Historical and Bibliographical Notes

Chapter 1: Introduction

The history of probability theory up to the time of Laplace is described by Todhunter[96]. The period from Laplace to the end of the nineteenth century is covered byGnedenko and Sheinin in [53]. Stigler [95] provides very detailed exposition ofthe history of probability theory and mathematical statistics up to 1900. Maistrov[66] discusses the history of probability theory from the beginning to the thirtiesof the twentieth century. There is a brief survey of the history of probability theoryin Gnedenko [32]. For the origin of much of the terminology of the subject seeAleksandrova [2].

For the basic concepts see Kolmogorov [51], Gnedenko [32], Borovkov [12],Gnedenko and Khinchin [33], A. M. and I. M. Yaglom [97], Prokhorov and Rozanov[77], handbook [54], Feller [30, 31], Neyman [70], Loeve [64], and Doob [22].We also mention [38, 90] and [91] which contain a large number of problems onprobability theory.

In putting this text together, the author has consulted a wide range of sources.We mention particularly the books by Breiman [14], Billingsley [10], Ash [3, 4],Ash and Gardner [5], Durrett [24, 25], and Lamperti [56], which (in the author’sopinion) contain an excellent selection and presentation of material.

The reader can find useful reference material in Great Soviet Encyclopedia, En-cyclopedia of Mathematics [40] and Encyclopedia of Probability Theory and Math-ematical Statistics [78].

The basic journal on probability theory and mathematical statistics in our countryis Teoriya Veroyatnostei i ee Primeneniya published since 1956 (translated as Theoryof Probability and its Applications).

Referativny Zhournal published in Moscow as well as Mathematical Reviews andZentralblatt fur Mathematik contain abstracts of current papers on probability andmathematical statistics from all over the world.

© Springer Science+Business Media New York 2016A.N. Shiryaev, Probability-1, Graduate Textsin Mathematics 95, DOI 10.1007/978-0-387-72206-1

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462 Historical and Bibliographical Notes

A useful source for many applications, where statistical tables are needed, isTablicy Matematicheskoy Statistiki (Tables of Mathematical Statistics) by Bol’shevand Smirnov [11]. Nowadays statistical computations are mostly performed usingcomputer packages.

Chapter 2

Section 1. Concerning the construction of probabilistic models see Kolmogorov [49]and Gnedenko [32]. For further material on problems of distributing objects amongboxes see, e.g., Kolchin, Sevastyanov, and Chistyakov [47].Section 2. For other probabilistic models (in particular, the one-dimensional Isingmodel) that are used in statistical physics, see Isihara [42].Section 3. Bayes’s formula and theorem form the basis for the “Bayesian approach”to mathematical statistics. See, for example, De Groot [20] and Zacks [98].Section 4. A variety of problems about random variables and their probabilisticdescription can be found in Meshalkin [68], Shiryayev [90], Shiryayev, Erlich andYaskov [91], Grimmet and Stirzaker [38].Section 6. For sharper forms of the local and integral theorems, and of Poisson’stheorem, see Borovkov [12] and Prokhorov [75].Section 7. The examples of Bernoulli schemes illustrate some of the basic conceptsand methods of mathematical statistics. For more detailed treatment of mathematicalstatistics see, for example, Lehmann [59] and Lehmann and Romano [60] amongmany others.Section 8. Conditional probability and conditional expectation with respect to a de-composition will help the reader understand the concepts of conditional probabilityand conditional expectation with respect to σ-algebras, which will be introducedlater.Section 9. The ruin problem was considered in essentially the present form byLaplace. See Gnedenko and Sheinin in [53]. Feller [30] contains extensive mate-rial from the same circle of ideas.Section 10. Our presentation essentially follows Feller [30]. The method of proving(10) and (11) is taken from Doherty [21].Section 11. Martingale theory is thoroughly covered in Doob [22]. A different proofof the ballot theorem is given, for instance, in Feller [30].Section 12. There is extensive material on Markov chains in the books by Feller [30],Dynkin [26], Dynkin and Yushkevich [27], Chung [18, 19], Revuz [81], Kemenyand Snell [44], Sarymsakov [84], and Sirazhdinov [93]. The theory of branchingprocesses is discussed by Sevastyanov [85].

Historical and Bibliographical Notes 463

Chapter 2

Section 1. Kolmogorov’s axioms are presented in his book [51].Section 2. Further material on algebras and σ-algebras can be found in, for example,Kolmogorov and Fomin [52], Neveu [69], Breiman [14], and Ash [4].Section 3. For a proof of Caratheodory’s theorem see Loeve [64] or Halmos [39].Sections 4–5. More material on measurable functions is available in Halmos [39].Section 6. See also Kolmogorov and Fomin [51], Halmos [39], and Ash [4]. TheRadon–Nikodym theorem is proved in these books.

