Bulletin of the American Mathematical Society

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Book Information:

Authors: Galen R. Shorack and Jon A. Wellner
Title: Empirical processes with applications to statistics
Additional book information: Wiley Series in Probability and Mathematical Statistics, John Wiley and Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1986, xxvii + 938 pp., $59.95. ISBN 0-471-86725-X.

R. R. Bahadur, A note on quantiles in large samples, Ann. Math. Statist. 37 (1966), 577–580. MR 189095, DOI 10.1214/aoms/1177699450

J. Bernoulli (1713), Ars Coniectandi. I-II, III-IV, Oswald's Klassiker der Exacten Wissenschaften, No. 108, W. Engelmann, Leipzig, 1899.

Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396

L. Brieman (1968), Probability, Addison-Wesley, Reading, Mass.

J. Bretagnolle, Statistique de Kolmogorov-Smirnov pour un échantillon non équiréparti, Statistical and physical aspects of Gaussian processes (Saint-Flour, 1980), Colloq. Internat. CNRS, vol. 307, CNRS, Paris, 1981, pp. 39–44 (French, with English summary). MR 716526

David R. Brillinger, An asymptotic representation of the sample distribution function, Bull. Amer. Math. Soc. 75 (1969), 545–547. MR 243659, DOI 10.1090/S0002-9904-1969-12237-8

F. P. Cantelli (1933), Sulla determinazione empirica delle leggi di probabilità, Giorn. 1st. Ital. Attuari 4, 421-424.

Miklós Csörgő, Quantile processes with statistical applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 42, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983. MR 745130, DOI 10.1137/1.9781611970289

Miklós Csörgő, Sándor Csörgő, and Lajos Horváth, An asymptotic theory for empirical reliability and concentration processes, Lecture Notes in Statistics, vol. 33, Springer-Verlag, Berlin, 1986. MR 856407, DOI 10.1007/978-1-4615-6420-1

M. Csörgő and P. Révész, A new method to prove strassen type laws of invariance principle. II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1975), no. 4, 261–269. MR 1554018, DOI 10.1007/BF00532866

M. Csörgő and P. Révész, Strong approximations in probability and statistics, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 666546

Monroe D. Donsker, An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc. 6 (1951), 12. MR 40613

Monroe D. Donsker, Justification and extension of Doob’s heuristic approach to the Komogorov-Smirnov theorems, Ann. Math. Statistics 23 (1952), 277–281. MR 47288

J. L. Doob, Heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Statistics 20 (1949), 393–403. MR 30732, DOI 10.1214/aoms/1177729991

R. M. Dudley, A course on empirical processes, École d’été de probabilités de Saint-Flour, XII—1982, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1984, pp. 1–142. MR 876079, DOI 10.1007/BFb0099432

P. Erdös and M. Kac, On certain limit theorems of the theory of probability, Bull. Amer. Math. Soc. 52 (1946), 292–302. MR 15705, DOI 10.1090/S0002-9904-1946-08560-2

W. Feller, On the Kolmogorov-Smirnov limit theorems for empirical distributions, Ann. Math. Statistics 19 (1948), 177–189. MR 25108, DOI 10.1214/aoms/1177730243

Peter Gänssler, Empirical processes, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 3, Institute of Mathematical Statistics, Hayward, CA, 1983. MR 744668

V. Glivenko (1933), Sulla determinazione empirica della legge di probabilità, Giorn. Ist. Ital. Attuari 4, 92-99.

B. V. Gnedenko and V. S. Koroljuk, On the maximum discrepancy between two empirical distributions, Select. Transl. Math. Statist. and Probability, Vol. 1, Inst. Math. Statist. and Amer. Math. Soc., Providence, R.I., 1961, pp. 13–16. MR 0116418

M. Kac, On the average of a certain Wiener functional and a related limit theorem in calculus of probability, Trans. Amer. Math. Soc. 59 (1946), 401–414. MR 16570, DOI 10.1090/S0002-9947-1946-0016570-3

M. Kac and A. J. F. Siegert, An explicit representation of a stationary Gaussian process, Ann. Math. Statistics 18 (1947), 438–442. MR 21672, DOI 10.1214/aoms/1177730391

J. Kiefer, On Bahadur’s representation of sample quantiles, Ann. Math. Statist. 38 (1967), 1323–1342. MR 217844, DOI 10.1214/aoms/1177698690

J. Kiefer, On the deviations in the Skorokhod-Strassen approximation scheme, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13 (1969), 321–332. MR 256461, DOI 10.1007/BF00539208

J. Kiefer, Deviations between the sample quantile process and the sample $\textrm {df}$, Nonparametric Techniques in Statistical Inference (Proc. Sympos., Indiana Univ., Bloomington, Ind., 1969) Cambridge Univ. Press, London, 1970, pp. 299–319. MR 0277071

J. Kiefer, Skorohod embedding of multivariate RV’s, and the sample DF, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 (1972), no. 1, 1–35. MR 1554013, DOI 10.1007/BF00532460

A. N. Kolmogorov (1931), Eine Verallgemeinerung des Laplace-Liapunovschen Stazes, Izv. Akad. Nauk SSSR Ser. Fiz-Mat., 959-962.

