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The work of Lajos Takács on probability theory

Published online by Cambridge University Press:  14 July 2016

N. H. Bingham
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Extract

In more than four decades of prolific scientific activity, Lajos Takács has produced so much that no one contributor to this Festschrift can hope to cover all of it in a balanced way. I thus propose to make a virtue of necessity, and concentrate on those aspects of Takács' work which have particularly interested or influenced me, and on the impact Takács' ideas in these areas have had on the subsequent development of the subject.

Type
Part 2 Probabilistic Methods
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 29 - 39
Copyright
Copyright © Applied Probability Trust 1994 

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References

References

Askey, R. (1975) Orthogonal Polynomials and Special Functions. SIAM, Philadelphia.Google Scholar
Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
Bartlett, M. S. (1953) Recurrence and first-passage times. Proc. Camb. Phil. Soc. 49, 263–275.CrossRefGoogle Scholar
Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press (2nd edn 1966).Google Scholar
Barton, D. E. and Mallows, C. L. (1965) Some aspects of the random sequence. Ann. Math. Statist. 36, 236–260.Google Scholar
Ben Arous, G. (1991) Géométrie de la courbe brownienne plane. Astérisque 201, 7–42.Google Scholar
Beneš, V. E. (1957) On queues with Poisson arrivals. Ann. Math. Statist. 28, 670–677.Google Scholar
Beneš, V. E. (1960a) General stochastic processes in traffic systems with one server. Bell Systems Tech. J. 39, 127–160. (MR 22#3036).CrossRefGoogle Scholar
Beneš, V. E. (1960b) Combinatory methods and stochastic Kolmogorov equations in the theory of queues with one server. Trans. Amer. Math. Soc. 94, 282–294.Google Scholar
Beneš, V. E. (1963) General Stochastic Processes in the Theory of Queues. Addison-Wesley, Reading, Mass.Google Scholar
Bingham, N. H. (1975) Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.Google Scholar
Bingham, N. H. (1991) Fluctuation theory for the Ehrenfest urn. Adv. Appl. Prob. 23, 598–611.CrossRefGoogle Scholar
Bloomfield, P. (1973) Stochastic inequalities for regenerative phenomena. In Stochastic Analysis , ed. Kendall, D. G. and Harding, E. F., pp. 226–233. Wiley, New York.Google Scholar
Cohen, J. W. (1969) The Single-Server Queue. North-Holland, Amsterdam.Google Scholar
Coxeter, H. S. M. (1963) Regular Polytopes , 3rd edn. Cambridge University Press (Dover edn 1973).Google Scholar
Davidson, R. (1968) Arithmetic and other properties of certain Delphic semigroups: II. Z. Wahrscheinlichkeitsth. 10, 146–172 (reprinted as pp. 150-182 in Stochastic Analysis, ed. Kendall, D. G. and Harding, E. F., Wiley, New York, 1973).Google Scholar
Diaconis, P. (1988) Group Representations in Probability and Statistics. Lecture Notes Monographs 11, Institute of Mathematical Statistics.Google Scholar
Dvoretzky, A. and Motzkin, Th. (1947) A problem of arrangements. Duke Math. J. 14, 305–313.Google Scholar
Feller, W. (1957) An Introduction to Probability Theory and its Applications , Volume 1, 2nd edn. Wiley, New York (3rd edn 1968).Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Volume 2. Wiley (1st edn 1966).Google Scholar
Gani, J. and Prabhu, N. U. (1963) A storage model with continuous infinitely-divisible inputs. Proc. Camb. Phil. Soc. 59, 417–429.Google Scholar
