BIBLIOGRAPHY

Titles of articles the results of which are included in cited books by the same author are omitted from the bibliography. Roman numerals designate books, and arabic numbers designate articles.

The references below may pertain to more than one part or chapter of this book.

INTRODUCTORY PART: ELEMENTARY PROBABILITY THEORY

I. Bernoulli, J. Ars conjectandi (1713). II. Bernstein, S. Theory of probabilities-in Russian (1946).

III. Borel, E. Principes et formules classiques (1925). IV. Chung, K. L. Notes on Markov chains-mimeographed, Columbia Uni­

versity (1951). V. Delteil, R. Probabilites geometriques (1926).

VI. Feller, W. An Introduction to probability theory and its applications (1950). VII. Frechet, M. Methode des fonctions arbitraires. Theorie des erenements en

chaine dans Ie cas d'un nombrejini d'etats possibles (1938). VIII. Frechet, M. us probabilites associees a un systeme d' evenements com­

patibles et dependants I (1940), II (1943). IX. Gnedenko, B. Course in theory of probabilities-in Russian (1950). X. Laplace. Traite des probabilites (1801).

XI. Loeve, M. Probability methods in physics-mimeographed, University of California (1948).

XII. Markov. Wahrscheinlichkeitsrechnung-translated from Russian (1912). XIII. Parzen, E., Modern probability theory and its applications (1960). XIV. Perrin, F. Mecanique statistique quantique (1939). XV. Neyman, J. First course on probability and statistics (1950).

XVI. Uspensky, J. Introduction to mathematical probability.

1. Kac, M. Random walk in the presence of ahsorbing barriers. Ann. Math. Stat. 16 (1945).

2. Kolmogorov, A. Markov chains with a countable number of possible states-in Russian. Bull. Math. Univ. Moscow 1 (1937).

3. Loeve, M. Sur les systemes d'evenements. Ann. Univ. Lyon (1942). 4. Polya, G. Sur quelques points de la theorie des probabilites. Ann. Inst.

H. Poincare 1 (1931).

PART ONE: NOTIONS OF MEASURE THEORY

I. Banach, S. Theorie des operations lineaires (1932). II. Bourbaki, N. Elements de mathematique (1939-).

407

408 BIBLIOGRAPHY

III. Glivenko, J. Stieltjes integral--in Russian (1936). IV. Frechet, M. Espaces abstraits (1928). V. Hahn and Rosenthal. Set functions (1948).

VI. Halmos, P. Measure theory (1950). VII. Hausdorff, F. Mengenlehre (1914).

VIII. Lebesgue, H. Le(ons sur I'integration et la recherche des fonc/ions primi­tives (1928).

IX. Riesz, F. and Sz-Nagy, B. Le(ons d'analysefonctionnelle (1952). X. Saks, S. Theory of the integral (1937).

CHAPTER I. SETS, SPACES, AND MEASURES

1. Andersen, E. S. and Jessen, B. On the introduction of measures in in­finite product sets. Danske Vid. Selsk, Mat-Fys. Medd. 22 (1946).

2. Daniell, P. J. Functions of limited variation in an infinite number of dimensions. Ann. Math. 21 (1919-1920).

3. Doubrosky, V. On some properties of completely additive set-functions and passage to the limit under the integral sign. Izv. Ak. Nauk SSSR 9 (1945).

4. Kelley, J. L. Convergence in topology. Duke Math. J. (1952).

CHAPTER II. MEASURABLE FUNCTIONS AND INTEGRATION

5. Bochner, S. Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind. Fund. Math. 20 (1933).

6. Dunford, N. Integration in general analysis. Trans. Amer. Math. Soc. 37 (1942).

7. Kolmogorov, A. Untersuchungen iiber den Integralbegriff. Math. Ann. 103 (1930).

8. Nikodym, O. Sur une generalisation des integrales de Radon. Fund. Math. 15 (1930).

9. Robbins, H. Convergence of distributions. Ann. Math. Stat. 19 (1948). 10. Tulcea, I. Mesures dans les espaces produits. Alii Accad. Naz. Lincei

Rend. 7 (1949, 1950). 11. Scheffe, H. A useful convergence theorem for probability distributions.

Ann. Math. Stat. 18 (1947).

PART TWO: GENERAL CONCEPTS AND TOOLS OF PROBABILITY THEORY

I. Bochner, S. Fourierische integrale (1932). II. Cramer, H. Mathematical methods of statistics (1946).

III. Frechet, M. Generalitis sur les probabilitEs. ElEments alEatoires (1937). IV. Kolmogorov, A. Grundbegriffe der Wahrscheinlichkeitsrechnung (1933). V. Levy, P. Calcul des probabilitis (1925).

VI. Lukacz, E. Characteristic functions (1960).

CHAPTER III. PROBABILITY CONCEPTS

1. Bienayme. Considerations a l'appui de la decouverte de Laplace sur la loi des probabilites dans la methode des moindres carres. C. R. Acad. Sci. Paris 37 (1853).