Inequality (23) was first stated without proof by Bienayme [8] in 1853 and provedby Chebyshev [16] in 1867. Inequality (21) and the proof given here are due toMarkov [67] (1884). This inequality together with its corollaries (22), (23) is usuallyreferred to as Chebyshev’s inequality. However sometimes inequality (21) is calledMarkov’s inequality, whereas Chebyshev’s name is attributed to inequality (23).Section 7. The definitions of conditional probability and conditional expectationwith respect to a σ-algebra were given by Kolmogorov [51]. For additional materialsee Breiman [14] and Ash [4].Section 8. See also Borovkov [12], Ash [4], Cramer [17], and Gnedenko [32].Section 9. Kolmogorov’s theorem on the existence of a process with given finite-dimensional distributions is in his book [51]. For Ionescu-Tulcea’s theorem see alsoNeveu [69] and Ash [4]. The proof in the text follows [4].Sections 10–11. See also Kolmogorov and Fomin [52], Ash [4], Doob [22], andLoeve [64].Section 12. The theory of characteristic functions is presented in many books. See,for example, Gnedenko [32], Gnedenko and Kolmogorov [34], and Ramachan-dran [79]. Our presentation of the connection between moments and semi-invariantsfollows Leonov and Shiryaev [61].Section 13. See also Ibragimov and Rozanov [41], Breiman [14], Liptser and Shi-ryaev [62], Grimmet and Stirzaker [37], and Lamperti [56].

Chapter 3

Section 1. Detailed investigations of problems on weak convergence of probabilitymeasures are given in Gnedenko and Kolmogorov [34] and Billingsley [9].Section 2. Prokhorov’s theorem appears in his paper [76].Section 3. The monograph [34] by Gnedenko and Kolmogorov studies the limittheorems of probability theory by the method of characteristic functions. See alsoBillingsley [9]. Problem 2 includes both Bernoulli’s law of large numbers andPoisson’s law of large numbers (which assumes that ξ1, ξ2, . . . are independentand take only two values (1 and 0), but in general are differently distributed:Ppξi “ 1q “ pi, Ppξi “ 0q “ 1´ pi, i ≥ 1q.

464 Historical and Bibliographical Notes

Section 4. Here we give the standard proof of the central limit theorem for sumsof independent random variables under the Lindeberg condition. Compare [34]and [72].Section 5. Questions of the validity of the central limit theorem without the hypoth-esis of asymptotic negligibility have already attracted the attention of P. Levy. Adetailed account of the current state of the theory of limit theorems in the nonclas-sical setting is contained in Zolotarev [99]. The statement and proof of Theorem 1were given by Rotar [82].Section 6. The presentation uses material from Gnedenko and Kolmogorov [34],Ash [4], and Petrov [71, 72].Section 7. The Levy–Prohorov metric was introduced in the well-known paper byProhorov [76], to whom the results on metrizability of weak convergence of mea-sures given on metric spaces are also due. Concerning the metric }P ´ P}BL, seeDudley [23] and Pollard [73].Section 8. Theorem 1 is due to Skorokhod. Useful material on the method of a singleprobability space may be found in Borovkov [12] and in Pollard [73].Sections 9–10. A number of books contain a great deal of material touching on thesequestions: Jacod and Shiryaev [43], LeCam [58], Greenwood and Shiryaev [36].Section 11. Petrov [72] contains a lot of material on estimates of the rate of con-vergence in the central limit theorem. The proof of the Berry–Esseen theorem givenhere is contained in Gnedenko and Kolmogorov [34].Section 12. The proof follows Presman [74].Section 13. For additional material on fundamental theorems of mathematical statis-tics, see Breiman [14], Cramer [17], Renyi [80], Billingsley [10], and Borovkov [13].

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[42] A. Isihara. Statistical Physics. Academic Press, New York, 1971.[43] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes.

Springer-Verlag, Berlin–Heidelberg, 1987.[44] J. Kemeny and L. J. Snell. Finite Markov Chains. Van Nostrand, Princeton,

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goodness-of-fit tests. Probability Theory and its Applications, 26, 2 (1981),240–257.

