Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T00:09:22.443Z Has data issue: false hasContentIssue false
  • English
  • Français

Doob: A Half-Century on

Part of: History of mathematics and mathematicians Probability theory and stochastic processes

Published online by Cambridge University Press:  14 July 2016

N. H. Bingham
N. H. Bingham*
Affiliation:
University of Sheffield
*
∗Postal address: Department of Probability and Statistics, University of Sheffield, Sheffield S3 7RH, UK. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Probability theory, and its dynamic aspect stochastic process theory, is both a venerable subject, in that its roots go back to the mid-seventeenth century, and a young one, in that its modern formulation happened comparatively recently - well within living memory. The year 2003 marked the seventieth anniversary of Kolmogorov's Grundbegriffe der Wahrscheinlichkeitsrechnung, usually regarded as having inaugurated modern (measure-theoretic) probability theory. It also marked the fiftieth anniversary of Doob's Stochastic Processes. The profound and continuing influence of this classic work prompts the present piece.

Keywords

J. L. Doobstochastic processmartingalehistory

MSC classification

Secondary: 60-03: Historical (must also be assigned at least one classification number from Section 01) 01A60: 20th century
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 257 - 266
Copyright
© Applied Probability Trust 2005 

References

Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Palm-Martingale Calculus and Stochastic Recurrences (Appl. Math. (New York) 26). Springer, Berlin.Google Scholar
Bartlett, M. S. (1955). Stochastic Processes, with Special Reference to Methods and Applications. Cambridge University Press.Google Scholar
Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
Billingsley, P. (1979). Probability and Measure. John Wiley, New York.Google Scholar
Bingham, N. H. (2000). Studies in the history of probability and statistics. XLVI. Measure into probability: from Lebesgue to Kolmogorov. Biometrika 87, 145–156.CrossRefGoogle Scholar
Bingham, N. H. and Kiesel, R. (2004). Risk-Neutral Valuation. Pricing and Hedging of Financial Derivatives, 2nd edn. Springer, London.Google Scholar
Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Brémaud, P. (1981). Point Processes and Queues. Martingale Dynamics. Springer, New York.CrossRefGoogle Scholar
Brémaud, P. (1999). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York.Google Scholar
Brockwell, P. J. and Davis, R. A. (1987). Time Series: Theory and methods. Springer, New York.CrossRefGoogle Scholar
Chung, K.-L. (1960). Markov Chains with Stationary Transition Probabilities. Springer, New York.Google Scholar
Chung, K.-L. (1968). A Course in Probability Theory. Academic Press, Reading, MA.Google Scholar
Cohn, H. (ed.) (1993). Doeblin and Modern Probability. American Mathematical Society, Providence, RI.Google Scholar
Cramér, H. (1937). Random Variables and Probability Distributions (Camb. Tracts Math. 36). Cambridge University Press.Google Scholar
Cramér, H. (1976). Half a century with probability. Some personal recollections. Ann. Prob. 4, 509–546.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Davis, M. H. A. (1977). Linear Estimation and Stochastic Control. Chapman and Hall, London.Google Scholar
Doob, J. L. (1949). Time series and harmonic analysis. In Proc. Berkeley Symp. Math. Statist. Prob., University of California Press, Berkeley, CA, pp. 303–343.Google Scholar
Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.Google Scholar
Doob, J. L. (1984). Classical Potential Theory and Its Probabilistic Counterpart (Fundamental Principles Math. Sci. 262). Springer, New York.CrossRefGoogle Scholar