BIBLIOGRAPHY 409

2. Cantelli, F. P. Una teoria astratta del calcolo delle probabilita. 1st. Ital. Alluari 3 (1932).

3. Lomnicki, A. Nouveaux fondements du calcul des probabilites. Fund. Math. 4 (1923).

4. Steinhaus, H. Les probabilites denombrables et leur rapport a la theorie de la mesure. Fund. Math. 4 (1923).

5. Tchebichev. Des valeurs moyennes. J. de Math. 12 (1867).

CHAPTER IV. DISTRIBUTION FUNCTIONS AND CHARACTERISTIC FUNCTIONS

6. Alexandrov, A. D. Additive set functions in abstract spaces. Matem. Sbornik 8, 9, 13 (1940, 1941, 1943).

7. Billingsley, P. Weak convergence of measures: applications in probabil­ity. Soc. Indust. and App/. Math. (1971).

8. Bochner, S. Monotone Funktionen, Stieltjes Integrale, und harmonische Analyse. Math. Ann. 108 (1933).

9. Bray, H. E. Elementary properties of the Stieltjes Integral. Ann. Math. 20 (1919).

10. Dugue, D. Analyticite et convexite des fonctions caracteristiques. Ann. Inst. H. Poincare XXII (1952).

11. F ortet, R. Calcul des momen ts d'une fonction de reparti tion a partir de sa characteristic. Bull. Sc. Math. 88 (1944).

12. Frechet, M. and Shohat, J. A proof of the generalized second limit theorem in the theory of probability. Trans. Am. Math. Soc. 33 (1931).

13. Helly, E. Ueber lineare Funktionaloperationen. Sitz. Nat. Kais. Akad. Wiss. 121 (1949).

14. Kawata, T. and Udagawa, M. On infinite convolutions. Kadai Math. Sem. 3 (1949).

15. Le Cam. Convergence in distribution of stochastic processes. Univ. Cal. Publ. Stat. 2, no 4 (1957).

16. Marcienkiewicz. Sur les fonctions independantes. Fund. Math. 31 (1939).

17. Marcienkiewicz. Sur une propriete de la loi de Gauss. Math. Zeit. 44 (1939).

18. Parzen, E. On uniform convergence of families of sequences of random variables. Univ. Calif. Publ. Stat. 2 (1954).

19. Poly a, G. Remark on characteristic functions. Proc. First Berkeley Symp. on Stat. and Prob. (1949).

20. Prohorov, U. V. Convergence of random processes and limit theorems of probability theory in Russia. Teoria Veroyatnostey, 1 (1956).

21. Zygmund, A. A remark on characteristic functions. Ann. Math. Stat. 18 (1947).

PART THREE: INDEPENDENCE

I. Breiman, L. Probability (1968). II. Chung, K. L. A course in probability theory, second edition (1974).

III. Cramer, H. Random variables and probability distributions (1937). IV. Feller, An-introduction to probability theory and its applications, Vol. II,

second edition (1971). V. Gnedenko, B. and Kolmogorov, A. Limit distributions for sums of inde­

pendent random variables (1954).

410 BIBLIOGRAPHY

VI. Kac, M. Statistical independence in probability, analysis and number theory (1959).

VII. Kemperman, J. H. B. The passage problem for a stationary Markov chain (1961).

VIII. Khintchine, A. Asymptotische Gesetze der Wahrscheinlichkeitsrechnung (1933).

IX. Khintchine, A. Limit laws for sums of independent random variables-in Russian (1938).

X. Levy, P. Theorie de l'addition des variables aleatoires, second edition (1954).

XI. Petrov, V. V. Sums of independent random variables-in Russian (1972). XII. Spitzer, F. Principles of random walk, second edition (1976).

XIII. Tucker, H. G. A graduate course in probability (1967).

CHAPTER v. SUMS OF INDEPENDENT RANDOM VARIABLES

1. Birnbaum, Z. and Zuckerman, H. An inequality due to Hornich. Ann. Math. Stat. 15 (1944).

2. Borel, E. Les probabilites denombrables et leurs applications arithme­tiques. Rend. eire. Mat. Palermo 27 (1909).

3. Brunk, H. D. The strong law of large numbers. Duke Math. J. 15 (1948). 4. Cantelli, F. P. Su due applicazioni di un teorema di G. Boole. Rend.

Accad. Naz. Lined 26 (1917). 5. Chung, K. L. The strong law of large numbers. Proc. Second Berkeley

Symp. on Stat. and Prob. (1951). 6. Da.Jy, J. On the use of the sample range in an analogue of Student's

test. Ann. Math. Stat. 17 (1946). 7. Feller, W. The general form of the so-called law of the iterated logarithm.

Trans. Am. Math. Soc. 54 (1943). 8. Kolmogorov, A. Ueber die Summen durch den Zufall bestimmter unab­

hiingiger Grossen. Math. Ann. 99 (1928), 102 (1929). 9. Prokhorov, U. V. On the strong law of large numbers-in Russian.

Dokl. Akad. Nauk USSR 69 (1949). 10. Robbins, H. On the equidistribution of sums of independent random

variables. Proe. Am. Math. Soc. 4 (1953). 11. Tukey, J. An inequality for deviations from medians. Ann. Math.

Stat. 16 (1945).

CHAPTER VI. CENTRAL LIMIT PROBLEM

12. Bawly, G. Uber einige Verallgemeinerungen der Grenzwertsiitze der Wahrscheinlichkeitsrechnung. Mat. Sbornik 43 (1936).

