61] Ferguson TS (1996) A Course in Large Sample Theory. Chapman &Hall, London

References 419

[62] Fix E, Neyman J (1951) A simple stochastic model of recovery, relapse,death and loss of patients. Hum Biol 23:205–241

[63] Flanders H (2001) From Ford to Faa. Amer Math Monthly 108:559–561

[64] Flatto L, Konheim AG (1962) The random division of an interval andthe random covering of a circle. SIAM Review 4:211–222

[65] Folland GB (1992) Fourier Analysis and its Applications. Wadsworthand Brooks/Cole, Pacific Grove, CA

[66] Friedlen DM (1990) More socks in the laundry. Problem E 3265. AmerMath Monthly 97:242–244

[67] Galambos J, Simonelli I (1996) Bonferroni-type Inequalities with Ap-plications. Springer, New York

[68] Gani J, Saunders IW (1977) Fitting a model to the growth of yeastcolonies. Biometrics 33:113–120

[69] Gelfand AE, Smith AFM (1990) Sampling-based approaches to calcu-lating marginal densities. J Amer Stat Assoc 85:398–409

[70] Gelman A, Carlin JB, Stern HS, Rubin DB (1995) Bayesian DataAnalysis. Chapman & Hall, London

[71] Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributionsand the Bayesian restoration of images. IEEE Trans Pattern AnalMachine Intell 6:721–741

[72] Geman S, McClure D (1985) Bayesian image analysis: An applica-tion to single photon emission tomography. Proceedings of the Statisti-cal Computing Section, American Statistical Association, Washington,DC, pp 12–18

[73] Gilks WR, Richardson S, Spiegelhalter DJ (editors) (1996) MarkovChain Monte Carlo in Practice. Chapman & Hall, London

[74] Gillespie DT (1977) Exact stochastic simulation of coupled chemicalreactions. J Phys Chem 81:2340–2361

[75] Gillespie DT (2001) Approximate accelerated stochastic simulation ofchemically reacting systems. J Chem Phys 115:1716–1733

[76] Goradia TM (1991) Stochastic Models for Human Gene Mapping.Ph.D. Thesis, Division of Applied Sciences, Harvard University

[77] Goradia TM, Lange K (1988) Applications of coding theory to thedesign of somatic cell hybrid panels. Math Biosci 91:201–219

420 References

[78] Graham RL, Knuth DE, Patashnik O (1988) Concrete Mathematics:A Foundation for Computer Science. Addison-Wesley, Reading, MA

[79] Green P (1990) Bayesian reconstruction for emission tomography datausing a modified EM algorithm. IEEE Trans Med Imaging 9:84–94

[80] Grimmett GR, Stirzaker DR (2001) Probability and Random Processes,3rd ed. Oxford University Press, Oxford

[81] Guttorp P (1991) Statistical Inference for Branching Processes. Wiley,New York

[82] Gzyl H, Palacios JL (1997) The Weierstrass approximation theoremand large deviations. Amer Math Monthly 104:650–653

[83] Hardy GH, Wright EM (1960) An Introduction to the Theory of Num-bers, 4th ed. Clarendon Press, Oxford

[84] Harris TE (1989) The Theory of Branching Processes. Dover, NewYork

[85] Hastings WK (1970) Monte Carlo sampling methods using Markovchains and their applications. Biometrika 57:97–109

[86] Henrici P (1979) Fast Fourier transform methods in computationalcomplex analysis. SIAM Review 21:481–527

[87] Henrici P (1982) Essentials of Numerical Analysis with Pocket Calcu-lator Demonstrations. Wiley, New York

[88] Herman GT (1980) Image Reconstruction from Projections: The Fun-damentals of Computerized Tomography. Springer, New York

[89] Hestenes MR (1981) Optimization Theory: The Finite DimensionalCase. Robert E Krieger Publishing, Huntington, NY

[90] Hewitt E, Stromberg K (1965) Real and Abstract Analysis. Springer,New York

[91] Higham DJ, (2008) Modeling and simulating chemical reactions. SIAMReview 50:347–368

[92] Higham NJ (2009) The scaling and squaring method for matrix expo-nentiation. SIAM Review 51:747–764

