Bulletin of the American Mathematical Society

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Book Information:

Authors: Allen Gut and Klaus D. Schmidt
Title: Amarts and set function processes
Additional book information: Lecture Notes in Mathematics, vol. 1042, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983, 258 pp., $12.50. ISBN 3-540-12867-0.

Erik Sparre Andersen and Børge Jessen, Some limit theorems on set-functions, Danske Vid. Selsk. Mat.-Fys. Medd. 25 (1948), no. 5, 8. MR 28376

Kenneth A. Astbury, Amarts indexed by directed sets, Ann. Probability 6 (1978), no. 2, 267–278. MR 464394, DOI 10.1214/aop/1176995572

D. G. Austin, G. A. Edgar, and A. Ionescu Tulcea, Pointwise convergence in terms of expectations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 30 (1974), 17–26. MR 358945, DOI 10.1007/BF00532860

4.

S. D. Bagchi, Almost sure convergence of classes of multivalued asymptotic martingales, Ann. Inst. H. Poincaré Sect. B (to appear).

J. R. Baxter, Pointwise in terms of weak convergence, Proc. Amer. Math. Soc. 46 (1974), 395–398. MR 380968, DOI 10.1090/S0002-9939-1974-0380968-7

Alexandra Bellow, On vector-valued asymptotic martingales, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 6, 1798–1799. MR 407966, DOI 10.1073/pnas.73.6.1798

Alexandra Bellow, Uniform amarts: a class of asymptotic martingales for which strong almost sure convergence obtains, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), no. 3, 177–191. MR 471065, DOI 10.1007/BF00534238

8.

A. Bellow, For historical record, Proc. 1983 Oberwolfach Conf. on Measure Theory, Lecture Notes in Math., vol. 1089, Springer-Verlag (to appear).

Antoine Brunel and Ulrich Krengel, Parier avec un prophète dans le cas d’un processus sous-additif, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 1, A57–A60 (French, with English summary). MR 522020

Antoine Brunel and Louis Sucheston, Sur les amarts faibles à valeurs vectorielles, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 17, Aiii, A1011–A1014 (French, with English summary). MR 415761

Antoine Brunel and Louis Sucheston, Une caractérisation probabiliste de la séparabilité du dual d’un espace de Banach, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 22, A1469–A1472 (French, with English summary). MR 440694

R. V. Chacon, A “stopped” proof of convergence, Advances in Math. 14 (1974), 365–368. MR 365688, DOI 10.1016/0001-8708(74)90038-3

R. V. Chacon and L. Sucheston, On convergence of vector-valued asymptotic martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975/76), no. 1, 55–59. MR 394859, DOI 10.1007/BF00539861

S. D. Chatterji, Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scand. 22 (1968), 21–41. MR 246341, DOI 10.7146/math.scand.a-10868

Bong Dae Choi and Louis Sucheston, Continuous parameter uniform amarts, Probability in Banach spaces, III (Medford, Mass., 1980) Lecture Notes in Math., vol. 860, Springer, Berlin-New York, 1981, pp. 85–98. MR 647956

W. J. Davis and W. B. Johnson, Weakly convergent sequences of Banach space valued random variables, Banach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976) Lecture Notes in Math., Vol. 604, Springer, Berlin, 1977, pp. 29–31. MR 0458167

J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896

Aryeh Dvoretzky, On stopping time directed convergence, Bull. Amer. Math. Soc. 82 (1976), no. 2, 347–349. MR 415760, DOI 10.1090/S0002-9904-1976-14060-8

G. A. Edgar, Asplund operators and a.e. convergence, J. Multivariate Anal. 10 (1980), no. 3, 460–466. MR 588087, DOI 10.1016/0047-259X(80)90064-0

G. A. Edgar, Additive amarts, Ann. Probab. 10 (1982), no. 1, 199–206. MR 637386

Gerald A. Edgar and Louis Sucheston, Amarts: a class of asymptotic martingales. B. Continuous parameter, J. Multivariate Anal. 6 (1976), no. 4, 572–591. MR 431370, DOI 10.1016/0047-259X(76)90005-1

G. A. Edgar and L. Sucheston, The Riesz decomposition for vector-valued amarts, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), no. 1, 85–92. MR 413252, DOI 10.1007/BF00533211

Gerald A. Edgar and Louis Sucheston, Démonstrations de lois des grands nombres par les sous-martingales descendantes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 22, 967–969 (French, with English summary). MR 631108

Nikos E. Frangos, On regularity of Banach-valued processes, Ann. Probab. 13 (1985), no. 3, 985–990. MR 799433

Nassif Ghoussoub and Louis Sucheston, A refinement of the Riesz decomposition for amarts and semiamarts, J. Multivariate Anal. 8 (1978), no. 1, 146–150. MR 494474, DOI 10.1016/0047-259X(78)90027-1

Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea, Abstract ergodic theorems, Trans. Amer. Math. Soc. 107 (1963), 107–124. MR 150611, DOI 10.1090/S0002-9947-1963-0150611-8

K. Krickeberg, Convergence of martingales with a directed index set, Trans. Amer. Math. Soc. 83 (1956), 313–337. MR 91328, DOI 10.1090/S0002-9947-1956-0091328-4

Klaus Krickeberg, Notwendige Konvergenzbedingungen bei Martingalen und verwandten Prozessen, Trans. 2nd Prague Conf. Information Theory, Publ. House Czech. Acad. Sci., Prague; Academic Press, New York 1961, pp. 279–305 (German). MR 0126294

Charles W. Lamb, A ratio limit theorem for approximate martingales, Canadian J. Math. 25 (1973), 772–779. MR 331503, DOI 10.4153/CJM-1973-079-6

30.