13. Berry, A. The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Am. Math. Soc. 49 (1941).

14. Dohlin, W. Sur les sommes d'un grand nombre de variables alt~atoires independants. Bull. Sc. Math. 63 (1939).

15. Dohlin, W. Sur I'ensemble de puissances d'une loi de probabilite. Ann. Ec. Norm. Sup. (1947).

16. Esseen, G. Fourier Analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta Math. (1944).

BIBLIOGRAPHY 411

17. Feller, W. Ueber den Zentralen Grenzwertsatz der Wahrscheinlichkeits­rechnung. Math. Zeit. 40 (1935), 42 (1937).

18. Feller, W. Ueber das Gesetz der grossen Zahlen. Acta Univ. Szeged 8 (1937).

19. Feller, W. On the Kolmogoroff-P. Levy formula for infinitely divisible distribution functions. Proc. Yougoslav Acad. Sc. 82 (1937).

20. de Finetti, B. Sulle funzioni a incremento aleatorio. Rl'nd. Accad. Naz. Lincei 10 (1929).

21. Liapounov, A. Nouvelle forme du theoreme sur la limite de la probabi­lite. Mem. Acad. St-Petersbourg 11 (1900),12 (1901).

22. Lindeberg, J. W. Ueber das Exponentialgesetz in der Wahrscheinlich­keitsrechnung. Math. Zeit. 15 (1922).

23. Loeve, M. Nouvelles classes de lois limites. C. R. Ac. Sc. (1941), Bull. Soc. Math. (1945).

24. Loeve, M. Ranking limit problem. Second Berkeley Symp. (1956). 25. Loeve, M. A I'interieur du probleme limite central. Ann. Inst. Stat.

Paris (1958). 26. Raikov, D. A. On the decomposition of Gauss and Poisson laws-in

Russian. !ZTJ. Akad. Nauk USSR 2 (1938).

CHAPTER VII. INDEPENDENT IDENTICALLY DISTRIBUTED S.UMMANDS

27. Andersen, Sparre. On sums of symmetrically dependent random vari­ables. Skand. Akturaetid. 36 (1953).

28. Andersen, Sparre. On the fluctuations of sums of random variables. Math. Scand. 1 (1953),2 (1954).

29. Baxter, G. An analytic approach to finite fluctuation problems in prob­ability. 1. d'Anal. Math. 9 (1961).

30. Blackwell, D. Extension of a renewal theorem. Pac. 1. Math. 3 (1953). 31. Chung, K. L. and Fuchs, W. H. J. On the distribution of values of sums

of random variables. Mem. Am. Math. Soc. 6 (1951). 32. Chung, K. L. and Ornstein, D. On the recurrence of sums of random

variables. Bull. Am. Math. Soc. 68 (1962). 33. Doblin, W. Sur l'ensemble de puissances d'une loi de probabilite. Studia

Math. 9 (1940). 34. Gnedenko, B. V. Some theorems on the powers of distribution functions.

Uchenye Zapiski Moskow UniT.!. Mat. 45 (1940). 35. Hewitt, E. and Savage, E. L. Symmetric measures on Cartesian products.

Trans. Am. Math. Soc. (1955). 36. Katz, M. A note on the weak law of large numbers. Ann. Math. Statist.

39 (1968). 37. Owen, W. L. An estimate for EISnl for variables in the domain of normal

attraction of a stable law of index a, 1 < a < 2. Ann. of Prob. 1 (1973). 38. Pollaczec, F. Uber eine Aufgabe der Wahrscheilichkeitstheorie I, II.

Math. Zeitsch. 32 (1930). 39. Pollaczec, F. Fonctions characteristiques de certaines repartitions de­

finies au moyen de la notion d'ordre. Applications a la theorie des at­tentes. Comptes Rendus 234 (1952).

412 BIBLIOGRAPHY

40. Poly a, G. Uber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt in Strassennetz. Math. Ann. 84 (1921).

41. Port, S. C. An elementary probability approach to fluctiation theorie. J. Math. Anal. and Appl. 6 (1963).

42. Ray, D. Stable processes with an absorbing barrier. Trans. Am. Math. Soc. 89 (1958).

43. Spitzer, F. A combinatorial lemma and its application to probability theory. Trans. Am. Math. Soc. 82 (1956).

44. Wendel, J. G. Order statistics of partial sums. Ann. Math. Statist. 31 (1960).

INDEX

Abel theorem, 400 Addition property, 10 Additive set function, 83

continuity, 85 continuity theorem, 85 countably,83 decomposition, 87 decomposition theorem, 88 extension, 88 extension theorem, 88 fini te, 83, 111 finitely, 83 restriction, 88 u-additive, 83, 111 u-finite, 83, 111

Adherence, 66 Adherent point, 66 Alexandrov, 190, 409 Allard,44 Almost everywhere, 112

convergence, 114 mutual convergence, 114

Almost sure(ly), 148 convergence, 152, 248, 260 mutual convergence, 153 stabili ty, 244, 249, 274, 260 stability criterion, 264

