Probability, random processes, and ergodic properties

Probability, Random Processes, and Ergodic Properties

Volumn , Issue , 2009, Pages 1-322

  Gray, Robert M ^a  

^a Stanford University   (United States)

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EID: 84892214744     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-1-4419-1090-5     Document Type: Book

References (114)

1. R. B. Ash. Real Analysis and Probability. Academic Press, New York, 1972.
2. P. J. Bickel and D. A. Freedman. Some asymptotic theory for the bootstrap. Annals of Statistics, Vol. 9, No. 1 196-1217, 1981.
3. P. Billingsley. Ergodic Theory and Information. Wiley, New York, 1965.
4. P. Billingsley. Convergence of Probability Measures. Wiley Series in Probability and Mathematical Statistics. Wiley, New York, 1968.
5. P. Billingsley. Weak Convergence of Measures: Applications in Probability. SIAM, Philadelphia, 1971.
6. O. J. Bjornsson. A note on the characterization of standard borel spaces. Math. Scand., Vol. 47, 135-136, 1980.
7. D. Blackwell. On a class of probability spaces. In Proc. 3rd Berkeley Symposium on Math. Sci. and Prob., Vol. II, 1-6, Berkeley, 1956. Univ. California Press.
8. N. Bourbaki. Elements de Mathematique, Livre VI, Integration. Hermann, Paris, 1956-1965.
9. S. Boyd and L. Vandenberghe, Convex Optimization Cambridge University Press, Cambridge, 2004.
10. L. Breiman. Probability. Addison-Wesley, Menlo Park, Calif., 1968.
11. J. C. Candy. A use of limit cycle oscillations to obtain robust analog-To-digital converters. IEEE Trans. Comm., Vol. COM-22, 298-305, March 1974.
12. J. P. R. Christensen. Topology and Borel Structure, volume Mathematics Studies 10. North-Holland/American Elsevier, New York, 1974.
13. W. Chou, unpublished.
14. K. L. Chung. A Course in Probability Theory. Academic Press, New York, 1974.
15. D. C. Cohn. Measure Theory. Birkhauser, New York, 1980.
16. H. Cramér. Mathematical Methods of Statistics. Princeton University Press, Princeton, NJ, 1946.
17. M. Denker, C. Grillenberger, and K. Sigmund. Ergodic Theory on Compact Spaces, volume 57 of Lecture Notes in Mathematics. Springer-Verlag, New York, 1970.
18. R.L. Dobrushin Prescribing a system of random variables by conditional distributions. Theor. Prob. Appl. 15:458-486, 1970.
19. R. Dobrushin. private correspondence.
20. J. L. Doob. Measure Theory. Graduate Texts in Mathematics 143, Springer, New York, 1994.
21. J. L. Doob. Stochastic Processes. Wiley, New York, 1953.
22. Y. N. Dowker. Finite and sigma-finite invariant measures. Ann. of Math., Vol. 54, 595-608, November 1951.
23. Y. N. Dowker. On measurable transformations in finite measure spaces. Ann. of Math., Vol. 62, 504-516, November 1955.
24. R. M. Dudley. Distances of probability measures and random variables. Ann. of Math. Statist., Vol. 39, 1563-1572, 1968.
25. M. Fréchet, Sur la distance de deux lois de probabilité. C. R. Acad. Sci. Paris, Vol. 244, 689â̌A̧S692. 366. Frisch, 1957.
26. N. A. Friedman. Introduction to Ergodic Theory. Van Nostrand Reinhold Company, New York, 1970.
27. R. G. Gallager. Information Theory and Reliable Communication John Wiley and Sons, New York, 1968.
28. A. Garsia. A simple proof of E. Hoph's maximal ergodic theorem. J. Math. Mech., Vol. 14, 381-382, 1965.
29. C. Gini Sulla misura della concentrazione e della variabilit'a del caratteri. Atti del R. Instituto Veneto, 1913-1914, 1914.
30. R. M. Gray. Entropy and Information Theory. Springer-Verlag http:ee.stanford.edu/-gray/it.html. (Second edition in preparation.).
