Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions

Preprint English OPEN
Biane, P.; Pitman, J.; Yor, M.;
  • Subject: 60E07 | Mathematics - Probability | Mathematics - Classical Analysis and ODEs | 11M06 | 60J65

This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related... View more
  • References (43)
    43 references, page 1 of 5

    [7] Ph. Biane. Decompositions of Brownian trajectories and some applications. In A. Badrikian, P-A Meyer, and J-A Yan, editors, Probability and Statistics; Rencontres Franco-Chinoises en Probabilit´es et Statistiques; Proceedings of the Wuhan meeting, pages 51-76. World Scientific, 1993.

    [8] Ph. Biane and M. Yor. Valeurs principales associ´ees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23-101, 1987.

    [9] P. Billingsley. Convergence of Probability Measures. Wiley, 1968.

    [13] K. L. Chung. A Course in Probability Theory. Academic Press, 1974. 2nd ed.

    [21] R. Durrett, D. L. Iglehart, and D. R. Miller. Weak convergence to Brownian meander and Brownian excursion. Ann. Probab., 5:117-129, 1977.

    [22] H.M. Edwards. Riemann's Zeta Function. Academic Press, New York, 1974.

    [23] L. Ehrenpreis. Fourier analysis, partial differential equations, and automorphic functions. In Theta functions-Bowdoin 1987, Part 2 (Brunswick, ME, 1987), pages 45-100. Amer. Math. Soc., Providence, RI, 1989.

    [24] W. Feller. The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Stat., 22:427-432, 1951.

    [25] I. I. Gikhman. On a nonparametric criterion of homogeneity for k samples. Theory Probab. Appl., 2:369-373, 1957.

    [26] B. V. Gnedenko. Kriterien fu¨r die Unver¨andlichkeit der Wahrscheinlichkeitsverteilung von zwei unabh¨angigen Stichprobenreihen (in Russian). Math. Nachrichten., 12:29-66, 1954.

  • Metrics
Share - Bookmark