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Keyword Index

Symbolsλ-system, 171Λ (condition), 407π-λ-system, 171π-system, 170

Aabsolute continuity with respect to P,

232Absolute moment, 220, 230Absolutely continuous

distribution function, 190distributions, 189measures, 189probability measures, 232random variables, 207

Algebra, 8, 160generated by a set, 167induced by a decomposition, 8,

168of sets (events), 8, 160, 167smallest, 168trivial, 8

Allocation of objects among cells, 4Almost everywhere (a.e.), 221Almost surely (a.s.), 221Appropriate set of functions, 175Arcsine law, 94, 103Arrangements

with repetitions, 2without repetitions, 3

Asymptotic negligibility, 407Atom, 316

of a decomposition, 8Axioms, 164

BBackward equation, 119

matrix form, 119Ballot Theorem, 108Banach space, 315Basis, orthonormal, 323Bayes

formula, 24theorem, 24

generalized, 272Bernoulli, J., 44

distribution, 189law of large numbers, 46random variable, 32, 45scheme, 28, 34, 44, 54, 69

Bernstein, S. N., 52, 369inequality, 54polynomials, 52proof of Weierstrass theorem, 52

Berry–Esseen theorem, 62, 446Bienayme–Chebyshev inequality, 228Binary expansion, 160Binomial distribution, 14, 15, 189

negative, 189Bochner–Khinchin theorem, 343Bonferroni’s inequalities, 14

© Springer Science+Business Media New York 2016A.N. Shiryaev, Probability-1, Graduate Textsin Mathematics 95, DOI 10.1007/978-0-387-72206-1

471

472 Keyword Index

Borel, E.σ-algebra, 175function, 175, 206inequality, 370sets, 175space, 271

Borel–Cantelli lemma, 308Bose–Einstein, 5Bounded variation, 246Branching process, 117Brownian

bridge, 367, 370motion, 366, 370

construction, 367Buffon’s needle, 266Bunyakovskii, V. Ya., 37

CCanonical

probability space, 299Cantelli, F. P., 452Cantor, G.

diagonal process, 386function, 191

Caratheodory theorem, 185Carleman’s test, 353Cauchy

criterion foralmost sure convergence , 311convergence in mean-p, 314convergence in probability, 313

distribution, 190sequence, 306

Cauchy–Bunyakovskii inequality, 229Central Limit Theorem, 388, 407

Lindeberg condition, 395, 401non-classical conditions for, 406rate of convergence, 446

Cesaro summation, 316Change of variable in integral, 234Chapman, D. G., 118, 300Characteristic function, 331

examples of, 353

inversion formula, 340Marcinkiewicz’s theorem, 344of a set, 31of distribution, 332of random vector, 332Polya’s theorem, 344properties, 332, 334

Charlier, C. V. L., 325Chebyshev, P. L., 388

inequality, 46, 53, 227, 228, 388Classical

distributions, 14method, 11models, 14probability, 11

Closed linear manifold, 328Coin tossing, 1, 14, 31, 83, 159Combinations

with repetitions, 2without repetitions, 3

Combinatorics, 11Compact

relatively, 384sequentially, 385

Complement, 7, 160Complete

function space, 314, 315probability measure, 188probability space, 188

Completion of a probability space, 188Composition, 136Concentration function, 356Conditional distribution

density of, 264existence, 271

Conditional expectation, 75in the wide sense, 320, 330properties, 257with respect toσ-algebra, 255decomposition, 78event, 254, 262random variable, 256, 262set of random variables, 81

Keyword Index 473

Conditional probability, 22, 75, 254regular, 268with respect toσ-algebra, 256decomposition, 76, 254random variable, 77, 256

Conditional variance, 256Confidence interval, 69, 73Consistency of finite-dimensional

distributions, 197, 298Consistent estimator, 70Construction of a process, 297Contiguity of probability measures, 441Continuity theorem, 389Continuous at zero (∅), 186, 199Continuous from above or below, 162Continuous time, 214Convergence

of distributions, 373in general, 375, 376, 381in variation, 432weak, 375, 376

of random elementsin distribution, 425in law, 425in probability, 425with probability one, 425

of random variablesalmost everywhere, 306almost surely, 306, 420in distribution, 306, 392in mean, 306in mean of order p, 306in mean square, 306in probability, 305, 420with probability 1, 306

