Article contents
Density approximation and exact simulation of random variables that are solutions of fixed-point equations
Part of:
Probabilistic theory: distribution modulo $1$; metric theory of algorithms
Probabilistic methods, simulation and stochastic differential equations
Published online by Cambridge University Press: 01 July 2016
Abstract
An algorithm is developed for exact simulation from distributions that are defined as fixed points of maps between spaces of probability measures. The fixed points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixed points with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method.
Keywords
MSC classification
Primary:
65C10: Random number generation
- Type
- General Applied Probability
- Information
- Copyright
- Copyright © Applied Probability Trust 2002
References
Bickel, P. J. and Freedman, P. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist.
9, 1196–1217.Google Scholar
Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Prob.
20, 1714–1798.Google Scholar
Chen, R., Goodman, R. and Zame, A. (1984). Limiting distributions of two random sequences. J. Multivariate Anal.
14, 221–230.Google Scholar
De Bruijn, N. G. (1951). The asymptotic behaviour of a function occurring in the theory of primes. J. Indian Math. Soc. (N.S.)
15, 25–32.Google Scholar
Devroye, L. (2001). Simulating perpetuities. Methodology Comput. Appl. Prob.
3, 97–115.Google Scholar
Devroye, L., Fill, J. A. and Neininger, R. (2000). Perfect simulation from the Quicksort limit distribution. Electron. Commun. Prob.
5, 95–99.Google Scholar
Dobrow, R. P. and Fill, J. A. (1999). Total path length for random recursive trees. Combinatorics Prob. Comput.
8, 317–333.Google Scholar
Embrechts, P. and Goldie, C. M. (1994). Perpetuities and random equations. In Asymptotic Statistics (Prague, 1993), eds Mandl, P. and Husková, M., Physica, Heidelberg, pp. 75–86.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
Fill, J. A. and Janson, S. (2000). Smoothness and decay properties of the limiting Quicksort density function. In Mathematics and Computer Science (Versailles, 2000), eds Gardy, D. and Mokkadem, A., Birkhäuser, Basel, pp. 53–64.Google Scholar
Fill, J. A. and Janson, S. (2001). Approximating the limiting Quicksort distribution. Random Structures Algorithms
19, 376–406.Google Scholar
Fill, J. A. and Janson, S. (2002). Quicksort asymptotics. Tech. Rep. 597, Department of Mathematical Sciences, The Johns Hopkins University. Available at http://www.mts.jhu.edu/~fill/. To appear in J. Algorithms.Google Scholar
Flajolet, P., Labelle, G., Laforest, L. and Salvy, B. (1995). Hypergeometrics and the cost structure of quadtrees. Random Structures Algorithms
10, 117–144.Google Scholar
Goldie, C. M. and Grübel, R. (1996). Perpetuities with thin tails. Adv. Appl. Prob.
28, 463–480.CrossRefGoogle Scholar
Goldie, C. M. and Maller, R. A. (2000). Stability of perpetuities. Ann. Prob.
28, 1195–1218.Google Scholar
Grübel, R. and Rösler, U. (1996). Asymptotic distribution theory for Hoare's selection algorithm. Adv. Appl. Prob.
28, 252–269.Google Scholar
Hennequin, P. (1989). Combinatorial analysis of quicksort algorithm. RAIRO Inf. Théor. Appl.
23, 317–333.Google Scholar
Hennequin, P. (1991). Analyse en moyenne d'algorithme, tri rapide et arbres de recherche. , École Polytechnique, Palaiseau.Google Scholar
Hwang, H.-K. and Neininger, R. (2002). Phase change of limit laws in the quicksort recurrence under varying toll functions. Preprint. Available at http://neyman.mathematik.uni-freiburg.de/homepages/neininger/. To appear in SIAM J. Comput.Google Scholar
Hwang, H.-K. and Tsai, T.-H. (2002). Quickselect and Dickman function. To appear in Combinatorics Prob. Comput.Google Scholar
Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math.
131, 207–248.Google Scholar
Knuth, D. E. (1997). The Art of Computer Programming, Vol. 3, Sorting and Searching. Addison-Wesley, Reading, MA.Google Scholar
Mahmoud, H. M. (1991). Limiting distributions for path lengths in recursive trees. Prob. Eng. Inf. Sci.
5, 53–59.CrossRefGoogle Scholar
Mahmoud, H. M., Modarres, R. and Smythe, R. T. (1995). Analysis of quickselect: an algorithm for order statistics. RAIRO Inf. Théor. Appl.
29, 255–276.Google Scholar
Neininger, R. (2001). On a multivariate contraction method for random recursive structures with applications to quicksort. Random Structures Algorithms
19, 498–524.Google Scholar
Neininger, R. and Rüschendorf, L. (1999). On the internal path length of d-dimensional quad trees. Random Structures Algorithms
15, 25–41.Google Scholar
Neininger, R. and Rüschendorf, L. (2001). A general contraction theorem and asymptotic normality in combinatorial structures. Preprint. Available at http://neyman.mathematik.uni-freiburg.de/homepages/neininger/. Tech. Rep. 01-28, Mathematische Fakultät, Universität Freiburg.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. Appl. Prob.
27, 770–799.Google Scholar
Régnier, M., (1989). A limiting distribution for quicksort. RAIRO Inf. Théor. Appl.
23, 335–343.CrossRefGoogle Scholar
Rösler, U., (1991). A limit theorem for ‘Quicksort’. RAIRO Inf. Théor. Appl.
25, 85–100.Google Scholar
Rösler, U., (1992). A fixed point theorem for distributions. Stoch. Process. Appl.
42, 195–214.Google Scholar
Rösler, U., (2001). On the analysis of stochastic divide and conquer algorithms. Algorithmica
29, 238–261.Google Scholar
Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica
29, 3–33.Google Scholar
Sedgewick, R. and Flajolet, P. (1996). An Introduction to the Analysis of Algorithms. Addison-Wesley, Amsterdam.Google Scholar
Takács, L., (1955). On stochastic processes connected with certain physical recording apparatuses. Acta Math. Acad. Sci. Hungar.
6, 363–380.Google Scholar
Tenenbaum, G. (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press. (Translated from the second French edition (1995) by Thomas, C. B..)Google Scholar
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
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