[93] Hille E (1959) Analytic Function Theory, Vol 1. Blaisdell, New York

[94] Hirsch MW, Smale S (1974) Differential Equations, Dynamical Sys-tems, and Linear Algebra. Academic Press, New York

References 421

[95] Hochstadt H (1986) The Functions of Mathematical Physics. Dover,New York

[96] Hoel PG, Port SC, Stone CJ (1971) Introduction to Probability Theory.Houghton Mifflin, Boston

[97] Hoffman K (1975) Analysis in Euclidean Space. Prentice-Hall, Engle-wood Cliffs, NJ

[98] Hoppe FM (2008) Faa di Bruno’s formula and the distribution of ran-dom partitions in population genetics and physics. Theor Pop Biol73:543–551

[99] Horn RA, Johnson CR (1991) Topics in Matrix Analysis. CambridgeUniversity Press, Cambridge

[100] Hwang JT (1982) Improving on standard estimators in discrete expo-nential families with applications to the Poisson and negative binomialcases. Ann Statist 10:857–867

[101] Isaacson E, Keller HB (1966) Analysis of Numerical Methods. Wiley,New York

[102] Jacquez JA, Simon CP, Koopman JS (1991) The reproduction num-ber in deterministic models of contagious diseases. Comments TheorBiol 2:159–209

[103] Jagers P (1975) Branching Processes with Biological Applications.Wiley, New York

[104] Jones GA, Jones JM (1998) Elementary Number Theory. Springer,New York

[105] Kac M (1959) Statistical Independence in Probability, Analysis andNumber Theory. Mathematical Association of America, Washington,DC

[106] Karlin S, Taylor HM (1975) A First Course in Stochastic Processes,2nd ed. Academic Press, New York

[107] Karlin S, Taylor HM (1981) A Second Course in Stochastic Processes.Academic Press, New York

[108] Katz B (1966) Nerve, Muscle, and Synapse. McGraw-Hill, New York

[109] Keener JP (1993), The Perron-Frobenius theorem and the ranking offootball teams, SIAM Review, 35:80–93

[110] Keiding N (1991) Age-specific incidence and prevalence: a statisticalperspective. J Royal Stat Soc Series A 154:371–412

422 References

[111] Kelly FP (1979) Reversibility and Stochastic Networks. Wiley, NewYork

[112] Kennedy WJ Jr, Gentle JE (1980) Statistical Computing. MarcelDekker, New York

[113] Kimura M (1980) A simple method for estimating evolutionary ratesof base substitutions through comparative studies of nucleotide se-quences. J Mol Evol 16:111–120

[114] Kingman JFC (1993) Poisson Processes. Oxford University Press,Oxford

[115] Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simu-lated annealing. Science 220:671–680

[116] Klain DA, Rota G-C (1997) Introduction to Geometric Probability.Cambridge University Press, Cambridge

[117] Korner TW (1988) Fourier Analysis. Cambridge University Press,Cambridge

[118] Lamperti J (1977) Stochastic Processes. A Survey of the Mathemat-ical Theory. Springer, New York

[119] Lando SK (2003) Lectures on Generating Functions. American Math-ematical Society, Providence, RI

[120] Lange K (1982) Calculation of the equilibrium distribution for a dele-terious gene by the finite Fourier transform. Biometrics 38:79–86

[121] Lange K (1995) A gradient algorithm locally equivalent to the EMalgorithm. J Roy Stat Soc B 57:425–437

[122] Lange K (1999) Numerical Analysis for Statisticians. Springer, NewYork

[123] Lange K (2002) Mathematical and Statistical Methods for GeneticAnalysis, 2nd ed. Springer, New York

[124] Lange K, Boehnke M (1982) How many polymorphic genes will ittake to span the human genome? Amer J Hum Genet 34:842–845

[125] Lange K, Carson R (1984) EM reconstruction algorithms for emissionand transmission tomography. J Computer Assist Tomography 8:306–316

[126] Lange K, Fessler JA (1995) Globally convergent algorithms for max-imum a posteriori transmission tomography. IEEE Trans Image Pro-cessing 4:1430–1438

References 423

[127] Lange K, Gladstien K (1980) Further characterization of the long-runpopulation distribution of a deleterious gene. Theor Pop Bio 18:31-43

[128] Lange K, Hunter DR, Yang I (2000) Optimization transfer using sur-rogate objective functions (with discussion). J Comput Graph Statist9:1–59