Almost uniform convergence, 140 Andersen and Jessen, 92, 408 Andersen, 368, 379, 404, 411

equivalence, 377, 391 equivalence lemma, 378

Andre, Desire, 47 Arcsine law, 379, 404 Asymptotic

behaviour theorem, 399

Asymptotic (Cont.) passage theorem, 36 uniform negligibility, 302

Atom, 100 Attraction

domain of, 360 partial, 403 stability and-criteria, 364 standard,402

Axioms of the countable case, 16 of the finite case, 8

Baire category theorem, 75 functions, 109

Banach,407 space, 81

Base countable, 72 of a cylinder, 62

Baxter, 369, 390, 411 Bawly, 302, 410 Bernoulli, 407

case, 12, 244, 280 extended, 26 law of large numbers, 14,244,

282 Berry, 294,410 Billingsley, 196, 409 Bienayme, 408

equality, 12,246 Blackwell, 369, 411 Bochner, 408, 409

theorem, 220 Boltzmann, 42, 43 Bolzano-Weierstrass property, 70

413

414 INDEX

Borel(ian), 107,407,410 Can telli lemma, 240 cylinder, 93 field, 93, 104 function, 111, 156 functions theorem, 156 line, 93, 107 sets, 93, 104 space, 93, 107 strong law of large numbers, 18,

19, 26, 244 zero-one criterion, 240

Bose-Einstein statistics, 43, 44 Bounded

functional, 80 Liapounov theorem, 213, 282 set, 74 totally, 75 variances, 303 variances limit theorem, 305

Bourbaki, 407 Breiman, 409 Brey, 409 Brunk, 271, 410

Cantelli, 20, 240,409,410 Cantor theorem, 74 Caratheodory extension

theorem, 88 Category

first, 75 second,75 theorem, 75

Cauchy mutual convergence crite­rion, 74, 104, 114

Centering, 244 at expectations, 244 at medians, 256 function, 350

Central convergence criterion, 323, 326 inequalities, 316 limit problem, 302 limit theorem, 321, 322 statistical theorem, 20

Chain, 29 constant, 29 elemen tary, 29 stationary, 39

Chained classes, 28 events, 28 random variables, 29

Change of variable lemma, 190 Characteristic function(s), 198

composition theorem, 226 continuity theorem, 204, 224 convergence criterion, 204 and dichotomy, 386 extension theorem, 224 general properties, 207 infinitely decomposable, 306 in tegral, 202 inversion formula, 199 triangular, 386 self-decomposable, 334 stable, 338, 363 uniform convergence theorem,

204 Chung, 407, 409, 410

and Fuchs, 368, 383, 411 and Ornstein, 383, 411

Class closed under, 59 lower, 272 monotone, 60 of sets, 55 upper, 272

Classical degenerate convergence crite­

rion, 290 limit problem, 286 normal convergence criterion,

292 Closed

class of states, 36 model, 22 set, 66

Closure theorem infinitely decomposablelaws,309

INDEX 415

Combinatorial lemma, 378 method, 47

Compact locally, 71 set, 69 space, 69

Compactification, 71 Compactness

properties, 70 relative, 195 relative, criterion, 195

Compactness theorem for distribution functions, 181 metric spaces, 76 separated spaces, 70

Comparison convergences theorem, 117 lemma(s), 303, 320

Complement, 4, 56 Complete

convergence, 180 convergence criterion, 204 measure, 91 metric space, 74

Completeness theorem, L r- , 163 Completion

of a metric space, 77 of <T-field, 91

Complex random variable, 154 Composition, 204, 283

and decomposition theorem, 283 lemma, 227 theorem, 206

Compound(s) Poisson, 347 theorem, 237 <T-field, 237

Condi tional expectation, 24 probability, 24 regular probability, 138

Consistency theorem, 94 Consistent, 93 Constant chain, 29

Continuity additive set functions-theo­

rem, 85 characteristic functions theo­

rem, 204, 224 F- , interval, 187 p- , set, 189

Continuous F-function, 67 functional, 80 p-set function, 85

Convergence almost everywhere, 114 almost sure, 153 almost uniform, 140 complete, 180 essential, 262 in quadratic mean, 260 in rth mean, 159 of sequences of sets, 58 of types, 216 uniform, 114 weak, 180

Convergence criterion(ria) almost everywhere, 116 central, 323, 327 complete, 204 degenerate, 290, 329 iid,346 iid cen tral, 348 normal, 292, 328 Poisson, 308, 329 pr.'s on metric space, 190 weak, 203

Convergence theorem(s) comparison, 117 dominated, 126 Fatou-Lebesgue, 12 L r- ,165 momen ts, 186 monotone, 125 of types, 216 uniform, 204

416 INDEX

Convex function, 161 Correspondence

lemma, 344 theorem, 97

Countable(ly),16 base, 72 class, 57 set, 57 set operations, 57 valued, 64, 106

Covering open, 69 rule, 16

C,-inequality, 157 Cramer, 271, 408, 409 Cylinder

Borel,62 product, 62

Daniell, 94, 408 Decomposable

infini tely, 308 self- ,334

Decomposition theorem(s) chains, 37 composition and, theorem, 283 degenerate type, 283 distribution functions, 178,200 Hahn, 87 Lebesgue, 131 normal type, 283 Poisson type, 283

Degenerate characteristic function, 215 convergence criterion, 290, 329 distribution function, 215 law, 215 random time, 376 random variable, 215 random walk, 370 type, 215

