Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-27T19:09:57.878Z Has data issue: false hasContentIssue false
  • English
  • Français

Complete exponential convergence and some related topics

Published online by Cambridge University Press:  14 July 2016

C. R. Heathcote
C. R. Heathcote*
Affiliation:
Australian National University
Get access Rights & Permissions [Opens in a new window]

Abstract

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Review Paper
Information
Journal of Applied Probability , Volume 4 , Issue 2 , August 1967 , pp. 217 - 256
Copyright
Copyright © Sheffield: Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Note: The following titles do not by any means constitute a complete bibliography.Google Scholar
Bahadur, R. R. and Rao, R. R. (1960) On deviations of the sample mean. Ann. Math. Statist. 31, 1015–1027.Google Scholar
Barlow, R. E. and Proschan, F. (1964) Comparison of replacement policies, and renewal theory implications. Ann. Math. Statist. 35, 577–589.Google Scholar
Bartholomew, D. J. (1963) An approximate solution of the integral equation of renewal theory. J. R. Statist. Soc. B 25, 432–441.Google Scholar
Baum, L. E. and Katz, M. L. (1965) Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108–123.Google Scholar
Baum, L. E., Katz, M. L. and Read, R. R. (1962) Exponential convergence rates for the law of large numbers. Trans. Amer. Math. Soc. 102, 187–199.CrossRefGoogle Scholar
Belyaev, Yu. K. and Maksimov, V. M. (1963) Analytic properties of a generating function for the number of renewals. Theor. Probability Appl. (Translation) 8, 102–105.CrossRefGoogle Scholar
Blackwell, D. and Hodges, J. L. (1959) The probability in the extreme tail of a convolution. Ann. Math. Statist. 30, 1113–1120.Google Scholar
Brillinger, D. R. (1962) A note on the rate of convergence of a mean. Biometrika 49, 574–576.Google Scholar
Chernoff, H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23, 493–507.Google Scholar
Cheong, C. K. and Heathcote, C. R. (1965) The rate of convergence of waiting times. J. Aust. Math. Soc. 5, 365–373.Google Scholar
Cramer, H. (1938) Sur un nouveau théorème-limite de la théorie des probabilités. Actualités Sci. Indust. No. 736. Hermann, Paris.Google Scholar
Daniels, H. E. (1954) Saddlepoint approximations in statistics. Ann. Math. Statist. 25, 631–650.Google Scholar
Derman, C. and Robbins, H. (1955) The strong law of large numbers when the first moment does not exist. Proc. Nat. Acad. Sci. 41, 586–587.Google Scholar
Erdös, P. (1949) On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286–291.CrossRefGoogle Scholar
Esscher, F. (1932) On the probability function in the collective theory of risk. Skand. Aktuartidskr. 15, 175–195.Google Scholar
Evgrafov, M. A. (1961) Asymptotic Estimates and Entire Functions. Gordon and Breach. New York.Google Scholar
Feller, W. (1949) Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67, 98–119.Google Scholar
Feller, W. (1957) An Introduction to Probability Theory and its Applications. Vol. I, 2nd ed. Wiley. New York.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley. New York.Google Scholar
Gelfond, A. O. (1964) An estimate for the remainder term in a limit theorem for recurrent events. Theor. Probability Appl. (Translation) 9, 299–303.Google Scholar
Hanson, D. L. and Koopmans, L. H. (1965a) On the convergence rate of the law of large numbers for linear combinations of independent random variables. Ann. Math. Statist. 36, 559–564.Google Scholar
Hanson, D. L. and Koopmans, L. H. (1965b) Convergence rates for the law of large numbers for linear combinations of exchangeable and * mixing stochastic processes. Ann. Math. Statist. 36, 1840–1852.CrossRefGoogle Scholar
Heathcote, C. R. (1964) Divergent single server queues. Chapter 5 in Proceedings of the Symposium on Congestion Theory. University of North Carolina Press.Google Scholar
Heathcote, C. R. (1966) On transient random walk. (To appear).Google Scholar
Heyde, C. C. (1964) Two probability theorems and their application to some first passage problems. J. Aust. Math. Soc. 4, 214–222.Google Scholar
Heyde, C. C. (1966a) Some renewal theorems with application to a first passage problem. Ann. Math. Statist. 37, 699–710.Google Scholar
Heyde, C. C. (1966b) Small-deviation probability convergence rates. Canadian J. Math. 18, 656–665.CrossRefGoogle Scholar
Heyde, C. C. and Rohatgi, V. K. (1967) A pair of complementary theorems on convergence rates in the law of large numbers. Proc. Camb. Phil. Soc. 63, 73–82.Google Scholar
Hoeffding, W. (1965) On probabilities of large deviations. University of N. Carolina Institute of Statistics Mimeo Series No. 443.Google Scholar
Hsu, P. L. and Robbins, H. (1947) Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. 33, 25–31.Google Scholar
Karlin, S. and Mcgregor, J. (1966) Spectral theory of branching processes parts I and II. Z. Wahrscheinlichkeitsth. 4.Google Scholar