[129] Lawler GF (1995) Introduction to Stochastic Processes. Chapman &Hall, London

[130] Lawler GF, Coyle LN (1999) Lectures on Contemporary Probability.American Mathematical Society, Providence, RI

[131] Lazzeroni LC, Lange K (1997) Markov chains for Monte Carlo testsof genetic equilibrium in multidimensional contingency tables. AnnStatist 25:138–168

[132] Lehmann EL (2004) Elements of Large-Sample Theory, Springer,New York

[133] Li W-H, Graur D (1991) Fundamentals of Molecular Evolution. Sin-auer Associates, Sunderland, MA

[134] Liggett TM (1985) Interacting Particle Systems. Springer, New York

[135] Lighthill MJ (1958) An Introduction to Fourier Analysis and Gener-alized Functions. Cambridge University Press, Cambridge

[136] Lindvall T (1992) Lectures on the Coupling Method. Wiley, New York

[137] Liu JS (1996) Metropolized independent sampling with comparisonsto rejection sampling and importance sampling. Stat Comput 6:113–119

[138] Lotka AJ (1931) Population analysis—the extinction of families I. JWash Acad Sci 21:377–380

[139] Lozansky E, Rousseau C (1996) Winning Solutions. Springer, NewYork

[140] Luenberger DG (1984) Linear and Nonlinear Programming, 2nd ed.Addison-Wesley, Reading, MA

[141] McLeod RM (1980) The Generalized Riemann Integral. Mathemati-cal Association of America, Washington, DC

[142] Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E (1953)Equations of state calculations by fast computing machines. J ChemPhysics 21:1087–1092

424 References

[143] Minin VN, Suchard MA (2008) Counting labeled transitions incontinuous-time Markov models of evolution. Math Biol 56:391–412

[144] Mitzenmacher M, Upfal E (2005) Probability and Computing: Ran-domized Algorithms and Probabilistic Analysis. Cambridge UniversityPress, Cambridge

[145] Moler C, Van Loan C (1978) Nineteen dubious ways to compute theexponential of a matrix. SIAM Review 20:801–836

[146] Muldoon ME, Ungar AA (1996) Beyond sin and cos. MathematicsMagazine 69:3–14

[147] Murray JD (1984) Asymptotic Analysis. Springer, New York

[148] Nedelman J, Wallenuis T (1986) Bernoulli trials, Poisson trials, sur-prising variances, and Jensen’s inequality. Amer Statistician 40:286–289

[149] Neuts MF (1995) Matrix-Geometric Solutions in Stochastic Models:An Algorithmic Approach. Dover, New York

[150] Newman DJ (1980) Simple analytic proof of the prime number the-orem. Amer Math Monthly 87:693–696

[151] Niven I, Zuckerman HS, Montgomery HL (1991) An Introduction tothe Theory of Numbers. Wiley, New York

[152] Norris JR (1997) Markov Chains. Cambridge University Press, Cam-bridge

[153] Paige CC, Styan GPH, Wachter PG (1975) Computation of the sta-tionary distribution of a Markov chain. J Stat Comput Simul 4:173–186

[154] Perelson AS, Macken CA (1985) Branching Processes Applied to CellSurface Aggregation Phenomena. Springer, Berlin

[155] Peressini AL, Sullivan FE, Uhl JJ Jr (1988) The Mathematics ofNonlinear Programming. Springer, New York

[156] Phillips AN (1996) Reduction of HIV concentration during acuteinfection: Independence from a specific immune response. Science271:497–499

[157] Pinsker IS, Kipnis V, Grechanovsky E (1986) A recursive formulafor the probability of occurrence of at least m out of N events. AmerStatistician 40:275–276

[158] Pittenger AO (1985) The logarithmic mean in n dimensions. AmerMath Monthly 92:99–104

References 425

[159] Pollard JH (1973) Mathematical Models for the Growth of HumanPopulations. Cambridge University Press, Cambridge

[160] Port SC, Stone CJ (1978) Brownian Motion and Classical PotentialTheory. Academic Press, New York

[161] Post E (1930) Generalized differentiation. Trans Amer Math Soc32:723–781

[162] Powell MJD (1981) Approximation Theory and Methods. CambridgeUniversity Press, Cambridge