Dense set, 72 nowhere, 75

Denumerable, 16 class, 57

Denumerable (Cont.) set, 57 set operations, 57 valued, 64, 106

Diameter of a set, 74 Dichotomy, 380

criterion, 382 Difference

equations, 48 proper, 56 of sets, 56

Direction(ted),67 set, 68

Dirichlet, 187 Disjoint

class, 57 events, 4 sets, 57

Distance of points, 73 points and sets, 78 sets, 77

Distribution, 168, 172, 175 empirical distribution function,

20 function, 20, 96, 169, 177 invariant, 39 L d-, 370 probability, 168

Doblin, 30, 302, 354, 403, 410, 411 Domain, 62, 108

of attraction, 360 partial, 401 standard,402

Dominated convergence theorem, 126

Doubrovsky, 408 Duality rule, 57 Dugue, 409 Dunford, 408

Egorov theorem, 141 Einstein, 43, 44 Elementary

chain, 29

Elementary (Cont.) function, 64, lO7 probability field, 16 random variable, 17, 152

Empirical distribution function, 20

Empty set, 4, 54 Equivalence

Andersen, 377, 393 Andersen, lemma, 378 class, 154 convergence, 245 lemma, laws, 290 lemma, series, 245 tail, 245 theorem(s), 263,379

Equivalent distribution functions, 96 functions, 114 random variables, 154 states, 36

Erdos, 34 Esseen, 294, 4lO Essential

convergence, 262 divergence, 262

Euler, 47 Events, 3, 8, 151

disjoint, 3 elementary, 151 exchangeable, 45, 21 impossible, 4, 151 independent, 11, 73, 235 null, 152 random,S, 8 sure, 4, 151 tail, 241

Everreturn state, 36 Exchangeable

events, 45, 373 random variables, 373

Expectation centering at, 244 cri terion, 384 indefini te, 153

INDEX 417

Expectation (Cont.) of a random function, 156 of a random sequence, 154 of a random variable, lO, 17,

153, 154 Exponential

bounds, 266 identities, 388, 396

Extended Bernoulli law of large numbers,

26 Borel line, 93, lO7 Borel space, 93, lO7 Borel strong law of large num­

bers,26 central convergence criterion,

326 central limit theorem, 322 convergence criterion, 306 Helly-Bray lemma, 183 identities, 395

Extension, 88 of characteristic functions, 225 of linear functionals, 81 of measures, 88

Factorization(s) extreme, 396 sample space, 392, 396 unique, theorem, 389

Feller, 34, 292,302, 353, 354, 369, 371,383,407,409-411

Fermi-Dirac statistics, 42, 43 Field(s), 59

Borel, 93, lO4 compound, 156 Lebesgue, 129 of outcomes, 4 probabili ty, 8 product, 61, 62 U-, 59

Finite intersection property, 70 interval lemma, 45

Finetti, de, 302, 411

418

Finitely-valued, 64, 106 First

category, 75 limit theorems, 282

Fortet, 409 F rechet, 187, 408 Fubini theorem, 136 Function(s)

additive set, 83 Baire, 111 centering, 350 characteristic, 199, 202 continuous, 67 convex, 161 countably-valued, 64, 106 denumerably-valued, 64, 106 distribution, 20, 96, 169, 177 domain of, 62, 107 elementary, 64, 107 equivalent, 114 F-continuous, 187 finite, 105 finitely-valued,64,106 of function, 64, 106 inverse, 63, 106 . measurable, 65, 107 non-negative definite, 219 numerical, 105 P-continuous, 187 positive part of, 105 random, 152, 156 range of, 63 range space, 62, 105 simple, 64, 107 tail, 241

Functional, 80 bounded, 80 continuous, 80 linear, 80 norm ed, 80

Gambler's ruin, 48 Geometric probabilities, 49 Glivenko, 408

-Cantelli, 21

INDEX

Gnedenko, 302, 354, 407, 409, 411 Gumbel, 45

Hadamard, 30 Hahn and Rosenthal, 408 Hahn decomposition theorem, 87 Halmos, 196,408 Hausdorff, 408

space, 68 Heine-Borel property, 70 Helly, 409 Helly-Bray lemma, 182

extended, 183 generalized, 187

Helly-Bray theorem, 184 Herglotz lemma, 220 Hewitt-Savage, 374, 411

zero-one law, 374 Hil bert space, 80 Hitting time, 377

lemma, 374 Holder inequality, 158 Huygens principle, 28

Identification property, 73 Image, Inverse

of a class, 63, 106 of a set, 63, 106

Impossible event, 4, 110 Improper integral, 130 Increments inequality, 208 Indecomposable class of states,

36 Indefinite

expectation, 154 integral, 130

Independent classes, 11, 235 events, 11, 235 random functions, 237 random variables, 11, 237 random vectors, 237 u-fields, 236 trials, 7

Indicator(s), 9, 59 method of, 44

Induced parti tion, 64, 106 probability space, 168, 171 u-field, 64 topology, 66

Inequality(ties) basic, 159 central, 316 cr - ,157 Holder, 158 integral, 208 Kolmogorov, 25, 247, 275 Levy, 259 Liapounov, 177 Schwarz, 158 symmetrization, 259 Tchebichev, 11, 160 truncation, 209 weak symmetrization, 258

Inferior limi t, 58 Infimum, 56, 103 Infini te

decomposability, 308 num bers, 103

Infinitely often, 241 Integrable, 119

uniformly, 164 Integral

characteristic function, 202 inequality, 208 representation theorem, 166

Integral(s) Daniell, 146 Darboux-Young, 144 definitions, 119 elementary properties, 120 improper, 130 iterated, theorem, 137 Kolmogorov, 145 Lebesgue, 129, 143 Lebesgue-Stieltjes, 128 Riemann, 129 Riemann-Stieltjes, 129