Doob, J. L. (1990). Review of Masani (1990). Bull. London Math. Soc. 22, 621–623.CrossRefGoogle Scholar
Durrett, R. (1984). Brownian Motion and Martingales in Analysis. Wadsworth, Belmont, CA.Google Scholar
Dynkin, E. B. (1965). Markov Processes, Vols 1, 2. Springer, Heidelberg.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. John Wiley, New York.Google Scholar
Feller, W. (1950). An Introduction to Probability Theory and Its Applications, Vol. 1. John Wiley, New York.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.Google Scholar
Fernique, X. (1975). Regularité des trajectoires des fonctions aléatoires gaussiennes. In Ecole d'Eté de Probabilités de Saint-Flour, IV-1974 (Lecture Notes Math. 480), Springer, Berlin, pp. 1–96.Google Scholar
Geiringer, H. (ed.) (1964). Richard von Mises. Mathematical Theory of Probability and Statistics. Academic Press, New York.Google Scholar
Gerber, H. U. (1986). Lebensversicherungsmathematik. Springer, Berlin.CrossRefGoogle Scholar
Gīhman, I. Ī. and Skorokhod, A. V. (1969). Introduction to the Theory of Random Processes. Saunders, Philadelphia, PA.Google Scholar
Gīhman, Ì. Ī. and Skorokhod, A. V. (1974). The Theory of Stochastic Processes, Vol. I. Springer, New York.Google Scholar
Gīhman, Ì. Ī. and Skorokhod, A. V. (1975). The Theory of Stochastic Processes, Vol. II. Springer, New York.Google Scholar
Gīhman, Ì. Ī. and Skorokhod, A. V. (1979). The Theory of Stochastic Processes, Vol. III. Springer, New York.Google Scholar
Gnedenko, B. V. and Kolmogorov, A. N. (1954). Limit Theorems for Sums of Independent Random Variables. Addison-Wesley, Reading, MA.Google Scholar
Grimmett, G. R. and Stirzaker, D. (1982). Probability and Random Processes. Oxford University Press.Google Scholar
Hall, P. G. and Heyde, C. C. (1980). Martingale Limit Theory and Its Applications. Academic Press, Reading, MA.Google Scholar
Halmos, P. R. (1950). Measure Theory. Van Nostrand, New York.CrossRefGoogle Scholar
Harrison, J. M. and Pliska, S. R. (1981). Martingales and arbitrage in multi-period securities markets. Stoch. Process. Appl. 11, 215–260.Google Scholar
Heyde, C. C. (1997). Quasi-Likelihood and Its Applications. A General Approach to Optimal Parameter Estimation. Springer, New York.Google Scholar
Ibragimov, I. A. and Rozanov, Yu. A. (1978). Gaussian Random Processes. Springer, New York.Google Scholar
Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
Itô, K. and McKean, H. P. (1965). Diffusion Processes and Their Sample Paths. Springer, New York.Google Scholar
Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.CrossRefGoogle Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
Janson, S. (1997). Gaussian Hilbert Spaces (Camb. Tracts Math. 129). Cambridge University Press.CrossRefGoogle Scholar
Kaczmarz, S. and Steinhaus, H. (1935). Theorie der Orthogonalreihen. Monografje Matematyczne, Warsaw.Google Scholar
Kahane, J.-P. (1985). Some Random Series of Functions, 2nd edn. Cambridge University Press.Google Scholar
Kakutani, S. (1944). Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo 20, 648–652.Google Scholar
Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Karlin, S. and Taylor, H. (1975). A First Course in Stochastic Processes. Academic Press, Reading, MA.Google Scholar
Karlin, S. and Taylor, H. (1981). A Second Course in Stochastic Processes. Academic Press, Reading, MA.Google Scholar
Kingman, J. F. C. and Taylor, S. J. (1966). An Introduction to Measure and Probability. Cambridge University Press.Google Scholar
Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin.Google Scholar
Krengel, U. (1985). Ergodic Theorems. De Gruyter, Berlin.Google Scholar
Lévy, P. (1925). Calcul des Probabilités. Gauthier-Villars, Paris.Google Scholar