[163] Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Nu-merical Recipes in Fortran: The Art of Scientific Computing, 2nd ed.Cambridge University Press, Cambridge

[164] Rao CR (1973) Linear Statistical Inference and Its Applications, 2nded. Wiley, New York

[165] Reed TE, Neil JV (1959) Huntington’s chorea in Michigan. Amer JHum Genet 11:107–136

[166] Renyi A (1970) Probability Theory. North-Holland, Amsterdam

[167] Roman S (1984) The Umbral Calculus. Academic Press, New York

[168] Roman S (1997) Introduction to Coding and Information Theory.Academic Press, New York

[169] Rosenthal JS (1995) Convergence rates for Markov chains. SIAM Re-view 37:387–405

[170] Ross SM (1996) Stochastic Processes, 2nd ed. Wiley, New York

[171] Royden HL (1968) Real Analysis, 2nd ed. Macmillan, London

[172] Rubinow SI (1975) Introduction to Mathematical Biology. Wiley, NewYork

[173] Rudin W (1964) Principles of Mathematical Analysis, 2nd ed.McGraw-Hill, New York

[174] Rudin W (1973) Functional Analysis. McGraw-Hill, New York

[175] Rushton AR (1976) Quantitative analysis of human chromosome seg-regation in man-mouse somatic cell hybrids. Cytogenetics Cell Genet17:243–253

[176] Schadt EE, Lange K (2002) Codon and rate variation models inmolecular phylogeny. Mol Biol Evol 19:1534–1549

426 References

[177] Sedgewick R (1988) Algorithms, 2nd ed. Addison-Wesley, Reading,MA

[178] Sehl ME, Alexseyenko AV, Lange KL (2009) Accurate stochasticsimulation via the step anticipation (SAL) algorithm. J Comp Biol16:1195–1208

[179] Seneta E (1973) Non-negative Matrices: An Introduction to Theoryand Applications. Wiley, New York

[180] Serre D (2002) Matrices: Theory and Applications. Springer, NewYork

[181] Severini TA (2005) Elements of Distribution Theory. Cambridge Uni-versity Press, Cambridge

[182] Shargel BH, D’Orsogna MR, Chou T (2010) Arrival times in a zero-range process with injection and decay. J Phys A: Math Theor (inpress)

[183] Silver EA, Costa D (1997) A property of symmetric distributions anda related order statistic result. Amer Statistician 51:32–33

[184] Solomon H (1978) Geometric Probability. SIAM, Philadelphia

[185] Spivak M (1965) Calculus on Manifolds. Benjamin/Cummings Pub-lishing Co., Menlo Park, CA

[186] Steele JM (1997) Probability Theory and Combinatorial Optimiza-tion. SIAM, Philadelphia

[187] Stein C (1986) Approximate Computation of Expectations. Instituteof Mathematical Statistics, Hayward, CA

[188] Stein EM, Shakarchi R (2003) Complex Analysis. Princeton Univer-sity Press, Princeton, NJ

[189] Stein EM, Shakarchi R (2005) Real Analysis: Measure Theory, Inte-gration, and Hilbert Spaces. Princeton University Press, Princeton, NJ

[190] Steutel FW (1985) Poisson processes and a modified Bessel integral.SIAM Review 27:73–77

[191] Stewart WJ (1994) Introduction to the Numerical Solution of MarkovChains. Princeton University Press, Princeton, NJ

[192] Strichartz R (2000) Evaluating integrals using self-similarity. AmerMath Monthly 107::316–326

References 427

[193] Sumita U, Igaki N (1997) Necessary and sufficient conditions forglobal convergence of block Gauss-Seidel iteration algorithm appliedto Markov chains. J Operations Research 40:283–293

[194] Tanner MA (1993) Tools for Statistical Inference: Methods for Ex-ploration of Posterior Distributions and Likelihood Functions, 2nd ed.Springer, New York

[195] Tanny S (1973) A probabilistic interpretation of Eulerian numbers.Duke Math J 40:717–722

[196] Taylor HM, Karlin S (1984) An Introduction to Stochastic Modeling.Academic Press, Orlando, FL

[197] Tenenbaum G, France MM (2000) The Prime Numbers and TheirDistribution. translated by Spain PG, American Mathematical Society,Providence, RI

