Lester Dubins Explained

Lester Dubins (April 27, 1920 – February 11, 2010) was an American mathematician noted primarily for his research in probability theory. He was a faculty member at the University of California at Berkeley from 1962 through 2004, and in retirement was Professor Emeritus of Mathematics and Statistics.

It has been thought that, since classic red-and-black casino roulette is a game in which the house on average wins more than the gambler, that "bold play", i.e. betting one's whole purse on a single trial, is a uniquely optimal strategy. While a graduate student at the University of Chicago, Dubins surprised his teacher Leonard Jimmie Savage with a mathematical demonstration that this is not true. Dubins and Savage wrote a book that appeared in 1965 titled How to Gamble if You Must (Inequalities for Stochastic Processes) which presented a mathematical theory of gambling processes and optimal behavior in gambling situations, pointing out their relevance to traditional approaches to probability. Under the influence of the work of Bruno de Finetti, Dubins and Savage worked in the context of finitely additive rather than countably additive probability theory, thereby bypassing some technical difficulties.^[1]

Dubins was the author of nearly a hundred scholarly publications. Besides probability, some of these were on curves of minimal length under constraints on curvature and initial and final tangents (see Dubins path), Tarski's circle squaring problem, convex analysis, and geometry.

His doctoral students include Theodore Hill. Together with Gideon E. Schwarz he proved the Dubins–Schwarz theorem.