Gani, J. and Pyke, R. (1960) The content of the dam as the supremum of an infinitely-divisible process. J. Math. Mech. 9, 639–651.Google Scholar
Heyde, C. C. (1969) A derivation of the ballot theorem from the Spitzer-Pollaczek identity. Proc. Camb. Phil. Soc. 65, 755–757.Google Scholar
Kac, M. (1959) Probability and Related Topics in the Physical Sciences. Interscience, New York.Google Scholar
Kendall, D. G. (1951) Some problems in the theory of queues. J. R. Statist. Soc. B13, 151–185.Google Scholar
Kendall, D. G. (1957) Some problems in the theory of dams. J. R. Statist. Soc. B19, 207–212.Google Scholar
Kingman, J. F. C. (1962a) Spitzer's identity and its use in probability theory. J. London Math. Soc. 37, 309–316.Google Scholar
Kingman, J. F. C. (1962b) The use of Spitzer's identity in the investigation of the busy period and other quantities in the queue GI/G/1. J. Austral. Math. Soc. 2, 345–356.Google Scholar
Kingman, J. F. C. (1963) On continuous-time models in the theory of dams. J. Austral. Math. Soc. 3, 480–487.CrossRefGoogle Scholar
Kingman, J. F. C. (1964) The stochastic theory of regenerative events. Z. Wahrscheinlichkeitsth. 2, 180–224.Google Scholar
Kingman, J. F. C. (1966) The algebra of queues. J. London Math. Soc. 3, 285–326.Google Scholar
Kingman, J. F. C. (1971) Regenerative Phenomena. Wiley, New York.Google Scholar
Le Gall, J.-F. (1992) Some properties of planar Brownian motion. Ecole d'Eté de Saint-Flour XX (1990). Lecture Notes in Mathematics 1527, pp. 112–234. Springer-Verlag, Berlin.Google Scholar
Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.Google Scholar
Palacios, J. L. (1993) Fluctuation theory for the Ehrenfest urn via electric networks. Adv. Appl. Prob. 25, 472–476.Google Scholar
Prabhu, N. U. (1961) Applications of storage theory to queues with Poisson arrivals. Ann. Math. Statist. 31, 475–482.Google Scholar
Prabhu, N. U. and Rubinovitch, M. (1971) On a regenerative phenomenon arising in a storage model. J. R. Statist. Soc. B32, 354–361.Google Scholar
Prabhu, N. U. and Rubinovitch, M. (1973) Further results for ladder processes in continuous time. Stoch. Proc. Appl. 1, 151–168.Google Scholar
Reich, E. (1958) On the integro-differential equation of Takács, I. Ann. Math. Statist. 29, 563–570.CrossRefGoogle Scholar
Reich, E. (1959) On the integro-differential equation of Takács, II. Ann. Math. Statist. 30, 143–148.Google Scholar
Reich, E. (1961) Some combinatorial theorems for continuous-parameter stochastic processes. Math. Scand. 9, 243–257.CrossRefGoogle Scholar
Rogers, L. C. G. (1990) The two-sided exit problem for spectrally positive Lévy processes. Adv. Appl. Prob. 22, 486–487.Google Scholar
Rubinovitch, M. (1971) Ladder phenomena in stochastic processes with stationary independent increments. Z. Wahrscheinlichkeitsth. 20, 58–74.Google Scholar
Shorack, G. R. and Wellner, J. A. (1986) Empirical Processes with Applications to Statistics. Wiley, New York.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. (1957) The Wiener-Hopf equation whose kernel is a probability density. Duke Math. J. 24, 327–343.Google Scholar
Whitworth, W. A. (1886) Choice and chance , 4th edn. Deighton Bell, Cambridge (5th edn 1901, reprinted Stechert, G. E., New York, 1934).Google Scholar
Zeilberger, D. (1983) André's reflection proof generalized to the many-candidate ballot problem. Discrete Math. 44, 325–326.Google Scholar

