Historical and Bibliographical Notes978-0-387-72206-1/1.pdfChapter 1: Introduction The history of...
Transcript of Historical and Bibliographical Notes978-0-387-72206-1/1.pdfChapter 1: Introduction The history of...
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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Springer-Verlag, Berlin–Heidelberg, 1987.[44] J. Kemeny and L. J. Snell. Finite Markov Chains. Van Nostrand, Princeton,
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Springer-Verlag, New York, 2006.[61] V. P. Leonov and A. N. Shiryaev. On a method of calculation of semi-
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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
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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