A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin-New York, 1977 (German). Reprint of the 1933 original. MR 0494348

A. N. Kolmogorov (1933b), Sulla determinazione empirica di una legge di distribuzione, Giorn. 1st. Ital. Attuari 4, 83-91.

A. N. Kolmogorov (1933c), Über die Grenzwertsätze der Wahrscheinlichkeitsrechnung, Izv. Akad. Nauk SSSR Ser. Fiz-Mat., 363-372.

J. Komlós, P. Major and G. Tusnády (1975; 1976), An approximation of partial sums of independent rv's and the sample df. I, II, Z. Wahrsch. Verw. Gebiete 32, 111-131; 34, 33-58.

Michel Loève, Probability theory, 3rd ed., D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748

K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684

E. J. G. Pitman, Some basic theory for statistical inference, Monographs on Applied Probability and Statistics, Chapman and Hall, London; John Wiley & Sons, Inc., New York, 1979. MR 549771

David Pollard, Convergence of stochastic processes, Springer Series in Statistics, Springer-Verlag, New York, 1984. MR 762984, DOI 10.1007/978-1-4612-5254-2

Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen. 1 (1956), 177–238 (Russian, with English summary). MR 0084896

A. V. Skorohod, Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen. 1 (1956), 289–319 (Russian, with English summary). MR 0084897

A. V. Skorohod, Issledovaniya po teorii sluchaĭ nykh protsessov (Stokhasticheskie differentsial′nye uravneniya i predel′nye teoremy dlya protsessov Markova), Izdat. Kiev. Univ., Kiev, 1961 (Russian). MR 0185619

N. V. Smirnov (1939a), Ob uklonenijah empiričeskoi krivoi raspredelenija, Recueil Mathématique (Matematičeskii Sbornik) N. S. 6 (48), 3-26.

N. V. Smirnov (1939b), An estimate of divergence between empirical curves of a distribution in two independent samples, Vestnik Moskov. Univ. 2, 3-14. (Russian)

N. V. Smirnov, Approximate laws of distribution of random variables from empirical data, Uspehi Matem. Nauk 10 (1944), 179–206 (Russian). MR 0012387

V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 211–226 (1964). MR 175194, DOI 10.1007/BF00534910

Volker Strassen, Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 315–343. MR 0214118

R. R. Bahadur (1966), A note on quantiles in large samples, Ann. Math. Statist. 37, 577-580. MR 0189095




J. Bernoulli (1713), Ars Coniectandi. I-II, III-IV, Oswald's Klassiker der Exacten Wissenschaften, No. 108, W. Engelmann, Leipzig, 1899.




P. Billingsley (1968), Convergence of probability measures, Wiley, New York. MR 233396




L. Brieman (1968), Probability, Addison-Wesley, Reading, Mass.




J. Bretagnolle (1980), Statistique de Kolmogorov-Smirnov pour un enchantillon nonequireparti, Colloq. Internat. CNRS 307, 39-44. MR 716526




D. R. Brillinger (1969), The asymptotic representation of the sample distribution function, Bull. Amer. Math. Soc. 75, 545-547. MR 243659




F. P. Cantelli (1933), Sulla determinazione empirica delle leggi di probabilità, Giorn. 1st. Ital. Attuari 4, 421-424.




M. Csörgő (1983), Quantile processes with statistical applications, Regional Conf. Ser. on Appl. Math. vol. 42, SIAM, Philadelphia, Pa. MR 745130




M. Csörgő, S. Csörgő and L. Horváth (1986), An asymptotic theory for empirical reliability and concentration processes, Lecture Notes in Statistics, vol. 33, Springer-Verlag, Berlin and New York. MR 856407




M. Csörgő and P. Révész (1975), A new method to prove Strassen-type laws of invariance principle. I, II, Z. Wahrsch. Verw. Gebiete 31, 255-260; 261-269. MR 1554018




M. Csörgő and P. Révész (1981), Strong approximations in probability and statistics, Akadémiai Kiadó, Budapest-Academic Press, New York, 1981. MR 666546