INDEX 419

Integration by parts lemma, 358 Interior, 66

point, 66 Intermediate value theorem, 102 Intersection(s), 4, 56

fini te-property, 70 Interval(s), 61, 62, 104

finite-lemma, 397 Invariance theorem, 39 Invariant distribution, 39 Inverse

function, 63, 106 image, 63, 106

Inversion formula, 199 Iterated

logari thm, law of, 219 regular conditional probability

theorem, 138

Kac, 407, 410 Katz, 411 Karamata, 354

main theorem, 356 Kawata, 210, 409 Kelley, 408 Kemperman, 369, 393, 395, 410 Khintchine, 28,302,410

measure, 343 representation, 343

Kolmogorov, 30, 94, 302, 407, 408, 410

approach, 145 inequalities, 25, 247, 275 strong law of large numbers, 251 three series criterion, 249 zero-one law, 241

Kronecker lemma, 250

Lambert, 46 Laplace, 22, 281, 286, 287, 407 Law of large numbers

Bernoulli, 14, 26, 244, 282 Borel, strong, 18, 19,26,244 classical, 290 Kolmogorov, strong, 251

420 INDEX

Law(s), 174 degenerate, 215, 281 equivalence lemma, 290 infinitely decomposable, 308 normal, 213, 281 of the iterated logarithm, 219 Poisson, 282 probability, 174,214 self-decomposable, 334 stable, 326, 363 types of, 215 universal, 403 zero-one, 241,374

Lebesgue, 408 approach, 143 decomposition theorem, 131 field, 129 in tegral, 129 measure, 128 sets, 129

Lebesgue-Stiel tj es field, 128 integral, 128 measure, 128

Le Cam, 193,409 Levy, P., 199,301,302,408,410

continuity theorem, 204 inequalities, 259 function(s), 361 measure, 343 representation, 343

Liapounov, 411 inequality, 174 theorem, 213, 287, 289

Limit of a directed set, 68 along a direction, 68 inferior, 58 superior, 58

Limit of a sequence of functions, 113 laws, 214 numbers, 104 sets, 58

Limit problem central, 302 classical, 286

Lindeberg, 292,411 Line

Borel, 93, 107 extended real, 104 real, 93, 103

Linear closure, 79 functional, 80 space, 70

Linearly ordered, 67 Liouville theorem, 369 Lomnicki, 409 Lower

class, 272 variation, 87

L r-

completeness theorem, 163 convergence theorem, 164 spaces, 162

Lusin theorem, 140

Marcinkiewitz, 225, 254, 302, 409 Markov, 407

chain, 28 dependence, 28 inequality, 160

Lukacz,408 Matrices, method of, 48 Matrix, transition probability, 29 Mean

rth mean, 159 Measurable

function, 107 sets, 60, 64, 107 space, 60, 64, 107

Measure, 84, 112 convergence in, 116 Khin tchine, 343 Lebesgue, 129 Lebesgue-Stieltjes, 128 Levy, 343 normed, 91, 151

Measure (Cont.) outer, 88 outer extension of, 89 product, 136 signed,87 space, 112

Median, 256 centering at, 256

Metric compactness theorem, 76 linear space, 79 space, 73 topology, 73

Minimal class over, 60 Minkowski inequality, 158 Moment(s)

convergence problem, 187 convergence theorem, 186 kth, 157, 186 lemma, 254 Tth absolute, 157, 186

Monotone class, 60 convergence theorem, 125 sequences of sets, 58

Mon tmort, 46 1'0 -measurable, 88 Multiplication

lemma, 238 property, 11 rule, 24 theorem, 238

INDEX

Normal approximation theorem, 300 convergence criterion, 307 decomposition theorem, 283 law, 213, 281 type, 283

421

Normalized distribution function, 199

Normed functional, 80 linear space, 79 measure, 91, 151 sums, 331

Nowhere dense, 75 Null

set, 91, 112 state, 32

Numerical function, 105

Open covering, 69 set, 66

Ordering, partial, 67 Orthogonal random variables, 246 Outcome(s),4

of an experiment, 4 field of, 4

Outer extension, 89 measure, 88

Owen, 411

Negligibility, uniform asymptotic, Parseval relation, 386 Parzen, 407 Petrov,410 302,314

Neighborhood, 66 Neyman, 407 Nikodym, 133, 408 Nonhereditary systems, 28 Nonrecurrent state, 31 No-return state, 31 Norm

of a functional, 79 Hilbert, 80 of a mapping, 79

Physical statistics, 42 Planck, 44 Poincare recurrence theorem, 28 Poisson

compound, 347 convergence criterion, 229,329 decomposition theorem, 283 law, 282 theorem, 15 type, 283

422 INDEX

Pollaczec, 396, 394, 400, 411 -Spitzer identity, 393, 400

Pollard,34 Polya, 368, 409 Port, 369, 393, 412 Positive

part, 105 state, 32

Positivity criterion, 33 Possible

state(s),370 value(s),370 values theorem, 371

Probability, 5, 8, 16, 91, 151, 152 condi tional, 6, 24 convergence in, 153 convergence on metric spaces,