Lévy, P. (1937). Théorie de l'addition des Variables Aléatoires. Gauthier-Villars, Paris.Google Scholar
Lévy, P. (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris.Google Scholar
Lindvall, T. (1991). Doeblin, W. (1915–1940). Ann. Prob. 19, 929–934.Google Scholar
Loève, M. (1955). Probability Theory. Van Nostrand, New York.Google Scholar
Loève, M. (1973). Obituary: Paul Lévy, 1886–1971. Ann. Prob. 1, 1–18.Google Scholar
Loève, M. (1977). Probability Theory. I., 4th edn. Springer, New York.Google Scholar
Loève, M. (1978). Probability Theory. II, 4th edn. Springer, New York.Google Scholar
Lynch, J. D. (2000). The Galton–Watson process revisited: some martingale relationships and applications. J. Appl. Prob. 37, 322–328.Google Scholar
Lyons, T. J. and Qian, Z. (2002). System Control and Rough Paths. Oxford University Press.Google Scholar
McKean, H. P. (1969). Stochastic Integrals. Academic Press, New York.Google Scholar
Masani, P. R. (1990). Norbert Wiener. Birkhäuser, Basel.Google Scholar
Meyer, P.-A. (1966). Probability and Potentials. Blaisdell, Waltham, MA.Google Scholar
Meyer, P.-A. (1976). Un cours sur les intégrales stochastiques. In Séminaire de Probabilités (Lecture Notes Math. 511), Vol. X, Springer, Heidelberg, pp. 245–398.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.Google Scholar
Neveu, J. (1975). Discrete-Parameter Martingales. North-Holland, Amsterdam.Google Scholar
Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge University Press.Google Scholar
Øksendal, B. (1985). Stochastic Differential Equations. An Introduction with Applications. Springer, New York.Google Scholar
Protter, P. (1990). Stochastic Integration and Differential Equations. A New Approach (Appl. Math. (New York) 27). Springer, New York.Google Scholar
Révész, P. (1968). The Laws of Large Numbers. Academic Press, New York.Google Scholar
Revuz, D. (1984). Markov Chains, 2nd edn. North-Holland, Amsterdam.Google Scholar
Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.CrossRefGoogle Scholar
Rogers, L. C. G. (1989). Review of Sharpe (1988). Bull. London Math. Soc. 21, 622–623.Google Scholar
Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes and Martingales, Vol. 1, Foundations. John Wiley, Chichester.Google Scholar
Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes and Martingales, Vol. 2, Itô Calculus. John Wiley, Chichester.Google Scholar
Sato, K.-I. (1999). Infinite Divisibility and Lévy Processes. Cambridge University Press.Google Scholar
Sharpe, M. J. (1988). General Theory of Markov Processes. John Wiley, New York.Google Scholar
Spitzer, F. (1964). Principles of Random Walk. Van Nostrand, Princeton, NJ.CrossRefGoogle Scholar
Stone, M. H. (1932). Linear Transformations in Hilbert Space and Their Applications to Analysis (AMS Colloq. Pub. 15). American Mathematical Society, Providence, RI.Google Scholar
Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusions. Springer, New York.Google Scholar
Whittle, P. (1963). Prediction and Regulation by Linear Least-Squares Methods. English Universities Press, London.Google Scholar
Wiener, N. (1949). Extrapolation, Interpolation and Smoothing of Stationary Time Series. John Wiley, New York.Google Scholar
Wiener, N. (1976). Collected Works, Vol. I. MIT Press, Cambridge, MA.Google Scholar
Wiener, N. (1979). Collected Works with Commentaries, Vol. II. MIT Press, Cambridge, MA.Google Scholar
Wiener, N. (1981). Collected Works with Commentaries, Vol. III. MIT Press, Cambridge, MA.Google Scholar
Wiener, N. (1985). Collected Works with Commentaries, Vol. IV. MIT Press, Cambridge, MA.Google Scholar
Zygmund, A. (1935). Trigonometric Series. Monografje Matematyczne, Warsaw (reprinted: Dover, New York, 1955).Google Scholar