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The Distribution of the Maximum of Partial Sums of Independent Random Variables

  • Mark Kac (a1) and Harry Pollard

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1. The problem. It has been shown [1] that if Xi, + X2, … are independent random variables each of density then

(1.1)

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References

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[1] Bachelier, L., Les lois des grands nombres du calcul des probabilités (Paris, Gauthier-Villars), 1937.
[2] Erdös, P. and Kac, M., On certain limit theorems of the theory of probability, Bull. Amer. Math. Soc, vol. 52 (1946), 292302.
[3] Levy, P., Sur les intégrales dont les éléments sont des variables aléatoires indépendantes, Annali délie Universita Toscane, Pisa (2) 3 (1934), 337366.
[4] Doob, J. L., Regularity properties of certain families of chance variables, Trans Amer. Math. Soc., vol. 47 (1940), 455486.
[5] Banach, S. and Saks, S., Sur la convergence forte dans les champs Lp , Stud. Math., vol. 2 (1930), 5157.
[6] Hille, E., Functional Analysis and Semigroups, Amer. Soc. Coll. Publ. 1948.
[7] Hamel, G., Integralgleichungen (Berlin, 1937), 145148.
[8] Reissner, E., Solution of a class of singular integral equations, Bull. Amer. Math. Soc., vol. 51 (1945), 920922.
[9] Kac, M., On the distributions of certain Wiener functionals, Trans. Amer. Math. Soc., vol.68 (1949), 113.
[10] Widder, D. V., The Laplace Transform (Princeton, 1940).
[11] Söhngen, H., Lösung einer Integralgleichung in der Tragfliigeltheorie, Math. Zeit., vol. 45 (1939), 245264.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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