P. A. Meyer, Probabilities and potential, Blaisdell, Waltham, Mass., 1966.

Annie Millet, Sur la caractérisation des conditions de Vitali par la convergence essentielle des martingales, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 13, A887–A890 (French, with English summary). MR 551772

Annie Millet, Convergence and regularity of strong submartingales, Two-index random processes (Paris, 1980) Lecture Notes in Math., vol. 863, Springer, Berlin, 1981, pp. 50–58. MR 630305

Annie Millet and Louis Sucheston, La convergence essentielle des martingales bornées dans $L^{1}$ n’implique pas la condition de Vitali $V$, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 11, A595–A598 (French, with English summary). MR 531590

Annie Millet and Louis Sucheston, Characterizations of Vitali conditions with overlap in terms of convergence of classes of amarts, Canadian J. Math. 31 (1979), no. 5, 1033–1046. MR 546956, DOI 10.4153/CJM-1979-095-2

Annie Millet and Louis Sucheston, Convergence of classes of amarts indexed by directed sets, Canadian J. Math. 32 (1980), no. 1, 86–125. MR 559789, DOI 10.4153/CJM-1980-009-1

Annie Millet and Louis Sucheston, On convergence of $L_{1}$-bounded martingales indexed by directed sets, Probab. Math. Statist. 1 (1980), no. 2, 151–169 (1981). MR 626308

Annie Millet and Louis Sucheston, On regularity of multiparameter amarts and martingales, Z. Wahrsch. Verw. Gebiete 56 (1981), no. 1, 21–45. MR 612158, DOI 10.1007/BF00531972

J. Neveu, Convergence presque sûre de martingales multivoques, Ann. Inst. H. Poincaré Sect. B (N.S.) 8 (1972), 1–7 (French, with English summary). MR 0331504

J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Mathematical Library, Vol. 10, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed. MR 0402915

William D. Sudderth, A “Fatou equation” for randomly stopped variables, Ann. Math. Statist. 42 (1971), 2143–2146. MR 300370, DOI 10.1214/aoms/1177693082

Michel Talagrand, Some structure results for martingales in the limit and pramarts, Ann. Probab. 13 (1985), no. 4, 1192–1203. MR 806217

Michel Talagrand, Moyennes invariantes s’annulant sur des idéaux, Compositio Math. 42 (1980/81), no. 2, 213–216 (French). MR 596876

1.

E. S. Andersen and B. Jessen, Some limit theorems on set-functions, Danske Vid. Selsk. Math.-fys. Medd. 25, no. 5 (1948). MR 0028376

2.

K. A. Astbury, Amarts indexed by directed sets, Ann. Probab. 6 (1978), 267-278. MR 464394

3.

D. G. Austin, G. A. Edgar and A. Ionescu Tulcea, Pointwise convergence in terms of expectations, Z. Wahrsch. verw. Gebiete 30 (1974), 17-26. MR 358945

4.

S. D. Bagchi, Almost sure convergence of classes of multivalued asymptotic martingales, Ann. Inst. H. Poincaré Sect. B (to appear).

5.

J. R. Baxter, Pointwise in terms of weak convergence, Proc. Amer. Math. Soc. 46 (1974), 395-398. MR 380968

6.

A. Bellow, On vector-valued asymptotic martingales, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 1798-1799. MR 407966

7.

A. Bellow, Uniform amarts: A class of asymptotic martingales for which strong almost sure convergence obtains, Z. Wahrsch. verw. Gebiete 41 (1978), 177-191. MR 471065

8.

A. Bellow, For historical record, Proc. 1983 Oberwolfach Conf. on Measure Theory, Lecture Notes in Math., vol. 1089, Springer-Verlag (to appear).

9.

A. Brunel and U. Krengel, Parier avec un prophète dans le cas d'un processus sous-additif, C. R. Acad. Sci. Paris Sér. A 288 (1979), 57-60. MR 522020

10.

A. Brunel and L. Sucheston, Sur les amarts faibles à valeurs vectorielles, C. R. Acad. Sci. Paris Sér. A 282 (1976), 1011-1014. MR 415761

11.

A. Brunel and L. Sucheston, Une caractérisation probabiliste de la séparabilité du dual d 'un espace de Banach, C. R. Acad. Sci. Paris Sér. A 282 (1977), 1469-1472. MR 440694

12.