Convergence in measure, 306Convergence-determining class, 380Convolution, 291Coordinate method, 299Correlation

coefficient, 40, 284maximal, 294

Counting measure, 274

Covariance, 40, 284, 350function, 366matrix, 285

Cumulant, 346Curve

U-shaped, 103Cylinder sets, 178

DDe Moivre, A., 47, 60De Moivre–Laplace integral theorem,

60Decomposition, 8

of Ω, 8countable, 168

of set, 8, 349trivial, 9

Degenerate random variable, 345Delta function, 358Delta, Kronecker, 324Density, 190, 207

n-dimensional, 195normal (Gaussian), 65, 190, 195,

284n-dimensional, 358two-dimensional, 286

of measure with respect to ameasure, 233

Derivative, Radon–Nikodym, 233Determining class, 380De Morgan’s laws, 13, 151, 160Difference of sets, 7, 164Direct product

of σ-algebras, 176of measurable spaces, 176, 183of probability spaces, 28

Dirichlet’s function, 250Discrete

measure, 188random variable, 206time, 214uniform distribution, 189

Discrimination between twohypotheses, 433

474 Keyword Index

Disjoint, 7, 161Distance in variation, 431, 436Distribution

F, 190Bernoulli, 32, 189Beta, 190bilateral exponential, 190binomial, 14, 15, 189Cauchy, 190, 415chi, 293chi-square, 190, 293conditional

regular, 269, 281discrete uniform, 189double exponential, 296entropy of, 49ergodic, 121exponential, 190Gamma, 190geometric, 189hypergeometric, 19

multivariate, 18infinitely divisible, 411initial, 115, 300invariant, 123lognormal, 290multinomial, 18multivariate, 34negative binomial, 189, 205normal (Gaussian), 65, 190

n-dimensional, 358density of, 65, 195semi-invariants, 350

Pascal, 189Poisson, 62, 189polynomial, 18Rayleigh, 293singular, 192stable, 416stationary, 123Student’s, t, 293Student, t, 190uniform, 190Weibull, 295

Distribution function, 32, 185, 206n-dimensional, 194absolutely continuous, 190, 204discrete, 204empirical, 452finite-dimensional, 298generalized, 192of a random vector, 34of function of random variables,

34, 289of sum, 34, 291singular continuous, 204

Dominated convergence, 224Doubling stakes, 89Dynkin’s d-system, 171

EElectric circuit, 30Elementary

events, 1, 164probability theory, Chapter I, 1

Empty set, 7, 164Entropy, 49Ergodic theorem, 121Ergodicity, 121Error

function, 65mean-square, 42

Errors of 1st and 2nd kind, 433Esseen’s inequality, 353Essential supremum, 315Estimation, 69, 287

of success probability, 69Estimator, 41, 70, 287

best (optimal), 83in mean-square, 41, 83, 287, 363

best linear, 41, 320, 330consistent, 70efficient, 70maximum likelihood, 21unbiased, 70, 280

Events, 1, 5, 160, 164certain, 7, 164elementary, 1, 164impossible, 7, 164

Keyword Index 475

independent, 26mutually exclusive, 164

Expectation, 36, 217–219conditional

of function, 83with respect to σ-algebra, 254with respect to decomposition,

78inequalities for, 220, 228properties, 36, 220

Expected (mean) value, 36Exponential distribution, 190Exponential family, 279Exponential random variable, 190, 294Extended random variable, 208Extension of a measure, 186, 197, 301

FFactorization theorem, 277Family

of probability measuresrelatively compact, 384tight, 384

Fatou’s lemma, 223for conditional expectations, 283

Fermi–Dirac, 5Fibonacci numbers, 134, 140Finer decomposition, 80Finite second moment, 318Finite-dimensional distributions, 214,

297First

arrival, 132exit, 126return, 94, 132

Fisher information, 71Formula

Bayes, 24for total probability, 23, 76, 79multiplication of probabilities, 24

Forward equation, 119matrix form, 119

Fourier transform, 332Frequency, 45

Fubini’s Theorem, 235Fundamental sequence, 306

GGalton–Watson model, 117, 145

extinction, 145Gamma

distribution, 190Gauss–Markov

process, 368Gaussian

density, 65, 190, 195, 284, 358multidimensional, 195

distribution, 65, 190, 195, 284multidimensional, 358two-dimensional, 286

distribution function, 61, 65measure, 324random variables, 284, 288, 358random vector, 358

characteristic function of, 358covariance matrix, 361mean-value vector, 361with independent components,

361sequence, 364, 366systems, 357, 365

Generating function, 251exponential, 135of a random variable, 135of a sequence, 135

Geometric distribution, 189Glivenko, V. I., 452Glivenko–Cantelli theorem, 452Goodness-of-fit test, 459Gram determinant, 321Gram–Schmidt process, 322Graph, 115

HHolder inequality, 230Haar functions, 327Hahn decomposition, 432Heine–Borel lemma, 186