[198] The Huntington’s Disease Collaborative Research Group (1993). Anovel gene containing a trinucleotide repeat that is expanded and un-stable on Huntington’s disease chromosomes. Cell 72:971–983

[199] Tuckwell HC (1989) Stochastic Processes in the Neurosciences. SIAM,Philadelphia

[200] van den Driessche P, Watmough J (2002) Reproduction numbers andsub-threshold endemic equilibria for compartmental models of diseasetransmission. Math Biosciences 180:29–48

[201] Waterman MS (1995) Introduction to Computational Biology: Maps,Sequences, and Genomes. Chapman & Hall, London

[202] Weiss M, Green H (1967) Human-mouse hybrid cell lines containingpartial complements of human chromosomes and functioning humangenes. Proc Natl Acad Sci USA 58:1104–1111

[203] Whitmore GA, Seshadri V (1987) A heuristic derivation of the inverseGaussian distribution. Amer Statistician 41:280–281

[204] Wilf HS (1978) Mathematics for the Physical Sciences. Dover, NewYork

[205] Wilf HS (1985) Some examples of combinatorial averaging. AmerMath Monthly 92:250–260

[206] Wilf HS (1986) Algorithms and Complexity. Prentice-Hall, New York

[207] Wilf HS (1990) generatingfunctionology. Academic Press, San Diego

428 References

[208] Williams D (1991) Probability with Martingales. Cambridge Univer-sity Press, Cambridge

[209] Williams D (2001) Weighing the Odds: A Course in Probability andStatistics. Cambridge University Press, Cambridge

[210] Yee PL, Vyborny R (2000) The Integral: An Easy Approach afterKurzweil and Henstock. Cambridge University Press, Cambridge

[211] Zagier D (1997) Newman’s short proof of the prime number theorem.Amer Math Monthly 104:705–708

Index

Abel’s summation formula, 383Absorbing state, 165Acceptance function, 184Adapted random variable, 255Aperiodicity, 152Arithmetic function, 375

periodic, 385Arithmetic-geometric mean inequal-

ity, 58Asymptotic expansions, 298

incomplete gamma function,303

Laplace’s method, 304–308,410–411

order statistic moments, 305Poincare’s definition, 303Stieltjes function, 318Stirling’s formula, 306Taylor expansions, 299

Asymptotic functions, 299examples, 318–320

Azuma-Hoeffding bound, 264Azuma-Hoeffding theorem, 260

Backtracking, 109

Backward equations, 189Balance equation, 191Barker’s function, 184Bayes’ rule, 15Bell numbers, 77, 119Bernoulli functions, 308–310, 408–

410Bernoulli numbers, 409

Euler-Maclaurin formula, in,309

Bernoulli polynomials, 308–310, 321,408–410

Bernoulli-Laplace model, 175Bernstein polynomial, 67, 72Bessel process, 275, 284Beta distribution, 12, 90, 179

asymptotics, 308mean, 39

Beta-binomial distribution, 29, 47Biggest random gap, 358Binary expansions, 316Binomial distribution, 12, 51, 178

factorial moments, 33Biorthogonality, 164Bipartite graph, 155

K. Lange, Applied Probability,DOI 10.1007/978-1-4419-7165-4, © Springer Science+Business Media, LLC 2010

430 Index

Birthday problem, 305, 357Block matrix decomposition, 166Bonferroni inequality, 82Borel sets, 3Borel-Cantelli lemma, 5, 324

partial converse of, 20Bradley-Terry ranking model, 64Branching process, 217, 340

convergence, 254criticality, 229irreducible, 229martingale, 251multitype, 229–231

Brownian motion, 270, 272–275Buffon needle problem, 28

Campbell’s moment formulas, 139–142

Cancer models, 167, 243–245Cantelli’s inequality, 71Catalan numbers, 84–85

asymptotics, 308Cauchy distribution, 17Cauchy-Schwarz inequality, 66, 256Cell division, 218Central limit theorem, 316Change of variables formula, 16Chapman-Kolmogorov relation, 191Characteristic function, 6

example of oscillating, 35table of common, 12

Chebyshev’s bound, 268Chebyshev’s inequality, 67Chen’s lemma, 41Chen-Stein method, 355

proof, 363–368Chernoff’s bound, 67, 72Chi-square distribution, 47, 51Cholesky decomposition, 19Circuit model, 196Circulant matrix, 351Cofinite subset, 20Coin tossing, waiting time, 352Compact set, 192Composition chain, 213

