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Probability metrics and recursive algorithms

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  01 July 2016

S. T. Rachev  and
L. Rüschendorf
S. T. Rachev*
Affiliation:
University of California at Santa Barbara
L. Rüschendorf*
Affiliation:
University of Freiburg
*
* Postal address: University of California, Statistics Department, Santa Barbara, CA 93106–3110, USA.
** Postal address: Universität Freiburg, Institüt Mathematische Stochastik, Hebelstr. 27, 79104 Freiburg, Germany.
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Abstract

It is shown by means of several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a ‘suitable' probability metric which yields contraction properties of the transformations describing the limits of the algorithm. In order to demonstrate the wide range of applicability of this contraction method we investigate examples from various fields, some of which have already been analysed in the literature.

Keywords

SEARCH ALGORITHMTRIECONTRACTION PROPERTIESPROBABILITY METRICSIMAGE ENCODING

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 3 , September 1995 , pp. 770 - 799
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Research supported by a DFG Grant and NATO Grant CRG 900798.

References

[1] Aldous, D. (1991) The continuum random tree II: An overview. In Stochastic Analysis, ed. Barlow, M. T. and Bingham, N. H., pp. 23–70. London Math. Soc. Lecture Notes Series 167, Cambridge University Press.Google Scholar
[2] Barnsley, M. F. and Elton, J. H. (1988) A new class of Markov processes for image encoding. Adv. Appl. Prob. 20, 14–32.Google Scholar
[3] Bickel, P. and Freedman, D. A. (1981) Some asymptotic theory for the bootstrap. Ann. Statist. 9, 1196–1217.Google Scholar
[4] De Haan, L. and Rachev, S. (1989) Estimates of the rate of convergence for max-stable processes. Ann. Prob. 17, 651–677.Google Scholar
[5] Deheuvels, P. and Pfeifer, D. (1988) On a relationship between Uspensky's theorem and Poisson approximation. Ann. Inst. Statist. Math. 40, 671–681.Google Scholar
[6] Devroye, L. (1986) Lecture Notes on Bucket Algorithms. Birkhäuser, Boston.Google Scholar
[7] De Jong, P. (1989) Central Limit Theorems for Generalized Multilinear Forms. CWI Tract 61, Amsterdam.Google Scholar
[8] Fayolle, G., Flajolet, P., Hofri, M. and Jacquet, P. (1985) Analysis of a stack algorithm for random multiple-access communication. IEEE Trans. Inf. Theory 31, 244–254.Google Scholar
[9] Fayolle, G., Flajolet, P. and Hofri, M. (1986) On a functional equation arising in the analysis of a protocol for a multi-access broadcast channel. Adv. Appl. Prob. 18, 441–472.Google Scholar
[10] Gutjahr, W. and Pflug, G. Ch. (1992) The asymptotic contour process of a binary tree is a Brownian excursion. Stoch. Proc. Appl. 41, 69–89.Google Scholar
[11] Hofri, M. (1987) Probabilistic Analysis of Algorithms. Springer Verlag, New York.Google Scholar
[12] Jacquet, P. and Regnier, M. (1988) Normal limiting distribution of the size of tries. In Proc. Performance 87, pp. 209–223. North-Holland, Amsterdam.Google Scholar
[13] Kemp, R. (1984) Fundamentals of the Average Case Analysis of Particular Algorithms. Wiley, New York.Google Scholar
[14] Knuth, D. E. (1969) The Art of Computer Programming II. Addison-Wesley, Reading, MA.Google Scholar
[15] Maejima, M. and Rachev, S. T. (1987) An ideal metric and the rate of convergence to a self similar process. Ann. Prob. 15, 708–727.Google Scholar
[16] Mahmoud, H. M. (1992) Evolution of Random Search Trees. Wiley, New York.Google Scholar
[17] Mathar, R. and Pfeifer, D. (1990) Stochastik für Informatiker. Teubner, Stuttgart.Google Scholar
[18] Mehlhorn, K. (1986) Datenstrukturen und effiziente Algorithmen, Vol. I. Teubner, Stuttgart.Google Scholar
[19] Nevzorov, V. B. (1988) Records. Theory Prob. Appl. 32, 201–228.Google Scholar
[20] Pfeifer, D. (1991) Some remarks on Nevzorov's record model. Adv. Appl. Prob. 23, 823–834.Google Scholar
[21] Pflug, G. (1986) Stochastische Moelle in der Informatik. Teubner, Stuttgart.Google Scholar
[22] Pittel, B. (1986) Paths in a random digital tree: Limiting distributions. Adv. Appl. Prob. 18, 139–155.Google Scholar
[23] Rachev, S. T. (1984) The Monge-Kantorovich mass transference problem and its stochastic applications. Theory Prob. Appl. 29, 647–676.Google Scholar
[24] Rachev, S. T. and Yukich, J. E. (1989) Rates for the CLT via new ideal metrics. Ann. Prob. 17, 775–788.Google Scholar
[25] Rachev, S. T. and Rüschendorf, L. (1992) Rate of convergence for sums and maxima and doubly ideal metrics. Theory Prob. Appl. 37, 276–289.Google Scholar
[26] Rachev, S. T. and Rüschendorf, L. (1990) Approximation of sums by compound Poisson distributions with respect to stop-loss distances. Adv. Appl. Prob. 22, 350–374.Google Scholar
[27] Rachev, S. T. and Todorovic, P. (1990) On the rate of convergence of some functionals of a stochastic process. J. Appl. Prob. 28, 805–814.Google Scholar
[28] Rachev, S. T. and Samorodnitski, G. (1990) Classes of distributions arising in modelling environmental processes. Tech. Report No. 904 (1990), School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY.Google Scholar
[29] Regnier, M. and Jacquet, P. (1989) New results on the size of tries. IEEE Trans. Inf. Theory 35, 203–205.Google Scholar
[30] Robbins, H. and Siegmund, D. (1971) A convergence theorem for nonnegative almost supermartingales. In Optimization Methods in Statistics, ed. Rustagi, J., pp. 233–258. Academic Press, New York.Google Scholar
[31] Rösler, U. (1991) A limit theorem for Quicksort. Informatique Théorique et Applications 25, 85–100.Google Scholar
[32] Ross, S. M. (1982) A simple heuristic approach to simplex efficiency. Eur. J. Operat. Res. 9, 344–346.CrossRefGoogle Scholar
[33] Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[34] Seidel, L. (1988) On limit distributions of random symmetric polynomials. Theory Prob. Appl. 23, 266–278.Google Scholar
[35] Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar
[36] Zolotarev, V. M. (1976) Approximation of distributions of sums of independent random variables with values in infinite dimensional spaces. Theory Prob. Appl. 21, 721–737.Google Scholar
[37] Zolotarev, V. M. and Rachev, S. T. (1987) Rate of convergence in limit theorems for the max scheme. In Stability Problems for Stochastic Models. Lecture Notes in Mathematics 1155, pp. 415–442. Springer-Verlag, Berlin.Google Scholar