M. Donsker (1951), An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc. 6, 1-12. MR 40613




M. Donsker (1952), Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Statist. 23, 277-281. MR 47288




J. L. Doob (1949), Heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Statist. 20, 393-403. MR 30732




R. M. Dudley (1984), A course on empirical processes, Lecture Notes in Math., vol. 1097, Springer-Verlag, Berlin and New York. MR 876079




P. Erdős and M. Kac (1946), On certain limit theorems of the theory of probability, Bull. Amer. Math. Soc. 52, 292-302. MR 15705




W. Feller (1948), On the Kolmogorov-Smirnov limit theorems for empirical distributions, Ann. Math. Statist. 19, 177-189. MR 25108




P. Gaenssler (1983), Empirical processes, IMS Lecture Notes-Monograph Series, vol. 3, Institute of Mathematical Statistics, Hayward, Calif. MR 744668




V. Glivenko (1933), Sulla determinazione empirica della legge di probabilità, Giorn. Ist. Ital. Attuari 4, 92-99.




B. V. Gnedenko and V. S. Korolyuk (1951), On the maximum discrepancy between two empirical distributions, Selected Transl. Math. Statist. Prob. 1 (1961), 13-16; original in Dokl. Akad. Nauk SSSR 80, 525. MR 116418




M. Kac (1946), On the average of a certain Wiener functional and a related limit theorem in calculus of probability, Trans. Amer. Math. Soc. 59, 404-414. MR 16570




M. Kac and A. J. F. Siegert (1947), An explicit representation of a stationary Gaussian process, Ann. Math. Statist. 18, 438-442. MR 21672




J. Kiefer (1967), On Bahadur's representation of sample quantiles, Ann. Math. Statist. 38, 1323-1342. MR 217844




J. Kiefer (1969), On the deviations in the Skorokhod-Strassen approximation scheme, Z. Wahrsch. Verw. Gebiete 13, 321-332. MR 256461




J. Kiefer (1970), Deviations between the sample quantile process and the sample df, Nonparametric Techniques in Statistical Inference (M. L. Puri, éd. ), Cambridge Univ. Press, Cambridge. MR 277071




J. Kiefer (1972), Skorohod embedding of multivariate RV's and the sample DF, Z. Wahrsch. Verw. Gebiete 24, 1-35. MR 1554013




A. N. Kolmogorov (1931), Eine Verallgemeinerung des Laplace-Liapunovschen Stazes, Izv. Akad. Nauk SSSR Ser. Fiz-Mat., 959-962.




A. N. Kolmogorov (1933a), Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin and New York. MR 494348




A. N. Kolmogorov (1933b), Sulla determinazione empirica di una legge di distribuzione, Giorn. 1st. Ital. Attuari 4, 83-91.




A. N. Kolmogorov (1933c), Über die Grenzwertsätze der Wahrscheinlichkeitsrechnung, Izv. Akad. Nauk SSSR Ser. Fiz-Mat., 363-372.




J. Komlós, P. Major and G. Tusnády (1975; 1976), An approximation of partial sums of independent rv's and the sample df. I, II, Z. Wahrsch. Verw. Gebiete 32, 111-131; 34, 33-58.




M. Loève (1955), Probability theory, Van Nostrand, New York. MR 203748




K. R. Parthasarathy (1967), Probability measures on metric spaces, Academic Press, New York. MR 226684




E. Pitman (1979), Some basic theory for statistical inference, Chapman & Hall, London. MR 549771




D. Pollard (1984), Convergence of stochastic processes, Springer-Verlag, Berlin and New York. MR 762984




Yu. V. Prohorov (1956), Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl. 1, 157-214. MR 84896




A. V. Skorokhod (1956), Limit theorems for stochastic processes, Theory Probab. Appl. 1, 261-290. MR 84897




A. V. Skorokhod (1961), Studies in the theory of random processes, Kiev Univ.; Addison-Wesley, Reading, Mass., 1965 (translation). MR 185620




N. V. Smirnov (1939a), Ob uklonenijah empiričeskoi krivoi raspredelenija, Recueil Mathématique (Matematičeskii Sbornik) N. S. 6 (48), 3-26.




N. V. Smirnov (1939b), An estimate of divergence between empirical curves of a distribution in two independent samples, Vestnik Moskov. Univ. 2, 3-14. (Russian)




N. V. Smirnov (1944), Approximate laws of distribution of random variables from empirical data, Uspekhi Mat. Nauk 10, 179-206. (Russian) MR 12387




V. Strassen (1964), An invariance principle for the law of the iterated logarithm, Z. Wahrsch. Verw. Gebiete 3, 211-226. MR 175194




V. Strassen (1967), Almost sure behaviour of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. and Probab., vol. 2, pp. 315-343, Univ. of California Press, Berkeley, Calif. MR 214118