Conditional probability, 6Connected graph, 155Convergence

almost sure, 314, 324in distribution, 315, 324in probability, 314, 324

Convex functions, 56–60minimization, 61–63

Convex set, 56Convolution

finite Fourier transform, 401integral equation, 336

Coupling, 158–163applications, 178–179independence sampler, 171

Covariance, see Variance

Densityas likelihood, 13conditional, 15marginal, 15table of common densities, 12

Detailed balance, 153, 192Hasting-Metropolis algorithm,

in, 169Diagonally dominant matrix, 337,

353Differentiable functions, 57Differential, see JacobianDiffusion process, 270–272

first passage, 282moments, 280numerical method, 343–347

Dirichlet distribution, 44as sum of gammas, 52variance and covariance, 52

Dirichlet product, 379Distribution, 9

and symmetric densities, 22continuous, 10convolution of, 13discrete, 9marginal, 15of a random vector, 14of a transformation, 16

Index 431

table of common, 12DNA sequence analysis, 115Doob’s martingale, 250

Ehrenfest diffusion, 156, 180, 211Eigenvalues and eigenvectors, 163–

165, 172, 207Epidemics, 219Equilibrium distribution, 152, 190

existence of, 160Ergodic theorem, 153ESP example, 50Euclidean norm, 57Euler’s constant, 88, 310Euler’s totient function, 81, 377Euler-Maclaurin formula, 308–311Eulerian numbers, 120, 324Ewens’ sampling distribution, 91Exchangeable random variable, 80Expectation, 4

and subadditivity, 114–117conditional, 6, 29–31

sum of i.i.d. random vari-ables, 7

differentiation and, 5of a random vector, 14

Exponential distribution, 12bilateral, 49convolution of gammas, 32lack of memory, 129

Exponential integral, 302Extinction, 221Extinction probability, 217, 224

geometric distribution, 223

Faa di Bruno’s formula, 91Family name survival, 219Family planning model, 31, 135Fast Fourier transform, 402–403

applications, 331–335Fatou’s lemma, 4Fejer’s theorem, 406Fibonacci numbers, 77, 119

asymptotics, 312Filter, 248

Finite Fourier transformcomputing, see Fast Fourier

transformdefinition, 401examples, 350–353inversion, 401properties, 402

Finnish population growth, 348Flux, 271Formation of polymers, 238Forward equations, 200Four-color theorem, 184Fourier coefficients, 320–321, 407Fourier inversion, see Inversion for-

mulaFourier series, 332, 406

Bernoulli polynomials, 408pointwise convergence, 407

Fourier transformdefinition, 403function pairs, table of, 404inversion, 405Riemann-Lebesgue lemma, 405

Fractional linear transformation,236

Fubini’s theorem, 9Fundamental theorem of arithmetic,

374, 397

Gambler’s ruin, 258Gamma distribution, 12, 49, 91

as convolution, 32characteristic function, 32inverse, 38

Gamma function, 60asymptotic behavior, 306

Gauss-Seidel algorithm, 329–331block version, 331

Gaussian, see Normal distributionGaussian elimination, 328Generalized hyperbolic functions,

145Generating function, see Progeny

generating functioncoin toss wait time, 352

432 Index

convolution, 332jump counting, 336

Genetic drift, see Wright-Fishermodel

Geometric distribution, 12Geometric progeny, 220Gibbs prior, 133Gibbs sampling, 170Gillespie’s algorithm, 340Graph coloring, 108–112Group homomorphism, 236

Holder’s inequality, 69Hadamard product, 23Hamming distance, 362Harmonic series, 310Hastings-Metropolis algorithm, 168–

171aperiodicity, 184Gibbs sampler, 170independence sampler, 169

convergence, 171–172random walk sampling, 169

Heron’s formula, 70Hessian matrix, 58Hitting probability, 165

matrix decomposition, 166Hitting time, 165–167

expectation, 166HIV

new cases of AIDS, 138viral reproduction, 231

Huffman bit string, 106Huffman coding, 106–108

string truncation, 108vowel tree, 106

Huntington’s disease, 227Hurwitz’s zeta function, 385Hyperbolic trigonometric functions,