Katz, M. L. (1963) The probability in the tail of a distribution. Ann. Math. Statist. 34, 312–318.Google Scholar
Katz, M. L. and Thomasian, A. J. (1960) An exponential bound for functions of a Markov chain. Ann. Math. Statist. 31, 470–474.Google Scholar
Keilson, J. (1965) Green's Function Methods in Probability Theory. Griffin. London.Google Scholar
Kendall, D. G. (1959) Unitary dilations of Markov transition operators and the corresponding integral representations for transition probability matrices, in Probability and Statistics (ed. Grenander, U.). Wiley, New York.Google Scholar
Kendall, D. G. (1960) Geometric ergodicity and the theory of queues, in Mathematical Methods in the Social Sciences (ed. Arrow, K. J., Karlin, S., Suppes, P.). Stanford University Press.Google Scholar
Kendall, D. G. (1964) Some recent work and further problems in the theory of queues. Theor. Probability Appl. (Translation) 9, 1–13.Google Scholar
Kingman, J. F. C. (1962) Some inequalities for the queue G/G/1. Biometrika 49, 315–324.Google Scholar
Kingman, J. F. C. (1963a) The exponential decay of Markov transition probabilities. Proc. Lond. Math. Soc. (3) 13, 337–358.Google Scholar
Kingman, J. F. C. (1963b) Ergodic properties of continuous time Markov processes and their discrete skeletons. Proc. Lond. Math. Soc. (3) 13, 593–604.Google Scholar
Kingman, J. F. C. (1964) The stochastic theory of regenerative events. Z. Wahrscheinlichkeitsth. 2, 180–224.Google Scholar
Koopmans, L. H. (1960) Asymptotic rate of discrimination for Markov processes. Ann. Math. Statist. 31, 982–984.Google Scholar
Koopmans, L. H. (1961) An exponential bound on the strong law of large numbers for linear stochastic processes with absolutely convergent coefficients. Ann. Math. Statist. 32, 583–586.Google Scholar
Koopmans, L. H. (1966) Convergence rates for the law of large numbers for linear combinations of Markov processes. Ann. Math. Statist. 37, 711–719.Google Scholar
Leadbetter, M. R. (1964) Bounds on the error in the linear approximation to the renewal function. Biometrika 51, 355–364.Google Scholar
Miller, H. D. (1961) A generalisation of Wald's Identity with applications to random walk. Ann. Math. Statist. 32, 549–560.Google Scholar
Miller, H. D. (1966) Geometric ergodicity in a class of denumerable Markov chains. Z. Wahrscheinlichkeitsth. 4, 354–373.CrossRefGoogle Scholar
Nagaev, S. V. (1965) Some theorems for large deviations. Theor. Probability Appl. (Translation) 10, 214–235.Google Scholar
Petrov, V. V. (1965) On the probabilities of large deviations for sums of independent random variables. Theor. Probability Appl. (Translation) 10, 287–298.Google Scholar
Port, S. C. (1963) An elementary approach to fluctuation theory. J. Math. Analysis and Appl. 6, 109–151.Google Scholar
Port, S. C. (1964) Some theorems on functionals of Markov chains. Ann. Math. Statist 35, 1275–1290.Google Scholar
Pruitt, W. E. (1964) Eigenvalues of non negative matrices. Ann. Math. Statist. 35, 1797–1800.Google Scholar
Richter, W. (1957) Local limit theorems for large deviations. Theor. Probability Appl. (Translation) 2, 206–220.CrossRefGoogle Scholar
Rosén, B. (1960) On the asymptotic distribution of sums of random variables. Ark. Mat. 4, 323–332.Google Scholar
Rubin, H. and Sethuraman, (1965) Probabilities of moderate deviations. Sankhya (A) 27, 325–346.Google Scholar
Seneta, E. (1966) Quasi-stationary distributions and time-reversion in genetics. J. R. Statist. Soc. B 28, 253–277.Google Scholar
Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.Google Scholar
Sethuraman, J. (1964) On the probability of large deviations of families of sample means. Ann. Math. Statist. 35, 1304–1316.Google Scholar
Smith, W. L. (1958) Renewal theory and its ramifications. J. R. Statist. Soc. B 20, 243–302.Google Scholar
Smith, W. L. (1961) A note on the renewal function when the mean renewal lifetime is infinite. J. R. Statist. Soc. B 23, 230–237.Google Scholar
Smith, W. L. (1966) A theorem on functions of characteristic functions and its application to some renewal theoretic random walk problems. University of N. Carolina Institute of Statistics Mimeo Series No. 437.Google Scholar
Spitzer, F. (1956) A combinatorial lemma and its application to Probability Theory. Trans. Amer. Math. Soc. 82, 323–339.Google Scholar
Spitzer, F. (1964) Principles of Random Walk. Van Nostrand, New York.CrossRefGoogle Scholar
Stone, C. (1965) On characteristic functions and renewal theory. Trans. Amer. Math. Soc. 120, 327–342.Google Scholar
Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. (2) 13, 7–28.Google Scholar
Vere-Jones, D. (1963) On the spectra of some linear operators associated with queueing systems. Z. Wahrscheinlichkeitsth. 2, 12–21.Google Scholar
Vere-Jones, D. (1964) A rate of convergence problem in the theory of queues. Theor. Probability Appl. (Translation) 9, 96–103.Google Scholar
Vere-Jones, D. (1966) Ergodic properties of non-negative matrices I. (To appear).Google Scholar