Advertisement

BIT Numerical Mathematics

, Volume 6, Issue 4, pp 339–346 | Cite as

Crude Monte Carlo quadrature in infinite variance case and the Central Limit Theorem

  • Aimo Törn
Article
  • 62 Downloads

Abstract

After a short discussion of Monte Carlo integration the crude Monte Carlo method is tested by estimating the integrals
$$\int\limits_0^1 {\left( {\frac{1}{x}} \right)^{1/v} } dxas\bar f_n = \frac{1}{n}\sum\limits_{i = 1}^n {\left( {\frac{1}{{\xi _i }}} \right)^{1/v} ,} $$
where ξ i are independent uniformly distributed random numbers in [0, 1] andν ∈ [1, 2], in which interval\(\sigma (\bar f_n )\) is infinite. By the aid of the Central Limit Theorem an approximation for the distributions of the sums\(\bar f_n \) is obtained. The results of the Monte Carlo computations are then compared with the results obtained from the distributions of\(\bar f_n \).

Key words

Monte Carlo integration random approximation quadrature 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barnett, V. D., Math. of Comput. 16, no. 77 (1962).Google Scholar
  2. 2.
    Dahlbolm, T.; Fogel, K.-G.; Qvist, B. and Törn, A., Nuclear Physics 56, no. 2 (1964).Google Scholar
  3. 3.
    Gnedenko, B. V. and Kolmogorov, A. N.,Limit distributions for sums of independent random variables, Addison-Wesley (1954).Google Scholar
  4. 4.
    Hammersley, J. M. and Handscomb, D. C.,Monte Carlo methods, Methuen (1964).Google Scholar
  5. 5.
    Jansson, B.,Random number generators, FOA, Sweden (1966).Google Scholar
  6. 6.
    Khintchine, A. Ya., Bull. Univ. Etat Moscou, Sér. Int., Sect. A: Math. et Mécan. 1, Fasc. 5 (1937) (Zentralblatt 18 (1938)).Google Scholar
  7. 7.
    Khintchine, A. Ya. and Lévy, P., C. R. Acad. Paris 202 (1936).Google Scholar
  8. 8.
    Loève, M.,Probability theory, Van Nostrand, 3rd ed. (1963).Google Scholar
  9. 9.
    Qvist, B. and Törn, A., Acta Acad. Abo., Math. et Phys. 23, no. 8 (1963).Google Scholar

Copyright information

© BIT Foundations 1966

Authors and Affiliations

  • Aimo Törn
    • 1
  1. 1.Institute of MathematicsÅbo AkademiFinland

Personalised recommendations