146Hypergeometric distribution, 7, 178

Immigration, 225–229Importance ratio, 169, 171Inclusion-exclusion formula, 78–83

Incomplete gamma function, 303Independence, 8Independence sampler, 169

convergence, 171–172Indicator random variable, 4

sums of, 25Inequality, 66–69

arithmetic-geometric mean, 58Cantelli’s, 71Cauchy-Schwarz, 66Chebyshev’s, 67Holder’s, 69Jensen’s, 68Markov’s, 66Minkowski’s, 73Schlomilch’s, 68

Infinitesimal generator, 190Infinitesimal mean, 270Infinitesimal transition matrix, see

Infinitesimal generatorInfinitesimal transition probabil-

ities, see Transition in-tensity

Infinitesimal variance, 270Inner product in Rn, 112Integrable function, 404Integration by parts, 302–303Intensity, 188Intensity leaping, 339–343Inversion formula, 11Involution, 96Irreducibility, 153Ising model, 170

Jacobi algorithm, 329–331block version, 331

Jacobian, 16Jensen’s inequality, 68

Kendall’s birth-death-immigrationprocess, 200–206, 215, 276,342

Kimura’s model of DNA substitu-tion, 193, 199, 210, 339

Kirchhoff’s laws, 197

Index 433

Kolmogorov’s circulation criterion,154

Kolmogorov’s forward equation, 272,344

Laplace transform, 34, 37, 254, 320,337, 388

Laplace’s method, 304–308, 320,410–411

Law of rare events, 360Least absolute deviation, 65Left-to-right maximum, 88Liapunov function, 191Light bulb problem, 176Likelihood, 13Lindeberg’s condition, 316Liouville’s arithmetic function, 393Lipschitz condition, 262Logarithmic distribution, 301Logistic distribution, 21Longest common subsequence, 115,

263Longest increasing subsequence, 93Lotka’s surname data, 223

Mobius function, 236, 380Menage problem, 356Marking and coloring, 138–139Markov chain, 151–154

continuous timeequilibrium distribution, 328

counting jumps, 336–339ergodic assumptions, 152, 174intensity leaping, 339stationary distribution, 152

finite state, 160transition matrix, 151

Markov chain Monte Carlo, 168–172

Gibbs sampling, 170Hastings-Metropolis algorithm,

168–170simulated annealing, 173–174

Markov property, 20Markov’s inequality, 66, 261

Martingale, 247–251convergence, 251large deviations, 260

Master equations, 341Matrix exponentials, 197–199Maximum likelihood estimates, 13,

65MCMC, see Markov chain Monte

CarloMedian finding, 118Minkowski’s triangle inequality, 73MM algorithm, 63–66Moment, 11

asymptotics, 305, 318factorial, 33generating function, 11polynomials on a sphere, 43

Moment inequalities, 66–69Monotone convergence theorem,

4Moran’s genetics model, 353Multinomial sampling, 134Mutant gene survival, 219

Negative binomial distribution, 31,51, 179

Negative multinomial distribution,148

Neuron firing, see Ornstein-UhlenbeckNeutron chain reaction, 218Newton’s method, 237Normal distribution, 12

affine transforms of, 18characteristic function, 31characterization of, 36distribution function

asymptotic expansion, 303multivariate, 17

maximum likelihood, 62NP-completeness, 173Null recurrence, 158Number-theoretic density, 3

O-notation, see Order relations

434 Index

Optional stopping theorem, 255,257

Order relations, 298–299examples, 318

Order statistics, 83–84distribution function of, 83from an exponential sample,

130moments, 305–306

Ornstein-Uhlenbeck process, 277,290

Oxygen in hemoglobin, 193

Pareto distribution, 21Pascal’s triangle, 76Pattern matching, 26Permutation cycles, 87, 317Permutation inversions, 317Pigeonhole principle, 93–94Planar graph, 109Point sets with acute angles, 112–

113Poisson distribution, 12, 124, 179

birthday problem, 305factorial moments, 33

Poisson process, 124from given intensity function,

126inhomogeneous, 202–206, 341one-dimensional, 127restriction, 137superposition, 137transformations, 136–138transformed expectations, 137waiting time, 127waiting time paradox, 130

