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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A Feynman–Kac approach to a paper of Chung and Feller on fluctuations in the coin-tossing game
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by F. Alberto Grünbaum
Proc. Amer. Math. Soc. 150 (2022), 4027-4036
DOI: https://doi.org/10.1090/proc/14758
Published electronically: June 16, 2022

Abstract:

A classical result of K. L. Chung and W. Feller deals with the partial sums $S_k$ arising in a fair coin-tossing game. If $N_n$ is the number of “positive” terms among $S_1$, $S_2$, …, $S_n$ then the quantity $P(N_{2n} = 2r)$ takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for $P(N_{2n+1} = r)$, $r = 0$, $1$, $2$, …, $2n+1$. We get to this ansatz by adaptating the Feynman–Kac methodology.
References
  • Kai Lai Chung and W. Feller, On fluctuations in coin-tossing, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 605–608. MR 33459, DOI 10.1073/pnas.35.10.605
  • E. Csáki, A discrete Feynman-Kac formula, J. Statist. Plann. Inference 34 (1993), no. 1, 63–73. MR 1209990, DOI 10.1016/0378-3758(93)90034-4
  • P. Dirac, The Lagrangian in quantum mechanics, Phys. Z. Sowjetunion, 3 (1933), no.1, pp. 64–72.
  • P. Erdös and M. Kac, On the number of positive sums of independent random variables, Bull. Amer. Math. Soc. 53 (1947), 1011–1020. MR 23011, DOI 10.1090/S0002-9904-1947-08928-X
  • William Feller, An Introduction to Probability Theory and Its Applications. Vol. I, John Wiley & Sons, Inc., New York, N.Y., 1950. MR 0038583
  • R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Physics 20 (1948), 367–387. MR 0026940, DOI 10.1103/revmodphys.20.367
  • Richard P. Feynman, The concept of probability in quantum mechanics, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley-Los Angeles, Calif., 1951, pp. 533–541. MR 0047541
  • F. A. Grünbaum, L. Velázquez, and J. Wilkening, Occupation time for classical and quantum walks, From operator theory to orthogonal polynomials, combinatorics, and number theory—a volume in honor of Lance Littlejohn’s 70th birthday, Oper. Theory Adv. Appl., vol. 285, Birkhäuser/Springer, Cham, [2021] ©2021, pp. 197–212. MR 4367468, DOI 10.1007/978-3-030-75425-9_{1}1
  • F. Alberto Grünbaum and Caroline McGrouther, Occupation time for two dimensional Brownian motion in a wedge, Recent advances in nonlinear partial differential equations and applications, Proc. Sympos. Appl. Math., vol. 65, Amer. Math. Soc., Providence, RI, 2007, pp. 31–45. MR 2381872, DOI 10.1090/psapm/065/2381872
  • Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected. MR 0345224
  • M. Jeanblanc, J. Pitman, and M. Yor, The Feynman-Kac formula and decomposition of Brownian paths, Mat. Apl. Comput. 16 (1997), no. 1, 27–52 (English, with English and Portuguese summaries). MR 1458521
  • M. Kac, On some connections between probability theory and differential and integral equations, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley-Los Angeles, Calif., 1951, pp. 189–215. MR 0045333
  • Mark Kac, Probability and related topics in physical sciences, Lectures in Applied Mathematics, Interscience Publishers, London-New York, 1959. With special lectures by G. E. Uhlenbeck, A. R. Hibbs, and B. van der Pol. MR 0102849
  • M. Kac, Integration in function spaces and some of its applications, Accademia Nazionale dei Lincei, Pisa, 1980. Lezioni Fermiane. [Fermi Lectures]. MR 660839
  • J. L. Lagrange, Miscellanea Tuarinensia, t. V, 1770-1773 Problem 1 in Memoire sur l’utilité de la méthode de prendre le milieu entre les résultats de plusiers observations, pages 173–185 of Lagrange’s Complete works, vol. 2.
  • Paul Lévy, Sur certains processus stochastiques homogènes, Compositio Math. 7 (1939), 283–339 (French). MR 919
  • Henry McKean, Probability: the classical limit theorems, Cambridge University Press, Cambridge, 2014. MR 3445370, DOI 10.1017/CBO9781107282032
  • J. Pitman, Random weighted averages, partition structures and generalized arcsine laws, arXiv:1804.07896, to appear in Probability Surveys.
  • Jim Pitman, Partition structures derived from Brownian motion and stable subordinators, Bernoulli 3 (1997), no. 1, 79–96. MR 1466546, DOI 10.2307/3318653
  • Jim Pitman and Marc Yor, Arcsine laws and interval partitions derived from a stable subordinator, Proc. London Math. Soc. (3) 65 (1992), no. 2, 326–356. MR 1168191, DOI 10.1112/plms/s3-65.2.326
  • Jim Pitman, private communication, 2018.
  • Alfréd Rényi, Legendre polynomials and probability theory, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 3(4) (1960/61), 247–251. MR 133144
  • Barry Simon, Functional integration and quantum physics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. MR 2105995, DOI 10.1090/chel/351
  • Daniel W. Stroock, Probability theory, an analytic view, Cambridge University Press, Cambridge, 1993. MR 1267569
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Bibliographic Information
  • F. Alberto Grünbaum
  • Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
  • MR Author ID: 77695
  • ORCID: 0000-0001-9663-4283
  • Received by editor(s): October 25, 2018
  • Published electronically: June 16, 2022
  • Communicated by: Mourad Ismail
  • © Copyright 2022 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 150 (2022), 4027-4036
  • MSC (2020): Primary 60J10, 60J65, 81Q30
  • DOI: https://doi.org/10.1090/proc/14758
  • MathSciNet review: 4446249