476 Keyword Index

Hellinger integral, 431, 435of order α, 435

Helly’s theorem, 385Helly–Bray

lemma, 382theorem, 382

Hermite polynomials, 324normalized, 324

Hilbert space, 319separable, 323

Hypergeometricdistribution, 18

Hypotheses, 24statistical, 433

IImpossible event, 7, 164Increasing sequence of σ-algebras, 184Increments

independent, 366uncorrelated, 111, 366

Independence, 25Independent

algebras of events, 26events, 26, 173

pairwise, 27increments, 366random elements, 215, 438random variables, 34, 42, 44, 53,

65, 77, 116, 117, 147, 215,216, 228, 282, 291–293, 299,304, 311, 318, 333, 353, 356,366, 388, 392, 394, 395, 401,406, 407, 411, 415, 446, 449,452

pairwise, 41systems of events, 173

Indeterminacy, 50Indicator of a set, 31, 42Inequality

Bell, 44Bernstein, 54Berry–Esseen, 62, 400, 446Bessel, 320Bienayme–Chebyshev, 46, 228

Bonferroni, 14Cauchy–Bunyakovskii, 37, 229Cauchy–Schwarz, 37Chebyshev, 46, 228

two-dimensional, 53Esseen, 353for large deviations probability, 68Frechet, 14Gumbel, 14Holder, 230Jensen, 229

for conditional expectations,282

Lyapunov, 229Markov, 462Minkowski, 231Rao–Cramer, 71Schwarz, 37

Infinitely divisible, 411characteristic function, 412

Kolmogorov–Levy–Khinchinrepresentation, 415

distribution, 412random variable, 412

Infinitely many outcomes, 159Initial distribution, 115, 300Integral

Darboux–Young, 244Lebesgue, 217, 218

integration by parts, 245Lebesgue vs Riemann, 241Lebesgue–Stieltjes, 219lower, 244Riemann, 219, 242

lower, 244upper, 244

Riemann–Stieltjes, 219upper, 244

Integral limit theorem, 60Integration

by parts, 245by substitution, 250

Intersection of sets, 7, 164Invariance principle, 405

Keyword Index 477

Ionescu Tulcea, C. T.theorem, 301

Izing model, 20

JJensen’s inequality, 229

for conditional expectations, 282

KKakutani–Hellinger distance, 431, 435Khinchin, A. Ya., 383

Kolmogorov–Levy–Khinchinrepresentation, 415

Levy–Khinchin representation,419

law of large numbers, 383Kolmogorov, A. N.

axioms, 164goodness-of-fit test, 454Kolmogorov–Chapman equation,

118, 300backward, 118forward, 119

Kolmogorov–Levy–Khinchinrepresentation, 415

theoremon existence of process, 298on extension of measures, 197,

200Kronecker delta, 324Kullback information, 440Ky Fan distance, 425

LLevy, P., 382, 420

distance, 382Kolmogorov–Levy–Khinchin

representation, 415Levy–Khinchin representation,

419Levy–Prohorov metric, 420

Laplace, P. S., 60Large deviations, 68Law of errors, 357

Law of large numbers, 44, 47, 388Bernoulli, 47for Markov chains, 124

Lebesgue, H.decomposition, 439derivative, 439dominated convergence theorem,

224integral, 217–219

change of variable, 234integration by parts, 245

measure, 187, 193, 196n-dimensional, 194

Lebesgue–Stieltjesintegral, 219, 235measure, 192, 219, 241, 243

n-dimensional, 220probability measure, 187

Le Cam, L., 450rate of convergence in Poisson’s

theorem, 450Likelihood ratio, 111Limit theorem

integral, 47, 60local, 47, 55

Limits underexpectation sign, 222integral sign, 222

Lindeberg condition, 395, 401Linear manifold, 320, 323

closed, 323Linearly independent, 321, 322Local limit theorem, 54, 55Locally bounded variation, 247Lognormal distribution, 290Lottery, 12, 19Lyapunov, A. M., 399

condition, 399inequality, 229

MMacmillan’s theorem, 51Mann–Wald theorem, 428Marcinkiewicz’s theorem, 344

478 Keyword Index

Markov, A. A., 388chain, 112, 115, 303

homogeneous, 115stationary, 123

process, 300property, 115

strong, 129Martingale, 104

reversed, 106Mathematical foundations, 159Mathematical statistics, 48, 69, 452

fundamental theorems of, 452Matrix

covariance, 285of transition probabilities, 115orthogonal, 285, 321positive semi-definite, 285stochastic, 115

Maximal correlation coefficient, 294Maxwell–Boltzmann, 5Mean

duration of random walk, 83, 90value, 36

vector, 361, 363Mean-square, 41, 363

error, 287Measurable

function, 206set, 187space, 161pC,BpCqq, 182pD,BpDqq, 182pR,BpRqq, 175pRT ,BpRTqq, 180pR8,BpR8qq, 178pRn,BpRnqq, 176pśtPT Ωt,

