Summary
Let ν and τ be finite measures on the set of integers such that the Fourier transform of ν is an analytic function of the τ transform. The central result shows how ν may be approximated by linear combinations of convolution powers of τ. Applications are given to renewal theory, infinitely divisible measures and age-dependent branching processes.
Article PDF
Similar content being viewed by others
References
Athreya, K.B., Ney, P.E.: Branching Processes. Berlin-Heidelberg-New York: Springer 1972
Borovkov, A.A.: Remarks on Wiener's and Blackwell's theorems. Theory Probability Appl. 9, 303–312 (1964)
Borovkov, A.A.: Stochastic Processes in Queuing Theory. New York-Heidelberg-Berlin: Springer 1976
Chover, J., Ney, P., Wainger, S.: Functions of probability measures. J. Analyse Math. 26, 255–302 (1973)
Cotlar, M., Cignoli, R.: An Introduction to Functional Analysis. Amsterdam-London: North Holland 1974
Davies, L., Grübel, R.: Spaces of summable sequences in renewal theory and the theory of Markov chains. Math. Nachr. 104, 119–128 (1981)
Embrechts, P., Goldie, C.M., Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49, 335–347 (1979)
Embrechts, P., Hawkes, J.: A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory. J. Austral. Math. Soc. Ser. A 32, 412–422 (1982)
Embrechts, P., Omey, E.: Functions of power series. Preprint, 1982
Erdös, P., Feller, W., Pollard, H.: A property of power series with positive coefficients. Bull. Amer. Math. Soc. 55, 201–204 (1949)
Essén, M.: Banach algebra methods in renewal theory. J. Analyse Math. 26, 303–336 (1973)
Gelfand, I.M., Raikow, D.A., Schilow, G.E.: Kommutative normierte Algebren. Berlin: Deutscher Verlag der Wissenschaften 1964
Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. Providence: Amer. Math. Soc. Coll. Publ. XXXI 1957
Kalashnikov, V.V.: Uniform estimation of the convergence rate in a renewal theorem for the case of discrete time. Theory Probability Appl. 22, 390–394 (1977)
Lindvall, T.: On coupling of discrete renewal processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 28, 57–70 (1979)
Ney, P.: A refinement of the coupling method in renewal theory. Stochastic Processes Appl. 11, 11–26 (1981)
Rogozin, B.A.: An estimate of the remainder term in limit theorems in renewal theory. Theory Probability Appl. 18, 662–677 (1973)
Rogozin, B.A.: Asymptotic behaviour of the coefficients of functions of power series and Fourier series. Sb. Math. J. 17, 640–647 (1976)
Rudin, W.: Functional Analysis. New Delhi: Tata McGraw-Hill 1974
Stone, C., Wainger, S.: One-sided error estimates in renewal theory. J. Analyse Math. 20, 325–352 (1967)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grübel, R. Functions of discrete probability measures: Rates of convergence in the renewal theorem. Z. Wahrscheinlichkeitstheorie verw Gebiete 64, 341–357 (1983). https://doi.org/10.1007/BF00532966
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00532966