31. R. M. Gray, "Sliding-block source coding, "IEEE Trans. on Info. Theory, Vol. IT-21, No. 4, 357-368, July 1975.
32. R. M. Gray, "Time-invariant trellis encoding of ergodic discrete-Time sources with a fidelity criterion, " IEEE Trans. on Info. Theory, Vol. IT-23, 71-83, Jan. 1977.
33. R. M. Gray. Oversampled sigma-delta modulation. IEEE Trans. Comm., Vol. COM- 35, 481-489, April 1987.
34. R. M. Gray and L. D. Davisson. Source coding without the ergodic assumption. IEEE Trans. Inform. Theory, Vol. IT-20, 502-516, 1974.
35. R. M. Gray and L. D. Davisson, "Quantizer Mismatch, "IEEE Trans. on Comm., Vol. COM-23, No. 4, 439-442, April 1975.
36. R. M. Gray and L. D. Davisson. Random Processes: A Mathematical Approach for Engineers. Prentice-Hall, Englewood Cliffs, New Jersey, 1986.
37. R.M. Gray and L. D. Davisson, Introduction to Statistical Signal Processing, Cambridge University Press, Cambridge, UK, 2004.
38. R. M. Gray, R. M. Gray, and L. D. Davisson, editors. Ergodic and Information Theory, volume 19 of Benchmark Papers in Electrical Engineering and Computer Science. Dowden, Hutchinson, & Ross, Stroudsbug PA, 1977.
39. R. M. Gray and J. C. Kieffer. Asymptotically mean stationary measures. Ann. Probab., Vol. 8, 962-973, 1980.
40. R. M. Gray, D. L. Neuhoff and D. S. Ornstein, "Nonblock source coding with a fidelity criterion, "Annals of Prob., Vol. 3, No. 3, 478-491, June 1975.
41. R. M. Gray, D. L. Neuhoff and J. K. Omura, "Process definitions of distortion rate functions and source coding theorems, "IEEE Trans. on Info. Theory, Vol. IT-21, No. 5, 524-532, Sept. 1975.
42. R. M. Gray, D. L. Neuhoff, and P. C. Shields. A generalization of Ornstein's d-bar distance with applications to information theory. Ann. Probab., Vol. 3, 315-328, April 1975.
43. R. M. Gray and D. S. Ornstein, "Sliding-block joint source/noisy-channel coding theorems, "IEEE Trans. on Info. Theory, Vol. IT-22, 682-690, Nov. 1976.
44. P. R. Halmos. Invariant measures. Ann. of Math., Vol. 48, 735-754, 1947.
45. P. R. Halmos. Measure Theory. Van Nostrand Reinhold, New York, 1950.
46. P. R. Halmos. Lectures on Ergodic Theory. Chelsea, New York, 1956.
47. P. R. Halmos. Introduction to Hilbert Space. Chelsea, New York, 1957.
48. P.R. Halmos and J. von Neumann. "Operator methods in classical mechanics. II" Ann. of Math., 43 : 2 (1942) 332â̌A̧S-350.
49. J. M. Hammersley and D. J. A. Welsh. First-passage, percolation, subadditive processes, stochastic networks, and generalized renewal theory. Bernoulli-Bayes- Laplace Anniversary Volume, 1965.
50. F. R. Hample. A general qualitative definition of robustness. Ann. of Math. Statist., Vol. 42, 1887-1896, 1971.
51. W. Hoeffding. Maβstabinvariante korrelationstheory Schriften des Mathematischen sInstituts für Angewandte Mathematik der Universit [̈at Berlin Vol. 5, 181- 233, 1940.
52. E. Hoph. Ergodentheorie. Springer-Verlag, Berlin, 1937.
53. H. Inose and Y. Yasuda. A unity bit coding method by negative feedback. Proc. IEEE, Vol. 51, 1524-1535, November 1963.
54. K. Jacobs. The ergodic decomposition of the Kolmogorov-Sinai invariant. In F. B. Wright and F. B. Wright, editors, Ergodic Theory. Academic Press, New York, 1963.
55. R. Jones. New proof for the maximal ergodic theorem and the hardy-littlewood maximal inequality. Proc. Amer. Math. Soc., Vol. 87, 681-684, 1983.
56. L.V. Kantorovich, "On one effective method of solving certain classes of extremal problems, " Dokl. Akad. Nauk, Vol. 28, 212â̌A̧S-215, 1940.
57. L.V. Kantorovich, "On the transfer of masses, " Uspekhi Mat. Nauk, Vol. 37, 7-8, 1942.