śb

tPTFtq, 182Measure, 161

σ-finite, 161absolutely continuous, 189, 232atomic, 316complete, 188continuous at ∅, 162countably (σ-) additive, 161counting, 434

discrete, 188, 434dominating, 435finite, 161, 164finitely additive, 160, 166interior, 188Lebesgue, 187Lebesgue–Stieltjes, 192outer, 188probability, 161

Lebesgue–Stieltjes, 187restriction of, 199signed, 232, 432singular, 190Wiener, 202

Measure of scatter, 39Measures

absolutely continuous, 438consistent, 200equivalent, 438orthogonal, 438singular, 438

Median, 43Mercer’s theorem, 369Method

of a single probability space, 425of characteristic functions, 388of moments, 388

Minkowski inequality, 231Model

of experiment with finitely manyoutcomes, 1, 10

of experiment with infinitely manyoutcomes, 159

one-dimensional Izing, 20Moment, 219

absolute, 219mixed, 346

Momentsand semi-invariants, 347factorial, 150method of, 388problem, 350

Carleman’s test, 353Monotone convergence theorem, 222

Keyword Index 479

Monotonic class, 169smallest, 169theorem, 169

Monte Carlo method, 267Multinomial distribution, 18Multiplication formula, 24Multivariate hypergeometric

distribution, 18

NNeedle

Buffon, 266Negative binomial, 189Non-classical conditions, 406Norm, 313Normal

correlation, 363, 369density, 65, 190distribution function, 61, 65

Number ofarrangements, 3bijections, 158combinations, 2derangements, 155functions, 158injections, 158surjections, 158

OObjects

distinguishable, 4indistinguishable, 4

Optimal estimator, 287, 363Order statistics, 296Ordered sample, 2Orthogonal

decomposition, 320, 330matrix, 285, 321random variables, 319system, 319

Orthogonalization, 322Orthonormal system, 319–321, 323,

328Orthonormal systems

Haar functions, 327Hermite polynomials, 324

Poisson–Charlier polynomials,325

Rademacher functions, 326Outcome, 1, 159

PPolya’s theorem, 344Pairwise independence, 27, 41Parallelogram property, 330Parseval’s equation, 323Pascal’s triangle, 3Path, 47, 84, 95, 98Pauli exclusion principle, 5Perpendicular, 320Phase space, 115Poincare theorem, 406Point estimation, 69Point of increase, 191Poisson, D., 325

distribution, 62, 63, 189, 449theorem, 62

rate of convergence, 63, 449Poisson–Charlier polynomials, 325Polynomial distribution, 18Polynomials

Bernstein, 52Hermite, 324Poisson–Charlier, 325

Positive semi-definite, 195, 285, 321,343, 355, 359, 365

Pratt’s lemma, 251Principal value of logarithm, 398Principle

of appropriate sets, 169of inclusion–exclusion, 150

Probabilistic model, 1, 5, 10, 159, 164in the extended sense, 161of a Markov chain, 112

Probabilistic-statisticalexperiment, 276model, 276

Probabilities of 1st and 2nd kind errors,433

480 Keyword Index

Probability, 161, 162, 164classical, 11conditional, 22, 75, 254finitely additive, 161initial, 115measure, 161, 162, 185

complete, 188multiplication, 24of first arrival, 132of first return, 132of outcome, 9of ruin, 83, 88posterior (a posteriori), 25prior (a priori), 24space, 9

universal, 304transition, 115, 300

Probability distribution, 206discrete, 189of a random variable, 32of a random vector, 34of process, 214

Probability space, 9, 164canonical, 299complete, 188

Problembirthday, 11coincidence, 11Euler, 149Galileo’s, 134lucky tickets, 134of ruin, 83on derangements, 155

Processbranching, 117Brownian motion, 366

construction, 367construction of, 297Gauss–Markov, 368Gaussian, 366Markov, 300of creation–annihilation, 117

stochastic, 214with independent increments, 366

Projection, 320Prokhorov, Yu. V., 384

rate of convergence in Poisson’stheorem, 63, 450

theorem on tightness, 384Pseudoinverse, 369Pythagorean property, 330

QQuantile function, 427Queueing theory, 117

RRademacher system, 327Radon–Nikodym

derivative, 233theorem, 233

Randomelements, 212

equivalent in distribution, 426function, 213process, 214, 366

existence of, 298Gauss–Markov, 368Gaussian, 366with continuous time, 214, 366with discrete time, 214with independent increments,

