Renewal function asymptotics refined à la Feller

• Autorzy: Daley D.

Identyfikatory

DOI

10.19195/0208-4147.37.2.5

• Języki publikacji: EN

Abstrakty

EN

Feller’s volume 2 shows how to use the Key Renewal Theorem to prove that in the limit x → ∞, the renewal function U(x) of a renewal process with nonarithmetic generic lifetime X with finite mean E(X) = 1/λ and second moment differs from its linear asymptote λ_x by the quantity 1/2λ²E(X²). His first edition (1966) (but not the second in 1971) asserted that a similar approach would refine this asymptotic result when X has finite higher order moments. The paper shows how higher order moments may justify drawing conclusions from a recurrence relation that exploits a general renewal equation and further appeal to the Key Renewal Theorem.

Słowa kluczowe

EN

renewal function; asymptotics of refined iterates; moment condition

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2017
• Tom: Vol. 37, Fasc. 2
• Strony: 291--298
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Daley D.

• Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Melbourne, Australia

Bibliografia

• [1] J. Blanchet and P. Glynn, Uniform renewal theory with applications to expansions of random geometric sums, Adv. in Appl. Probab. 39 (2007), pp. 1070-1097.
• [2] H. Carlsson, Remainder term estimates of the renewal function, Ann. Probab. 11 (1983), pp. 123-157.
• [3] D. J. Daley and N. R. Mohan, Asymptotic behaviour of the variance of renewal processes and random walks, Ann. Probab. 6 (1978), pp. 516-521.
• [4] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Volume I: Elementary Theory and Methods, second edition, Springer, New York 2003.
• [5] C. Dombry and L. Rabehasaina, Asymptotic expansion for renewal functions and applications to ruin theory, arXiv, 2016.
• [6] W. Feller, An Introduction to the Theory of Probability and Its Applications, vol. 2, second edition, Wiley, New York 1971.
• [7] W. Feller and S. Orey, A renewal theorem, J. Math. Mech. 10 (1961), pp. 619-624.
• [8] C. Stone, On characteristic functions and renewal theory, Trans. Amer. Math. Soc. 120 (1965), pp. 327-342.

Uwagi

Dedicated to Tomasz Rolski, a constant colleague over forty years, distance notwithstanding.

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).