189, 190 distribution, 168 field, 8 law, 214 product-theorem, 92 rule, total, 24 stability in, 244 sub, 187 transition, 29

Probability space, 91, 151, 152 induced, 168 product, 92 transition, 29 product, 92 sample, 168

Product cylinder, 62 field, 61, 62 measurable space, 61, 62, 137 measure, 136 probability, 92 probability theorem, 242 scalar, 80 set, 61 u-field, 61, 62 space, 61, 62

Prohorov, 190, 193, 264, 409

Quadratic mean convergence in, 260

Radon-Nikodym theorem, 133 extension, 134

Raikov, 283, 411 Random

event, 5, 8 function, 152, 156 sequence, 152, 155 time, 375 time identities, 390 time translations, 376 trial, 6 variable, 6, 9, 17, 152 vector, 152, 155 walk, 47, 378, 379

Range, 63 space, 62, 105

Ranked random variables, 350 sums, 405

Ray, 369, 395, 412 Real

line, 93 line, extended, 93, 107 number, 93 number extended, 93 space, 93

Recurrence, 380 criterion, 32 theorem(s), 27, 384

Recurrent state(s), 31, 380 walk, 28

Regular variation, 354 criterion, 354

Relative compactness, 190 theorem, 195

Representation theorem, 313 integral, 166

Restriction, 88 Return

cri terion, 32 state, 31

Riemann integral, 129

Riemann-Stieltjes integral, 129 Riesz, F., 222, rth

absolute moment, 157, 186 mean, 159

Ruin, gambler's, 48

Saks, 408 Savage, 374 Scalar product, 80 Scheff€:, 408 Schwarz inequality, 158 Section, 61, 62, 135 Self-decomposable(bili ty), 334

criterion, 335 Separable space, 68 Separation theorem, 68 Sequence(s)

convergence equivalent, 245 random, 152, 155 tail of, 241 tail equivalent, 245

Series criterion three, 249 two, 263

Set function additive, 83 continuous, 85 countably additive, 83 fini te, 82, 111 finitely, 83 u-additive, 83, 111 u-finite, 83, 11

Set(s) Borel, 93, 104 bounded, 74 closed, 66 compact, 69 dense, 72 directed, 68 empty, 4, 54 Lebesgue, 129 measurable, 60, 64, 107 null, 91, 112 open, 66

INDEX

Set(s) (Cont.) product, 61 subdirected, 69 totally bounded, 75

Shohat, 187 u-additive, 83, 111 u-field(s), 59

compound, 156, 235 independent, 236 induced, 64 product, 61, 62 tail, 241

Signed measure, 87 Simple

function, 64, 107 random variable, 6, 152

Snell, 66 Space

adjoint, 81 Banach, 79 Borel, 93, 107 compact, 69 complete, 74 Hausdorff, 68 Hilbert, 80 induced probability, 168 linear, 79 measurable, 60, 64, 107 measure, 112 metric, 73 metric linear, 79 normal, 78 normed linear, 79 probability, 91, 151, 152 product, 61, 62

423

product measurable, 61, 62, 137 product measure, 136 product probability, 91 range, 62, 105 sample probability, 168 separated, 68 of sets, 55 topological, 66

Sphere, 73

424

Spitzer, 369, 393, 394, 404,410, 412

basic identity, 396 basic theorem, 401

Stability almost sure, 244 almost sure criterion, 264 and attraction criterion, 364 in probability, 244, 246

Stable characteristic function, 338 law, 338, 363

State(s) closed class of, 36 equivalent, 36 everreturn, 36 indecomposable class of, 36 nonrecurrent, 31, 380 no return, 31 null, 32 period of, 33 positive, 32 possible, 370 recurrent, 31, 380 return, 31 transient, 380

Stationary chain, 39 Steinhaus, 409 Stiel tj es, 128, 129 Stochastic variable, 174 Stochastically independent, 11 Strong law of large numbers

Borel, 18, 19, 26, 244 Kolmogorov, 241

Structure corollary, 348 theorem, 310

Subspace linear, 79 topological, 66

Sum of sets, 4, 51 Superior limit, 58 Supremum, 56, 103 Sure

almost, 151

INDEX

Sure (Cont.) event, 4, 151

Symmetrization, 257 inequalities, 259 inequalities, weak, 257

Tail equivalence, 245 event, 241 function, 241 of a sequence, 241 u-field, 241

Tchebichev, 409 inequality, 11, 160 theorem, 287

Tight(ness),194 lemma, 194 and relative compactness, 195 theorem, 194

Three alternatives, 375 alternatives criteria, 399 series criterion, 249

Toeplitz lemma, 250 Topological

space, 66 su bspace, 66

Topology, 66 metric, 73 reduced,66

Total(ly) bounded set, 75 probability rule, 24 variation, 87

Transition probability, 29 Trial(s)

deterministic,S identical,S, 6 independent,S, 6 random, 6 repeated,S, 6

Triangle property, 73 Triangular

characteristic function, 386 probability density, 386

Truncation, 245 inequality, 209

Tulcea, 138, 408 Tucker, 410 Two-series criterion, 251 Type(s),215

convergence of, 216 degenerate, 215, 282 normal, 282 Poisson, 282

Ugakawa, 102 Uniform

asymptotic negligibility, 302, 314

continuity, 77 convergence, 114 convergence theorem, 204

Union, 4, 56 Upper

class, 272 variation, 87 Urysohn,78

Uspensky, 407

Value(s), possible, 370 theorem, 371

Variable, random, 69,17, 152 Variance, 12, 244

INDEX

Variances, bounded, 302 limi t theorem, 305

Variation lower, 87 regular, 354 slow, 354 total, 87 of truncated moments, 359 upper, 87