Polar coordinates, 138Polya’s model, see Urn modelPolynomial

multiplication, 332Polynomial on Sn−1, 44Positive definite quadratic forms,

58Power method, 328–331Powers of integers, sum of, 322

Prime integer, 374Prime number theorem, 386Probabilistic embedding, 89Probability measure, 2Probability space, 2Product measure formula, 42Progeny generating function, 218Proposal distribution, 168

QR decomposition, 18Quick sort, 104–106

average-case performance, 105median finding, 118promotion process, 104

Random circlesin R2, 148in R3, 149

Random deviates, generatinglogistic, 21Pareto, 21Weibull, 21

Random permutation, 26and card shuffling, 158and Poisson distribution, 80and Sperner’s theorem, 113cycles in, 87fixed points, 46, 80successive, 259

Random sums, 33, 49Random variables

correlated, coupling, 158definition, 3measurability of, 3

Random walk, 85as a branching process, 236coupling, 178equilibrium, 156, 330eventual return, 182first return, 97hitting probability, 183, 210,

267hitting time, 183, 210, 267martingales, 258, 267on a graph, 155, 176

Index 435

renewal theory, 158sampling, 169self avoiding, 121

Reaction channel, 340Recessive gene equilibrium, 290Recurrence relations, 31

average-case quick sort, 105Bernoulli numbers, 409Bernoulli polynomials, 408family planning model, 47

Relatively prime integers, 374Renewal equation, 333–335Renewal process, 157Repeated uniform sampling, see

Uniform distributionResidual, 66Reversion of sequence, 402Riemann’s zeta function, 70, 374Riemann-Lebesgue lemma, 404Right-tail probability, 36Runs in coin tossing, 323, 335

Sampling without replacement, 27Scheffe’s lemma, 20Schlomilch’s inequality, 68, 256Schrodinger’s method, see Multi-

nomial samplingSelf-adjointness condition, 182Self-avoiding random walk, 121Sequential testing, 259Simulated annealing, 173–174Skorokhod representation theorem,

315Smoothing, 350Socks in the laundry, 89–91

asymptotics, 307Somatic cell hybrid panels, 360Sperner’s theorem, 113–114Splitting entry, 104Squares of integers, sum of, 322Starlight intensity, 141Stationary distribution, see Equi-

librium distributionStein’s lemma, 40Stieltjes function

asymptotic expansion, 318Stirling numbers, 86–89

first kind, 87second kind, 86

Stirling’s formula, 94, 306Euler-Maclaurin formula, de-

rived from, 310Stochastic domination, 178–179Stochastic simulation, 339–343Stone-Weierstrass theorem, 407Stopping time, 255Stretching of sequence, 402Strong law of large numbers, 253Strong stationary time, 162Subadditive sequence, 114Sudoku puzzle, 185Summation by parts, 301, 383Superadditive sequence, 114Superposition process, 90Surface integral, 42Surrogate function, 64Symmetric difference, 20

Tauberian lemma, 412Taylor expansion, 299–300Temperature, 173Top-in shuffling, 162Total variation norm, 159

binomial-Poisson, 180Chen-Stein method, 355–356Ehrenfest process, 211stopping time, 162

Tower property, 8, 248, 250Transient state, 165Transition intensity, 188Transition probabilities, 188Translation of sequence, 401Transmission tomography, 131–134

loglikelihood, 146Traveling salesman problem, 116,

173, 263Triangle

in random graph, 99random points on, 100

Turing’s morphogen model, 353

436 Index

Uniform distribution, 12continuous, 3discrete, 2on surfaces, 43products of, 30sums of, 34

Uniform process, 89Uniformization, 199Urn model, 89–91

Variance, 12as inner product, 12of a product of independent

random variables, 22von Mangoldt function, 379Von Mises distribution, 319

Waiting timeinsurance claim, 149paradox, 130train departures, 149

Wald’s identity, 256Watson’s lemma, 307–308, 412Weibull distribution, 21Weierstrass’s approximation the-

orem, 67and coupling, 159

Weighted mean, 68, 73Wright-Fisher model, 156, 251, 257,

278, 283numerical solution, 347

X-linked disease, 239

Yeast cell reproduction, 239

Zipf’s probability measure, 374