Hostname: page-component-5c6d5d7d68-qks25 Total loading time: 0 Render date: 2024-08-10T01:08:50.087Z Has data issue: false hasContentIssue false
  • English
  • Français

On simulation of random vectors by given densities in regions and on their boundaries

Part of: Probabilistic methods, simulation and stochastic differential equations Special processes

Published online by Cambridge University Press:  14 July 2016

K. A. Borovkov
K. A. Borovkov*
Affiliation:
Steklov Mathematical Institute
*
∗ Postal address: Steklov Mathematical Institute, Vavilov st. 42, 117966 Moscow GSP-1, Russia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We suggest a new universal method of stochastic simulation, allowing us to generate rather efficiently random vectors with arbitrary densities in a connected open region or on its boundary. Our method belongs to the class of dynamic Monte Carlo procedures and is based on a special construction of a Markov chain on the boundary of the region. Its remarkable feature is that this chain admits a simple simulation, based on a universal (depending only on the dimensionality of the space) stochastic driver.

Keywords

MONTE CARLO SIMULATIONMARKOV CHAINSEMI-MARKOV PROCESS

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 205 - 220
Copyright
Copyright © Applied Probability Trust 1994 

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.)

Footnotes

This work was done while the author was visiting the Carl von Ossietzky Universität Oldenburg. Research supported by the Alexander von Humboldt-Stiftung.

References

Athreya, K. B., Mcdonald, D. and Ney, P. (1978) Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Prob. 6, 788–797.Google Scholar
Borovkov, K. A. (1991) On a new variant of the Monte Carlo method. Theory Prob. Appl. 36, 355–360.CrossRefGoogle Scholar
Devroye, L. (1986) Non-uniform Random Variate Generation. Wiley, New York.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Frigess, A., Hwang, C.-R. and Younes, L. (1992) Optimal spectral structure of reversible stochastic matrices, Monte Carlo methods and the simulation of Markov random fields. Ann. Appl. Prob. 2, 610–628.Google Scholar
Hager, W. H. (1988) Applied Numerical Linear Algebra. Prentice-Hall, London.Google Scholar
Kalos, M. H. and Whillock, P. A. (1986) Monte Carlo Methods, Vol. 1: Basics. Wiley, New York.Google Scholar
Lalley, S. and Robbins, H. (1988) Stochastic search in a convex region. Prob. Theory Rel. Fields 77, 99–116.Google Scholar
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, H. (1953) Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092.Google Scholar
Ogata, Y. (1989) A Monte Carlo method for high dimensional integration. Numer. Math. 55, 137–157.Google Scholar
Ripley, B. R. (1987) Stochastic Simulation. Wiley, New York.CrossRefGoogle Scholar
Ritov, Y. (1989) Monte Carlo computation of the mean of a function with convex support. Comput. Statist. Data Anal. 7, 269–277.CrossRefGoogle Scholar
Robbins, H. (1948) Convergence of distributions. Ann. Math. Statist. 19, 72–76.Google Scholar
Rubinstein, R. Y. (1982) Generating random vectors uniformly distributed inside and on the surface of different regions. Eur. J. Operat. Res. 10, 205–209.CrossRefGoogle Scholar
Rubinstein, R. Y. (1986) Monte Carlo Optimization, Simulation and Sensitivity of Queueing Networks. Wiley, New York.Google Scholar
Smith, R. L. (1984) Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions. Operat. Res. 32, 1296–1308.CrossRefGoogle Scholar
Stefănesku, S. and VĂduva, I. (1987) On computer generation of random vectors by transformations of uniformly distributed vectors. Computing 39, 141–153.Google Scholar
Turcin, V. F. (1971) On the computation of multidimensional integrals by the Monte Carlo method. Theory Prob. Appl. 16, 720–724.Google Scholar
Valentine, F. A. (1964) Convex Sets. McGraw-Hill, New York.Google Scholar