Vector, random, 152, 155

Wald's relation, 377, 397 Weak

compactness theorem, 181 convergence, 180

425

convergence, to apr., 190 symmetrization inequalities, 257 convergence, to apr., 190

Weierstrass theorem,S Wendel,412

Zero-one criterion, Borel, 24 law, Kolmogorov, 241 law, Hewitt-Savage, 374

Zygmund, 409

Graduate Texts in Mathematics

Soft and hard cover editions are available for each volume up to Vol. 14, hard cover only from Vol. 15

1 TAKE UTI/ZARING. Introduction to Axiomatic Set Theory. vii, 250 pages. 1971.

2 OXTOBY. Measure and Category. viii, 95 pages. 1971.

3 SCHAEFFER. Topological Vector Spaces. xi, 294 pages. 1971.

4 HILTON/STAMMBACH. A Course in Homological Algebra. ix, 338 pages. 1971. (Hard cover edition only)

5 MACLANE. Categories for the Working Mathematician. ix, 262 pages. 1972.

6 HUGHES/PIPER. Projective Planes. xii, 291 pages. 1973.

7 SERRE. A Course in Arithmetic. x, 115 pages. 1973.

8 TAKEUTI/ZARING. Axiomatic Set Theory. viii, 238 pages. 1973.

9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory. xiv, 169 pages. 1972.

10 COHEN. A course in Simple Homotopy Theory. xii, 114 pages. 1973.

11 CONWAY. Functions of One Complex Variable. 2nd corrected reprint. xiii, 313 pages. 1975. (Hard cover edition only.)

12 BEALS. Advanced Mathematical Analysis. xi, 230 pages. 1973.

13 ANDERSON/FuLLER. Rings and Categories of Modules. ix, 339 pages. 1974.

14 GOLUBITSKY/GUILLEMIN. Stable Mappings and Their Singularities. x, 211 pages. 1974.

15 BERBERIAN. Lectures in Functional Analysis and Operator Theory. x, 356 pages. 1974.

16 WINTER. The Structure of Fields. xiii, 205 pages. 1974.

17 ROSENBLATT. Random Processes. 2nd ed. x, 228 pages. 1974.

18 HALMOS. Measure Theory. xi, 304 pages. 1974.

19 HALMOS. A Hilbert Space Problem Book. xvii, 365 pages. 1974.

20 HUSEMOLLER. Fibre Bundles. 2nd ed. xvi, 344 pages. 1975.

21 HUMPHREYS. Linear Algebraic Groups. xiv. 272 pages. 1975.

22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic. x, 137 pages. 1975.

23 GREUB. Linear Algebra. 4th ed. xvii, 451 pages. 1975.

24 HOLMES. Geometric Functional Analysis and Its Applications. x, 246 pages. 1975.

25 HEWITT/STROMBERG. Real and Abstract Analysis. 3rd printing. viii, 476 pages. 1975.

26 MANES. Algebraic Theories. x, 356 pages. 1976.

27 KELLEY. General Topology. xiv, 298 pages. 1975.

28 ZARISKI/SAMUEL. Commutative Algebra I. xi, 329 pages. 1975.

29 ZARISKI/SAMUEL. Commutative Algebra II. x, 414 pages. 1976.

30 JACOBSON. Lectures in Abstract Algebra I: Basic Concepts. xii, 205 pages. 1976.

31 JACOBSON. Lectures in Abstract Algebra II: Linear Algebra. xii, 280 pages. 1975.

32 JACOBSON. Lectures in Abstract Algebra III: Theory of Fields and Galois Theory. ix, 324 pages. 1976.

33 HIRSCH. Differential Topology. x, 222 pages. 1976.

34 SPITZER. Principles of Random Walk. 2nd ed. xiii, 408 pages. 1976.

35 WERMER. Banach Algebras and Several Complex Variables. 2nd ed. xiv, 162 pages. 1976.

36 KELLEY/NAMIOKA. Linear Topological Spaces. xv, 256 pages. 1976.

37 MONK. Mathematical Logic. x, 531 pages. 1976.

38 GRAUERT/FRITZSCHE. Several Complex Variables. viii, 207 pages. 1976.

39 ARVESON. An Invitation to C*-Algebras. x, 106 pages. 1976.

40 KEMENy/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed. xii, 484 pages. 1976.

41 ApOSTOL. Modular Functions and Dirichlet Series in Number Theory. x, 198 pages. 1976.

42 SERRE. Linear Representations of Finite Groups. approx. 176 pages. 1976.

43 GILLMAN!JERISON. Rings of Continuous Functions. xiii, 300 pages. 1976.

44 KENDIG. Elementary Algebraic Geometry. approx. 304 pages. 1977.

45 LOEVE. Probability Theory. 4th ed. Vol. 1. xvii, 425 pages. 1977.

46 LOEvE. Probability Theory. 4th ed. Vol. 2. In preparation. 1977.