Publications

Dubins–Spanier theorems
10.2307/2372560. 79. 3. 497–516. Dubins. L. E.. On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents. American Journal of Mathematics. 1957. 2372560.
10.2307/2032685. 8. 6. 1088–1092. Dubins. Lester E.. Conditional Probability Distributions in the Wide Sense. Proceedings of the American Mathematical Society. 1957. 2032685. free.
10.2307/2309181. 67. 6. 573–574. Dubins. Lester E.. Another Proof of the Four Vertex Theorem. The American Mathematical Monthly. 1960. 2309181.
10.2307/2308657. 67. 8. 771–772. Dubins. Lester E.. On Differentiation of Series Term-By-Term. The American Mathematical Monthly. 1960. 2308657.
46. 12. 1597–1598. Dubins. Lester E.. Savage. Leonard J.. Optimal Gambling Systems. Proceedings of the National Academy of Sciences of the United States of America. 1960. 70777. 10.1073/pnas.46.12.1597. 223086. 16590787. 1960PNAS...46.1597D. free.
9. 6. 782–801. Dubins. Lester E.. Morgenthaler. George W.. Inclusion of Detection in Probability of Survival Models. Operations Research. 1961. 167046. 10.1287/opre.9.6.782.
10.2307/2033819. 14. 3. 450–453. Blackwell. David. Dubins. Lester E.. Sharp Bounds on the Distribution of the Hardy-Littlewood Maximal Function. Proceedings of the American Mathematical Society. 1963. 2033819. free.
36. 3. 800–807. Dubins. Lester E.. Freedman. David A.. A Sharper Form of the Borel-Cantelli Lemma and the Strong Law. The Annals of Mathematical Statistics. 1965. 2238191. 10.1214/aoms/1177700054. free.
53. 2. 274–275. Dubins. Lester E.. Savage. Leonard J.. A Tchebycheff-Like Inequality for Stochastic Processes. Proceedings of the National Academy of Sciences of the United States of America. 1965. 72579. 10.1073/pnas.53.2.274. 219506. 16591259. 1965PNAS...53..274D. free.
53. 5. 913–916. Dubins. Lester E.. Schwarz. Gideon. On Continuous Martingales. Proceedings of the National Academy of Sciences of the United States of America. 1965. 72837. 10.1073/pnas.53.5.913. 301348. 16591279. 1965PNAS...53..913D. free.
37. 3. 728. Dubins. Lester E.. A Note on Upcrossings of Semimartingales. The Annals of Mathematical Statistics. 1966. 2238744. 10.1214/aoms/1177699469. free.
37. 4. 837–848. Dubins. Lester E.. Freedman. David A.. Invariant Probabilities for Certain Markov Processes. The Annals of Mathematical Statistics. 1966. 2238573. 10.1214/aoms/1177699364. free.
37. 6. 1505–1509. Dubins. Lester E.. Freedman. David A.. On the Expected Value of a Stopped Martingale. The Annals of Mathematical Statistics. 1966. 2238767. 10.1214/aoms/1177699142. free.
38. 2. 601–605. Dubins. Lester E.. Teicher. Henry. Optimal Stopping When the Future is Discounted. The Annals of Mathematical Statistics. 1967. 2239174. 10.1214/aoms/1177698978. free.
39. 2. 390–393. Dubins. Lester E.. A Simpler Proof of Smith's Roulette Theorem. The Annals of Mathematical Statistics. 1968. 2239033. 10.1214/aoms/1177698402. free.
39. 6. 2094–2097. Dubins. Lester E.. On a Theorem of Skorohod. The Annals of Mathematical Statistics. 1968. 2239305. 10.1214/aoms/1177698036. free.
43. 5. 1559–1565. Dubins. Lester E.. Sharp Bounds for the Total Variance of Uniformly Bounded Semimartingales. The Annals of Mathematical Statistics. 1972. 2240078. 10.1214/aoms/1177692388. free.
1. 2. 313–316. Dubins. Lester E.. Which Functions of Stopping Times are Stopping Times?. The Annals of Probability. 1973. 2959492. 10.1214/aop/1176996983. free.
2. 3. 456–463. Dubins. Lester E.. On Lebesgue-like Extensions of Finitely Additive Measures. The Annals of Probability. 1974. 2959178. 10.1214/aop/1176996660. free.
2. 2. 243–255. Dubins. Lester E.. Meilijson. Isaac. On Stability for Optimization Problems. The Annals of Probability. 1974. 2959222. 10.1214/aop/1176996706. free.
3. 5. 741–752. Blackwell. David. Dubins. Lester E.. On Existence and Non-Existence of Proper, Regular, Conditional Distributions. The Annals of Probability. 1975. 2959116. 10.1214/aop/1176996261. free.
3. 1. 89–99. Dubins. Lester E.. Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations. The Annals of Probability. 1975. 2959267. 10.1214/aop/1176996451. free.
3. 4. 722–725. Dubins. Lester E.. Sudderth. William D.. An Example in which Stationary Strategies are not Adequate. The Annals of Probability. 1975. 2959335. 10.1214/aop/1176996312. free.
10.2307/2319963. 84. 5. 350–356. Dubins. Lester E.. Group Decision Devices. The American Mathematical Monthly. 1977. 2319963.
10.2307/2041941. 62. 1. 34–36. Dubins. Lester E.. Measurable, Tail Disintegrations of the Haar Integral are Purely Finitely Additive. Proceedings of the American Mathematical Society. 1977. 2041941.
10.2307/1998933. 232. 187–194. Dubins. Lester E.. On Everywhere-Defined Integrals. Transactions of the American Mathematical Society. 1977. 1998933. free.
2. 2. 125–134. Dubins. Lester E.. Sudderth. William D.. Persistently ϵ-Optimal Strategies. Mathematics of Operations Research. 1977. 3689649. 10.1287/moor.2.2.125. 11299/199246. free.
10.2307/2043117. 68. 3. 337–338. Dubins. Lester E.. Gilat. David. On the Distribution of Maxima of Martingale. Proceedings of the American Mathematical Society. 1978. 2043117.
10.2307/1998695. 251. 299–308. Dubins. Lester E.. Pitman. Jim. A Pointwise Ergodic Theorem for the Group of Rational Rotations. Transactions of the American Mathematical Society. 1979. 1998695. free.
7. 3. 461–476. Dubins. Lester E.. Sudderth. William D.. On Stationary Strategies for Absolutely Continuous Houses. The Annals of Probability. 1979. 2243199. 10.1214/aop/1176995047. free.
10.2307/2042335. 78. 3. 414–416. Dubins. Lester E.. Pitman. Jim. A Divergent, Two-Parameter, Bounded Martingale. Proceedings of the American Mathematical Society. 1980. 2042335. free.
10.2307/2043394. 86. 2. 273–274. Dubins. Lester E.. A Zero-One, Borel Probability which Admits of No Countably Additive Extensions. Proceedings of the American Mathematical Society. 1982. 2043394. free.
10.2307/2044607. 89. 4. 691–692. Blackwell. David. Dubins. Lester E.. An Extension of Skorohod's Almost Sure Representation Theorem. Proceedings of the American Mathematical Society. 1983. 2044607.
10.2307/2045487. 89. 3. 421–422. Dubins. Lester E.. A Gloss on a Theorem of Furstenberg. Proceedings of the American Mathematical Society. 1983. 2045487. free.
10.2307/2044985. 88. 3. 416–418. Dubins. Lester E.. Heath. David. With Respect to Tail Sigma Fields, Standard Measures Possess Measurable Disintegrations. Proceedings of the American Mathematical Society. 1983. 2044985. free.
10.2307/2044640. 91. 2. 270–274. Dubins. Lester E.. Heath. David. On Means with Countably Additive Continuities. Proceedings of the American Mathematical Society. 1984. 2044640. free.
30. 7–12. Dubins. Lester E.. The Gambler's Ruin Problem for Periodic Walks. Lecture Notes-Monograph Series. Institute of Mathematical Statistics Lecture Notes - Monograph Series. 1996. 10.1214/lnms/1215453561. 4355934. 0-940600-42-0. free.
37. 1. 393–402. Dubins. Lester E.. Gilat. David. Meilijson. Isaac. On the Expected Diameter of an L₂-Bounded Martingale. The Annals of Probability. 2009. 30244283. 10.1214/08-aop406. 0807.3571. 115157104.
Book: How to gamble if you must: inequalities for stochastic processes . McGraw–Hill series in probability and statistics . Lester E. . Dubins . Leonard J.. Savage . McGraw–Hill. 1965 .

Notes and References

Web site: Obituary for Lester Eli Dubins. David Gilat . Ted Hill . Bill Sudderth . The Bernoulli Society for Mathematical Statistics and Probability. 27 May 2010.