Summary
Let μ be a probability and\(\nu _h = \sum\limits_{n = 1}^\infty {\frac{1}{n}\mu ^{*n} }\) the corresponding harmonic renewal measure. Complementing earlier results where μ is concentrated on a halfline we investigate the behaviour ofv h ([x, x + 1]) and the harmonic renewal functionG(x) =v h((−∞,x])asx→∞ ifm 1=∫xμ(dx)>0. We also consider the casem 1=0.
Article PDF
Similar content being viewed by others
References
Embrechts, P., Maejima, M., Omey, E.: A renewal theorem of Blackwell type. Ann. Probab.12, 561–570 (1984)
Embrechts, P., Goldie, C.M., Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheor. Verw. Geb.49, 335–347 (1979)
Esseen, C.G.: On the concentration function of a sum of independent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.9, 290–308 (1968)
Essén, M.: Banach algebra methods in renewal theory. J. Anal. Math.26, 303–336 (1973)
Feller, W.: An Introduction to Probability Theory and Its Applications II. (2nd ed.) New York: Wiley 1971
Gelfand, I.M., Raikow, D.A., Schilow, G.E.: Kommutative normierte Algebren. Berlin: Deutscher Verlag der Wissenschaften 1964
Greenwood, P., Omey, E., Teugels, J.L.: Harmonic renewal measures. Z. Wahrscheinlichkeitstheor. Verw. Geb.59, 391–409 (1982)
Greenwood, P., Omey, E., Teugels, J.L.: Harmonic renewal measures and bivariate domains of attractions in fluctuation theory. Z. Wahrscheinlichkeitstheor. Verw. Geb.61, 527–539 (1982)
Grübel, R.: Functions of discrete probability measures: rates of convergence in the renewal theorem. Z. Wahrscheinlichkeitstheor. Verw. Geb.64, 341–357 (1983)
Grübel, R.: Asymptotic analysis in probability theory with Banach algebra techniques. Habilitationsschrift, Essen 1984
Kalma, J.M.: Generalized renewal measures. Thesis, Groningen University 1972
Maejima, M., Omey, E.: A generalized Blackwell renewal theorem. Yokohama Math. J.32, 123–133 (1984)
Rogozin, B.A.: Asymptotics of renewal functions. Theory Probab. Appl.21, 669–686 (1976)
Rogozin, B.A., Sgibnev, M.S.: Banach algebras of measures on the line. Sib. Math. Zh.21, 160–169 (1980)
Rudin, W.: Functional Analysis. New Delhi: Tata-McGraw-Hill 1974
Whittaker, E.T., Watson, G.N.: A course of modern analysis (4th ed.) London: Cambridge University Press 1927
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grübel, R. On harmonic renewal measures. Probab. Th. Rel. Fields 71, 393–404 (1986). https://doi.org/10.1007/BF01000213
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01000213