366sequence, 214

existence of, 301vector, 33

Random variables, 31, 206absolutely continuous, 207Bernoulli, 32binomial, 32complex, 213continuous, 207degenerate, 345discrete, 206exponential, 190, 294extended, 208Gaussian, 284, 292

Keyword Index 481

independent, 34, 42, 44, 53, 65, 77,116, 117, 147, 228, 282,291–293, 299, 304, 311, 318,333, 353, 356, 366, 388, 394,395, 401, 406, 407, 411, 415,446, 449, 452

measurablerelative to a decomposition, 79

normally distributed, 284orthogonal, 319orthonormal, 319random number of, 117simple, 206uncorrelated, 284

Random vectors, 34, 213Gaussian, 358

Random walk, 15, 83, 94symmetric, 94

Rao–Cramer inequality, 71Rate of convergence

in Central Limit Theorem, 62, 446in Poisson’s theorem, 63, 449

Real line, extended, 184Realization of a process, 214Recalculation

of conditional expectations, 275of expectations, 233

Reflection principle, 94Regression, 288

curve, 288Regular

conditional distribution, 269, 271conditional probability, 268distribution function, 269

Relatively compact, 384Renewal

equation, 305function, 304process, 304

Restriction of a measure, 199Reversed

martingale, 106sequence, 133

Riemann integral, 242Riemann–Stieltjes integral, 242

Ruin, 83, 88

SSample

mean, 296space, 1, 14, 23, 31, 50, 84, 164,

167variance, 296

Samplesordered, 2, 5unordered, 2, 5

Samplingwith replacement, 2, 5without replacement, 3, 5, 18

Scalar product, 318Semi-invariant, 346Semi-norm, 313Semicontinuous, 378Separable

Hilbert space, 323metric space, 182, 199, 201, 271,

299, 300, 384, 387, 426, 428,429, 431

Separation of probability measures, 441Sequential compactness, 385Sigma-algebra, 161, 167

generated by ξ, 210generated by a decomposition, 211smallest, 168

Significance level, 73Simple

moments, 348random variable, 31semi-invariants, 348

Singular measure, 190Skorohod, A. V., 182

metric, 182Slepyan’s inequality, 370Slutsky’s lemma, 315Smallest

σ-algebra, 168algebra, 168monotonic class, 169

482 Keyword Index

Spaceof elementary events, 1, 164of outcomes, 1

Spitzer identity, 252Stable, 416

characteristic function, 416representation, 419

distribution, 416Standard deviation, 39, 284State of Markov chain, 115

absorbing, 116State space, 115Stationary

distribution, 123Markov chain, 123

Statisticalestimation, 70model, 70

Statistically independent, 26Statistics, 48Stieltjes, T. J., 219, 241, 245Stirling’s formula, 20Stochastic

exponent, 248matrix, 115

Stochastically dependent, 41Stopping time, 84, 107, 130, 459Strong Markov property, 129Student distribution, 190, 293, 296, 353Substitution

ntegration by, 250operation, 136

Sufficientσ-subalgebra, 276

minimal, 279statistic, 276, 277

Sum ofexponential random variables, 294Gaussian random variables, 292independent random variables, 40,

142, 291, 412intervals, 186Poisson random variables, 147,

294

random number of randomvariables, 83, 117, 296

sets (events), 7, 164Sums

lower, 241upper, 241

Symmetric difference �, 42, 164, 167System of events

exchangeable, 166interchangeable, 166

TTables

absolutely continuousdistributions, 190

discrete distributions, 189terms in set theory and probability,

164Taking limits

under the expectation sign, 259Taxi stand, 117Telescopic property, 80

first, 257second, 257

TheoremBayes, 24Beppo Levi, 253Caratheodory, 185ergodic, 121Glivenko–Cantelli, 452on monotonic classes

functional version, 174on normal correlation, 288, 363Poisson, 62Rao–Blackwell, 281Weierstrass, 52

Tight, 384, 389, 413Time

average, 265continuous, 214, 366discrete, 117, 214domain, 213of first return, 94stopping, 84, 107, 130, 459

