References
Aczél, J. (1966),Lectures on functional equations and their applications. Academic Press, New York.
Aczél, J. (1975),General solution of a functional equation connected with a characterization of statistical distributions. InStatistical Distributions in Scientific Work (Proc. NATO Advanced Study Institute, Univ. of Calgary, Calgary, 1974), Vol. 3, D. Reidel, Dordrecht, pp. 47–55.
Aczél, J. (1975a),Some recent applications of functional equations and inequalities to characterizations of probability distributions, combinatorics, information theory and mathematical economics. InStatistical Distributions in Scientific Work (Proc. NATO Advanced Study Institute, Univ. of Calgary, Calgary, 1974), Vol. 3, D. Reidel, Dordrecht, pp. 321–337.
Aczél, J. (1979), Personal communication.
Aczél, J. andDaróczy, Z. (1975),On measures of information and their characterizations. Academic Press, New York.
Ahsanullah, M. (1976),On a characterization of the exponential distribution by order statistics. J. Appl. Probab.13, 818–822.
Ahsanullah, M. (1977),A characteristic property of the exponential distribution. Ann. Statist.5, 580–582.
Ahsanullah, M. (1978)A characterization of the exponential distribution by spacings. J. Appl. Probab.15, 650–653.
Arnold, B.C. (1971),Two characterizations of the exponential distribution using order statistics. Technical Report, Iowa State University, Ames, Iowa.
Arnold, B.C. andMeeden, G. (1976),A characterization of the uniform distribution based on summation module one, with application to fractional backlogs. Austral. J. Statist.18, 173–175.
Bakstys, A. (1968),On the asymptotic distribution of multiplicative number theoretic functions, I (Russian). Litovsk. Mat. Sb.8, 5–20.
Basu, A.P. andBlock, H.W. (1975),On characterizing univariate and multivariate exponential distribution with applications. InStatistical Distributions in Scientific Work (Proc. NATO Advanced Study Institute, Univ. of Calgary, Calgary, 1974), Vol. 3, D. Reidel, Dordrecht, pp. 399–421.
Berk, R.H. (1977),Characterizations via conditional distributions. J. Appl. Probab.14, 806–816.
Block, H. W. andSavits, T. H. (1981),Multivariate distributions in reliability theory and life testing. InStatist. Distributions in Scientific Work, Vol. 5 (Trieste 1980, NATO Adv. Study Inst.), D. Reidel, Dordrecht, pp. 271–280.
Brandhofe, T. andDavies, L. (1980),On a functional equation in the theory of linear statistics. Ann. Inst. Statist. Math.32 (A), 17–23.
Daley, D. J. (1973),Poisson and alternating renewal processes with super-position a renewal process. Math. Nachr.57, 359–369.
Davies, P. L. andShimizu, R. (1976),On identically distributed linear statistics. Ann. Inst. Statist. Math.28, 469–489.
Deheuvels, P. (1980),The decomposition of infinite order and extreme multivariate distributions. InAsymptotic Theory of Statistical Tests and Estimation (Proc. Adv. Internat. Symp., Univ. North Carolina, Chapel Hill, N.C., 1979), Academic Press, New York, pp. 259–286.
Elliott, P. D. T. A. (1974),A conjecture of Kátai. Acta Arith.26, 11–20.
Elliott, P. D. T. A. (1979–1980),Probabilistic number theory, Vol. I, II. Springer-Verlag, Heidelberg.
Erdös, P. andKátai, I. (1977),On characterizing logn among additive functions. Manuscript, presented at the Internat. Conf. on Number Theory, Oberwolfach.
Erdös, P. andWintner, A. (1939),Additive arithmetical functions and statistical independence. Amer. J. Math.61, 713–721.
Galambos, J. (1971),On the distribution of strongly multiplicative functions. Bull. London Math. Soc.3, 307–312.
Galambos, J. (1971a),Limit distribution of sums of (dependent) random variables with applications to arithmetical functions. Z. Wahrsch. Verw. Gebiete18, 261–270.
Galambos, J. (1975),Characterizations of probability distributions by properties of order statistics, I. InStatistical Distributions in Scientific Work (Proc. NATO Advanced Study Institute, Univ. of Calgary, Calgary, 1974), Vol. 3, D. Reidel, Dordrecht, pp. 71–88.
Galambos, J. (1978),The asymptotic theory of extreme order statistics. Wiley, New York.
Galambos, J. (1981),Extreme value theory in applied probability. Math. Sci.6, 13–26.
Galambos, J. andKotz, S. (1978),Characterizations of probability distributions. Lecture Notes in Mathematics, Vol. 675, Springer, Heidelberg.