Works of Takács cited

[1955] Investigation of waiting-time problems by reduction to Markov processes. Acta. Math. Acad. Sci. Hungar. 6, 101–129 (in Hungarian).CrossRefGoogle Scholar
[1956] On a probability problem arising in the theory of counters. Proc. Camb. Phil. Soc. 52, 488–498.Google Scholar
[1961a] The transient behaviour of a single-server queueing process with a Poisson input. Proc. 4th Berkeley Symp. Math. Statist. Prob. II, 535–567.Google Scholar
[1961b] Charles Jordan, 1871-1959. Ann. Math. Statist. 32, 1–11.Google Scholar
[1962a] The time-dependence of a single-server queue with Poisson input and general service times. Ann. Math. Statist. 33, 1340-1348.Google Scholar
[1962b] A generalization of the ballot problem and its applications in the theory of queues. J. Amer. Statist. Assoc. 57, 327-337.Google Scholar
[1962c] Ballot problems. Z. Wahrscheinlichkeitsth. 1, 154-158.Google Scholar
[1962d] Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
[1963a] The stochastic law of the busy period for a single-server queue with Poisson input. J. Math. Anal. Appl. 6, 33–42.Google Scholar
[1963b] The distribution of majority times in a ballot. Z. Wahrscheinlichkeitsth. 2, 118–121.Google Scholar
[1964] Combinatorial methods in the theory of dams. J. Appl. Prob. 1, 69–76.Google Scholar
[1965a] Application of ballot theorems in the theory of queues. Proc. Symp. Congestion Theory, Chapel Hill, NC, ed. Smith, W. L. and Wilkinson, W. E., pp. 337–398, University of North Carolina Press.Google Scholar
[1965b] The distributions of some statistics depending on the deviations between empirical and theoretical distribution functions. Sankhya A27, 93–100.Google Scholar
[1967a] The distribution of the content of finite dams. J. Appl. Prob. 4, 151–161.Google Scholar
[1967b] Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar
[1968] On dams with finite capacity. J. Austral. Math. Soc. 8, 161–170.Google Scholar
[1970] On the distribution of the maxima of sums of mutually independent and identically distributed random variables. Adv. Appl. Prob. 2, 344–354.CrossRefGoogle Scholar
[1971a] On the comparison of a theoretical and an empirical distribution function. J. Appl. Prob. 8, 321–330.Google Scholar
[1971b] On the comparison of two empirical distribution functions. Ann. Math. Statist. 42, 1157–1166.Google Scholar
[1972] On a formula of Pollaczek and Spitzer. Studia Math. 41, 27–34.Google Scholar
[1973a] On a method of Pollaczek. Stoch. Proc. Appl. 1, 1–9.Google Scholar
[1973b] On an identity of Shih-Chieh Chu. Acta Sci. Math. (Szeged) 34, 383–391.Google Scholar
[1975a] On a problem of fluctuations of sums of independent random variables. In Perspectives in Probability and Statistics , ed. Gani, J., pp. 29–37, Applied Probability Trust/Academic Press, London.Google Scholar
[1975b] Combinatorial and analytic methods in the theory of queues. Adv. Appl. Prob. 7, 607–635.CrossRefGoogle Scholar
[1976] On fluctuation problems in the theory of queues. Adv. Appl. Prob. 8, 548–583.CrossRefGoogle Scholar
[1978] On fluctuations of sums of random variables. In Studies in Probability and Ergodic Theory , pp. 45–93, Adv. Math. Suppl. Studies 2, Academic Press, New York.Google Scholar
[1979a] On an urn problem of Paul and Tatiana Ehrenfest. Math. Proc. Camb. Phil. Soc. 86, 127–130.CrossRefGoogle Scholar
[1979b] (with G. Letac) Random walks on an m-dimensional cube. J. reine angew. Math. 310, 187-195.Google Scholar
[1980a] (with Letac, G.) Random walk on a dodecahedron. J. Appl. Prob. 17, 373–384.Google Scholar
[1980b] (with Letac, G.) Random walk on a 600-cell. SIAM J. Alg. Discrete Methods 1, 114–120.Google Scholar
[1980c] Expected perimeter length. Amer. Math. Monthly 87, 142.Google Scholar
[1981a] The arc-sine law of P. Lévy. In Contributions to Probability , ed. Gani, J. and Rohatgi, V. K., pp. 49–63, Academic Press, New York (MR 82h:60049).Google Scholar
[1981b] Random flights on regular polytopes. SIAM J. Alg. Discrete Methods 2, 153–171.Google Scholar
[1982] Random walks on groups. Linear Alg. Appl. 43, 49–67.Google Scholar
[1983] Random walk on a finite group. Acta Sci. Math. (Szeged) 45, 395–408.Google Scholar
[1984] Random flights on regular graphs. Adv. Appl. Prob. 16, 618–637.Google Scholar
[1986] Harmonic analysis on Schur algebras and its applications in the theory of probability. In Probability Theory in Harmonic Analysis , ed. Chao, J.-A. and Woyczynski, W. A., pp. 227–283, Dekker, New York.Google Scholar
[1988a] Queues, random graphs and branching processes. J. Appl. Math. Simul. 1, 223–243.Google Scholar
[1988b] On the limit distribution of the number of cycles in a random graph. J. Appl. Prob. 25A, 359–376.Google Scholar
[1989] Ballots, queues and random graphs. J. Appl. Prob. 26, 103–112.Google Scholar
[1991] A Bernoulli excursion and its various applications. Adv. Appl. Prob. 23, 557–585.Google Scholar