R. V. Chacon, A "stopped" proof of convergence, Adv. in Math. 14 (1974), 365-368. MR 365688

13.

R. V. Chacon and L. Sucheston, On convergence of vector-valued asymptotic martingales, Z. Wahrsch. verw Gebiete 33 (1975), 55-59. MR 394859

14.

S. D. Chatterji, Martingale convergence and Radon-Nikodym theorem in Banach spaces, Math. Scand. 22 (1968), 21-41. MR 246341

15.

B. D. Choi and L. Sucheston, Continuous parameter uniform amarts, Probability in Banach Spaces. III, Lecture Notes in Math., vol. 860, Springer-Verlag, 1981, pp. 85-98. MR 647956

16.

W. J. Davis and W. B. Johnson, Weakly convergent sequences of Banach space valued random variables, Banach Spaces of Analytic Functions, Lecture Notes in Math., vol. 604, Springer-Verlag, 1977, pp. 29-31. MR 458167

17.

J. L. Doob, Stochastic processes, Wiley, New York, 1953. MR 58896

18.

A. Dvoretzky, On stopping time directed convergence, Bull. Amer. Math. Soc. 82 (1976), 347-349. MR 415760

19.

G. A. Edgar, Asplund operators and a. e. convergence, J. Multivariate Anal. 10 (1980), 460-466. MR 588087

20.

G. A. Edgar, Additive amarts, Ann. Probab. 10 (1982), 199-206. MR 637386

21.

G. A. Edgar and L. Sucheston, Amarts: A class of asymptotic martingales. I, II, J. Multivariate Anal. 6 (1976), 196-221, 572-591. MR 431370

22.

G. A. Edgar, The Riesz decomposition for vector-valued amarts, Z. Wahrsch. verw. Gebiete 36 (1976), 85-92. MR 413252

23.

G. A. Edgar, Démonstrations de lois des grands nombres par les sous-martingales descendantes, C. R. Acad. Sci. Paris Sér. I 292 (1981), 967-969. MR 631108

24.

N. Frangos, On regularity of Banach valued processes, Ann. Probab. (to appear). MR 799433

25.

N. Ghoussoub and L. Sucheston, A refinement of the Riesz decomposition for amarts and semiamarts, J. Multivariate Anal. 8 (1978), 146-150. MR 494474

26.

A. and C. Ionescu Tulcea, Abstract ergodic theorems, Trans. Amer. Math. Soc. 107 (1963), 313-337. MR 150611

27.

K. Krickeberg, Convergence of martingales with a directed index set, Trans. Amer. Math. Soc. 83 (1956), 313-337. MR 91328

28.

K. Krickeberg, Notwendige Konvergenzbedingungen bei Martingalen und verwandten processen, Trans. 2nd Prague Conf. on Information theory (Czech. Acad. Sci., Prague, 1959), pp. 279-305. MR 126294

29.

C. W. Lamb, A ratio limit theorem for approximate martingales, Canad. J. Math. 25 (1973), 772-779. MR 331503

30.

P. A. Meyer, Probabilities and potential, Blaisdell, Waltham, Mass., 1966.

31.

A. Millet, Sur la caractérisation des conditions de Vitali par la convergence essentielle des martingales, C. R. Acad. Sci. Paris Sér. A 287 (1978), 887-890. MR 551772

32.

A. Millet, Convergence and regularity of strong submartingales, Processus Aleatoires à Deux Indices, Lecture Notes in Math., vol. 863, Springer-Verlag, 1981, pp. 50-58. MR 630305

33.

A. Millet and L. Sucheston, La convergence essentielle des martingales bornées dans L, C. R. Acad. Sci. Paris Sér. A 288 (1979), 595-598. MR 531590

34.

A. Millet and L. Sucheston, Characterizations of Vitali conditions with overlap in terms of convergence of classes at amarts, Canad. J. Math 31 (1979), 1033-1046. MR 546956

35.

A. Millet and L. Sucheston, Convergence of classes of amarts indexed by directed sets, Canad. J. Math. 32 (1980), 86-125. MR 559789

36.

A. Millet and L. Sucheston, On convergence of L, Probab. Math. Statist. 1 (1980), 151-169. MR 626308

37.

A. Millet and L. Sucheston, On regularity of multiparameter amarts and martingales, Z. Wahrsch. verw. Gebiete 56 (1981), 21-45. MR 612158

38.

J. Neveu, Convergence presque sur des martingales multivoques, Ann. Inst. Poncaré Sect. B. 8 (1972), 1-7. MR 331504

39.

J. Neveu, Discrete-parameter martingales, North-Holland, Amsterdam, 1975. MR 402915

40.

W. D. Sudderth, A "Fatou 's equation" for randomly stopped variables, Ann. Math. Statist. 42 (1971), 2143-2146. MR 300370

41.

M. Talagrand, Some structure results for martingales in the limit and pramarts, Ann. Probab. (to appear). MR 806217

42.

M. Talagrand, Derivation, L, Trans. Amer. Math. Soc. (to appear). MR 596876