Grosswald, E. andKotz, S. (1980),An integrated lack of memory characterization of the exponential distribution. To appear in Ann. Inst. Statist. Math.
Grosswald, E., Kotz, S. andJohnson, N. L. (1980),Characterizations of the exponential distribution by relevation-type equations. J. Appl. Probab.17, 874–877.
Gupta, R. C. (1973),A characteristic property of the exponential distribution. Sankhya, Ser. B,35, 365–366.
Huang, J. S. (1974),Characterizations of the exponential distribution by order statistics. J. Appl. Probab.11, 605–608.
Indlekofer, K.-H. (1976),On sets characterizing additive arithmetical functions. Math. Z.146, 285–290.
Indlekofer, K.-H. (1981),On sets characterizing additive and multiplicative arithmetical functions. Illinois J. Math.25, 251–257.
Jánossy, L., Rényi, A. andAczél, J. (1950),On composed Poisson distributions, I. Acta Math. Acad. Sci. Hungar.1, 209–224.
Kagan, A. M., Linnik, Yu. V. andRao, C. R. (1973),Characterization problems in mathematical statistics. Wiley, New York.
Kátai, I. (1969/70),On sets characterizing number-theoretical functions, II. Acta Arith.16, 1–4.
Klebanov, L. B. (1978),Some problems of characterization of distributions arising in reliability theory (Russian). Teor. Verojatnost. i Primenen.23, 828–831.
Klebanov, L. B. (1980),Some results connected with the characterization of the exponential distribution (Russian). Teor. Verojatnost. i Primenen.25, 628–633.
Kotz, S. (1974),Characterizations of statistical distribution: A supplement to recent surveys. Internat. Statist. Rev.42, 39–65.
Kubilius, J. (1964),Probabilistic methods in the theory of numbers. Amer. Math. Soc. Transl. Monographs, Vol. 11, Amer. Math. Soc., Providence, R. I.
Lau, Ka-Sing andRao, C. R. (1981),Solution to the integrated Cauchy functional equation and characterization of the exponential law. Bull. Inst. Math. Statist.6, 265.
Loève, M. (1963),Probability theory, 3rd. ed. Van Nostrand, New York.
Lukacs, E. andLaha, R. G. (1964),Applications of characteristic functions. Mafner, New York.
Patil, G. P. andSeshadri, V. (1964),Characterization theorems for some univariate probability distributions. J. Roy. Statist. Soc. Ser. B,26, 286–292.
Patil, G. P., Kotz, S. andOrd, J. K. (1975),Proceedings of the conference on characterizations, Calgary. Statistical Distributions in Scientific Work, Vol. 3, D. Reidel, Dordrecht.
Ramachandran, B. (1977),On the strong Markov property of the exponential laws. InAnalytic function methods in probability theory (1980), North-Holland, Amsterdam, pp. 277–292.
Rao, C. R. (1974),Functional equations and characterizations of probability distributions. In Proc. Internat. Congress of Mathematicians (Vancouver, 1974, Vol. 2), Canad. Math. Congress, 1975, pp. 163–168.
Rao, C. R. (1975),Some problems in the characterization of the multivariate normal distribution. InStatistical Distributions in Scientific Work (Proc. NATO Advanced Study Institute, Univ. of Calgary, Calgary, 1974), Vol. 3, D. Reidel, Dordrecht, pp. 1–13.
Seneta, E. (1975),Characterization by functional equations of branching process limit laws. InStatistical Distributions in Scientific Work (Proc. NATO Advanced Study Institute, Univ. of Calgary, Calgary, 1974), Vol. 3, D. Reidel, Dordrecht, pp. 249–254.
Seneta, E. (1976),Regularly varying functions. Lecture Notes in Mathematics, Vol. 508, Springer, Heidelberg.
Sethuraman, J. (1965),On a characterization of the three limiting types of extremes. Sankhya, Ser. A,27, 357–364.
Shimizu, R. (1978),Solution to a functional equation and its application to some characterization problems. Sankhya, Ser. A,40, 319–332.
Shimizu, R. (1979),On a lack of memory property of the exponential distribution. Ann. Inst. Statist. Math.31, 309–313.
Shimizu, R. andDavies, P. L. (1979),General characterization theorems for the Weibull and the stable distributions. Technical Report, The Institute of Statistical Mathematics, Tokyo.
Wirsing, E. (1977),Additive und vollständing additive Funktionen mit Wachstumsbeschränkungen. Manuscript, Presented at Internat. Conf. on Number Theory, Oberwolfach.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Galambos, J. The role of functional equations in stochastic model building. Aeq. Math. 25, 21–41 (1982). https://doi.org/10.1007/BF02189595
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02189595