Total probability, 23, 76, 118, 142

Keyword Index 483

Total variation distance, 431Trajectory

of a process, 214typical, 51

Transitionmatrix, 115operator, 133probabilities, 115

Trial, 6, 15, 27, 28, 48, 70, 205Triangle array, 401Trivial algebra, 8Typical

path, 49realization, 49

UUlam’s theorem, 387Unbiased estimator, 70, 280Uncorrelated

increments, 111random variables, 40, 284, 319,

358, 362, 371Unfavorable game, 88Uniform distribution, 190, 266, 282,

295, 406, 419, 455discrete, 189

Uniformlybounded, 226continuous, 53, 226, 334, 378, 395integrable, 225, 226

Union of sets, 7, 164Uniqueness of

binary expansion, 160, 325classes, 184decomposition, 8density, 233extension of measure, 186, 301probability measure, 194, 197product measure, 238projection, 328pseudoinverse, 369

solution of moments problem,338, 350

tests for, 352, 353, 356stationary distribution, 123

Universal probability space, 304

VVariance, 39, 284

conditional, 256of bivariate Gaussian

distribution, 288of estimator, 70of normal distribution, 284of sum, 40, 83, 288sample, 296

Vectormean of Gaussian distribution, 195of initial probabilities, 118of random variables, 213random

characteristic function of, 332covariance matrix, 285independence of components,

343Jensen’s inequality, 229mixed moments, 346normal (Gaussian), 358semi-invariants, 346

Venn diagram, 150

WWald’s identities, 108Wandermonde’s convolution, 144Weak convergence, 373, 389, 420, 425,

432, 456Weierstrass approximation theorem, 52Weierstrass–Stone theorem, 339Wiener, N.

measure, 202, 203process, 366

conditional, 367, 458

Symbol Index

SymbolspE,E , ρq, 376pMqn, 3pR,BpRqq, 175pR1,B1q, 176pRn,BpRnqq, 176pΩ,A ,Pq, 9, 161pa1, . . . , anq, 2A ` B, 7A X B, 7A Y B, 7A�B, 42Acptq, 248Ab, 369An

M , 3BzA, 7C, 182Cl

k, 2D, 182F ˚ G, 291Fξ, 32Fξn ñ Fξ, 306Fn ñ F, 375, 381Fn

wÑ F, 375HpP, rPq, 435Hpα;P, rPq, 435Hpxq, 56LpP, rPq, 420L2, 318Lp, 313

Lθpωq, 71LkpAq, 95NpAq, 10NpA q, 9NpΩq, 1Pξ, 206Pn

varÝÝÑ P, 432R1, 176RT , 180R8, 178Rnpxq, 326

X D“ Y , 426Xn

DÑ X, 425ra1, . . . , ans, 2Varpξ |G q, 256Var ξ, 39E ξ, 36, 218Epξ;Aq, 220Epξ |Dq, 78Epξ | ηq, 256Epξ |Dq, 78Epξ |G q, 255P, 162PpA | ηq, 77PpA |Dq, 76PpB |Aq, 22PpB | ηq, 256PpB |Dq, 254PpB |G q, 254, 256

© Springer Science+Business Media New York 2016A.N. Shiryaev, Probability-1, Graduate Textsin Mathematics 95, DOI 10.1007/978-0-387-72206-1

485

486 Symbol Index

PnwÑ P, 376

Pn ñ P, 376}P ´ rP}, 431}ppx, yq}, 115}pij}, 115αpDq, 8, 168FP, 188B1 bB2, 176E rpP, rPq, 433χ2, 293Covpξ, ηq, 284

ηndÑ η, 392

Epξ | η1, . . . , ηnq, 320şA ξ dP, 220şΩξ dP, 217

P, 118Ppkq, 118

CpFq, 375p, 118ppkq, 118a. s.Ñ , 306LpÑ, 306dÑ, 306varpP ´ rPq, 432A , 8, 160BpCq, 182BpDqq, 182BpRq, 175BpRTq, 180BpR8q, 178Bpr0, 1sq, 187BpRq, 176EtpAq, 248F , 161F {E , 212F˚, 167F˚, 167Fξ, 210

FA, 167L tη1, . . . , ηnu, 320N pm,Rq, 359N pm, σ2q, 284P “ tPα;α P Au , 384μ, 160μpAq, 160μpE q, 169μn ñ μ, 379μn

wÑ μ, 379erf , 66med, 383A, 7R, 176B, boundary, 376ď, 9ρpP, rPq, 435ρpξ, ηq, 40, 284ρ1 ˆ ρ2, 235σpE q, 168σpξq, 210(R–S)

şR ξpxqGpdxq, 220

ΔFξpxq, 36Φpxq, 65ϕpxq, 65

ξd“ η, 412

ξ K η, 319dPpX,Yq, 426fξ, 207

mpν1,...,νkqξ , 346

ppωq, 9

spν1,...,νkqξ , 346L tη1, η2 . . .u, 323śb

tPTFt, 183PÑ, 305

(L)ş8´8 ξpxq dx, 219

(L–S)ş

R ξpxqGpdxq, 219