58. L.V. Kantorovich, "On a problem of Monge, " Dokl. Akad. Nauk, Vol. 3, 225-226, 1948.
59. I. Katznelson and B. Weiss. A simple proof of some ergodic theorems. Israel Journal of Mathematics, Vol. 42, 291-296, 1982.
60. J. F. C. Kingman. The ergodic theory of subadditive stochastic processes. Ann. Probab., Vol. 1, 883-909, 1973.
61. A. N. Kolmogorov. Foundations of the Theory of Probability. Chelsea, New York, 1950.
62. U. Krengel. Ergodic Theorems. De Gruyter Series in Mathematics. De Gruyter, New York, 1985.
63. N. Kryloff and N. Bogoliouboff. La théorie générale de la mesure dans son application à l'étude des systèmes de la mecanique non linéaire. Ann. of Math., Vol. 38, 65-113, 1937.
64. V. I. Levenshtein. Binary codes capable of correcting deletions, insertions, and reversals. Sov. Phys.-Dokl., Vol. 10, 707-710, 1966.
65. E. Levina and P. Bickel. The earth mover's distance is the Mallows distance: Some insights from statistics. Proceedings Eighth IEEE International Conference on In Computer Vision (ICCV 2001), Vol.2, 251-256, 2001.
66. M. Loeve. Probability Theory. D. Van Nostrand, Princeton, New Jersey, 1963. Third Edition.
67. D. G. Luenberger. Optimization by Vector Space Methods. Wiley, New York, 1969.
68. G. Mackey. Borel structures in groups and their duals. Trans. Amer. Math. Soc., Vol. 85, 134-165, 1957.
69. L. E. Maistrov. Probability Theory: A Historical Sketch. Academic Press, New York, 1974. Translated by S. Kotz.
70. C.L. Mallows. A note on asymptotic joint normality Ann. Math.l Stat. Vol. 43, 508âǍŞ 515, 1972.
71. G. Monge. Mémoire sur la théorie des déblais et des remblais, 1781.
72. J. Moser, E. Phillips, and S. Varadhan. Ergodic Theory: A Seminar. Courant Institute of Math. Sciences, New York, 1975.
73. J. Nedoma. On the ergodicity and r-ergodicity of stationary probability measures. Z. Wahrsch. Verw. Gebiete, Vol. 2, 90-97, 1963.
74. David Lee Neuhoff Source coding and distance measures on random processes PhD dissertation, Stanford University August 1974.
75. D.L. Neuhoff, R. M. Gray and L. D. Davisson. Fixed rate universal block source coding with a fidelity criterion. IEEE Trans. on Info. Theory, Vol. IT-21, No. 5, 511-523, Sept. 1975.
76. D. S. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math., Vol. 4, 337-352, 1970.
77. D. Ornstein. An application of ergodic theory to probability theory. Ann. Probab., Vol. 1, 43-58, 1973.
78. D. Ornstein. Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, New Haven, 1975.
79. D. Ornstein and B. Weiss. The Shannon-McMillan-Breiman theorem for a class of amenable groups. Israel J. of Math, Vol. 44, 53-60, 1983.
80. J. C. Oxtoby. Ergodic sets. Bull. Amer. Math. Soc., Vol. 58, 116-136, 1952.
81. P. Papantoni-Kazakos and R. M. Gray. Robustness of estimators on stationary observations. Ann. Probab., Vol. 7, 989-1002, Dec. 1979.
82. K. R. Parthasarathy. Probability Measures on Metric Spaces. Academic Press, New York, 1967.
83. I. Paz. Stochastic Automata Theory. Academic Press, New York, 1971.
84. K. Petersen. Ergodic Theory. Cambridge University Press, Cambridge, 1983.
85. H. Poincaré. Les méthodes nouvelles de la mécanique céleste, volume I, II, III. Gauthiers-Villars, Paris, 1892, 1893, 1899. Also Dover, New York, 1957.
86. S.T. Rachev Probability metrics and the stability of stochastic models. John Wiley & Sons Ltd., Chichester, 1991.
87. S.T. Rachev and L. Rüschendorf Mass Transportation Problems Vol. I: Theory, Vol. II: Applications. Probability and its applications. Springer- Verlag, New York, 1998.
88. O. W. Rechard. Invariant measures for many-one transformations. Duke J. Math., Vol. 23, 477-488, 1956.
89. V. A. Rohlin. Selected topics from the metric theory of dynamical systems. Uspechi Mat. Nauk., Vol. 4, 57-120, 1949. English translation: Amer. Math. Soc. Transl. (2) Vol. 49, 171-240, 1966.
90. V. A. Rohlin. On the fundamental ideas of measure theory, Mat. Sb., Vol. 25, 107-150, 1949. English translation: Amer. Math. Soc. Transl.Amer. Math. Soc. Transl., .
91. Y. Rozanov. Stationary Random Processes. Holden Day, San Francisco, 1967.
92. Y. Rubner, C. Tomasi, and L.J. Guibas. A metric for distributions with applications to image databases. In Proceedings of the IEEE International Conference on Computer Vision, 59-66. Bombay, India, Jan. 1998.
93. W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, New York, 1964.
94. L. Schwartz. Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press, Oxford, 1973.
95. C. E. Shannon. A mathematical theory of communication. Bell Syst. Tech. J., Vol. 27, 379-423, 623-656, 1948.
96. C. E. Shannon. Coding theorems for a discrete source with a fidelity criterion. In IRE National Convention Record, Part 4, 142-163, 1959.
97. P. C. Shields. The Theory of Bernoulli Shifts. The University of Chicago Press, Chicago, III., 1973.
98. P. C. Shields. The ergodic and entropy theorems revisited. IEEE Trans. Inform. Theory, Vol. IT-33, 263-266, 1987.
99. P. C. Shields. The Ergodic Theory of Discrete Sample Paths. Graduate Studies in Mathematics, Vol. 13. American Mathematical Society, Providence, Rhode Island, 1991.
100. G. F. Simmons. Introduction to Topology and Modern Analysis. McGraw-Hill, New York, 1963.
101. Ya. G. Sinai. Introduction to Ergodic Theory. Mathematical Notes, Princeton University Press, Princeton, 1976.
102. M. Smorodinsky. A partition on a Bernoulli shift which is not weak Bernoulli Math. Systems Theory, Vol. 5, No. 3, 201-203, 1971.
103. M. Smorodinsky. Ergodic Theory, Entropy in Lecture Notes in Mathematics, Vol. 214 Springer-Verlag, New York, 1971.
104. V. Strassen. The existence of probability measures with given marginals. Ann. of Math. Statist., Vol. 36, 423-429, 1965.
105. E. Tanaka and T. Kasai. Synchronization and subsititution error correcting codes for the Levenshtein metric. IEEE Trans. Inform. Theory, Vol. IT-22, 156- 162, 1976.
106. A. H. Tchen. Inequalities for distributions with given marginals. The Annals of Probability, Vol. 8, No. 4, 814-827, August 1980.
107. S. S. Vallender. Computing the Wasserstein distance between probability distributions on the line. Theory Probab. Appl., Vol. 18, 824-827, 1973.
108. L. N. Vasershtein. Markov processes on countable product space describing large systems of automata. Problemy Peredachi Informatsii, Vol. 5, 64-73, 1969.
109. A.M. Vershik. The Kantorovich metric: The initial history and little-known applications. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 312, Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 11 (2004), 69â̌A̧S085, 311.
110. C. Villani. Topics in Optimal Transportation Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003.
111. C. Villani. Optimal Transport, Old and New Spring, 2009.
112. J. von Neumann. Zur operatorenmethode in der klassischen mechanik. Ann. of Math., Vol. 3, 587-642, 1932.
113. P. Walters. Ergodic Theory-Introductory Lectures. Lecture Notes in Mathematics No. 458. Springer-Verlag, New York, 1975.
114. F. B. Wright. The recurrence theorem. Amer. Math. Monthly, Vol. 68, 247-248, 1961.