Probability Theory and Related Fields

, Volume 77, Issue 2, pp 167–178 | Cite as

On a law of the iterated logarithm for sums mod 1 with application to Benford's law

  • Peter Schatte
Article

Summary

Let Z n be the sum mod 1 of n i.i.d.r.v. and let 1[0,x](·) be the indicator function of the interval [0, x]. Then the sequence 1[0,x](Z n ) does not converge for any x. But if arithmetic means are applied then under suitable suppositions convergence with probability one is obtained for all x as well-known. In the present paper the rate of this convergence is shown to be of order n-1/2 (loglogn)1/2 by using estimates of the remainder term in the CLT for m-dependent r.v.

Keywords

Stochastic Process Probability Theory Statistical Theory Indicator Function Remainder Term 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barlow J.L., Bareiss, E.H.: On roundoff error distributions in floating point and logarithmic arithmetic. Computing 34, 325–347 (1985)Google Scholar
  2. Benford F.: The law of anomalous numbers. Proc. Amer. Phil. Soc. 78, 552–572 (1938)Google Scholar
  3. Bhattacharya R.N.: Speed of convergence of the n-fold convolution of a probability measure on a compact group. Z. Wahrscheinlichkeitstheor. Verw. Geb. 25, 1–10 (1972)Google Scholar
  4. Chung K.L.: A course in probability theory. New York-London: Academic Press 1974Google Scholar
  5. Egorov V.A.: Some limit theorems for m-dependent random variables (Russian). Liet. mat. rink. 10, 51–59 (1970)Google Scholar
  6. Heinrich L.: A method for the derivation of limit theorems for sums of m-dependent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 501–515 (1982)Google Scholar
  7. Herrmann H.: Konvergenzgeschwindigkeit der Folge der Faltungspotenzen eines Wahrscheinlichkeitsmaßes auf einer kompakten topologischen Gruppe. Math. Nachr. 104, 49–59 (1981)Google Scholar
  8. Holewijn P.J.: On the uniform distribution of random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 14, 89–92 (1969)Google Scholar
  9. Kuipers L., Niederreiter H.: Uniform distribution of sequences. New York: Wiley 1974Google Scholar
  10. Loynes R.M.: Some results in the probability theory of asymptotic uniform distribution modulo 1. Z. Wahrscheinlichkeitstheor. Verw. Geb. 26, 33–41 (1973)Google Scholar
  11. Petrov V.V.: Sums of independent random variables. Berlin Heidelberg New York: Springer 1975Google Scholar
  12. Philipp W.: A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables. Ann. Probab. 5, 319–350 (1977)Google Scholar
  13. Raimi R.A.: The first digit problem. Amer. Math. Mon. 83, 521–538 (1976)Google Scholar
  14. Robbins H.: On the equidistribution of sums of independent random variables. Proc. Amer. Math. Soc. 4, 786–799 (1953)Google Scholar
  15. Schatte P.: Zur Verteilung der Mantisse in der Gleitkommadarstellung einer Zufallsgröße. Zeitschr. Angew. Math. Mech. 83, 553–565 (1973)Google Scholar
  16. Schatte P.: On the asymptotic uniform distribution of sums reduced mod 1. Math. Nachr. 115, 275–281 (1984)Google Scholar
  17. Schatte P.: The asymptotic uniform distribution modulo 1 of cumulative processes. Optimization 16, 783–786 (1985)Google Scholar
  18. Schatte P.: On the asymptotic uniform distribution of the n-fold convolution mod 1 of a lattice distribution. Math. Nachr. 128, 233–241 (1986)Google Scholar
  19. Schatte P.: On the asymptotic behaviour of the mantissa distributions of sums. J. Inf. Process. Cybern. ElK. 23, 353–360 (1987)Google Scholar
  20. Schatte P.: On the almost sure convergence of floating-point mantissas and Benford's law. Math. Nachr. 135 (1988)Google Scholar
  21. Schatte P.: On mantissa distributions in computing and Benford's law. J. Inf. Process. Cybern. ElK (in press)Google Scholar
  22. Schmidt V.: On the asymptotic uniform distribution of stochastic clearing processes. Optimization 17, 125–134 (1986)Google Scholar
  23. Shergin V.V.: On the speed of convergence in the central limit theorem for m-dependent random variables (Russian). Teor. Verotn. Primen. 24, 781–794 (1979)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Peter Schatte
    • 1
  1. 1.Sektion Mathematik der Bergakademie FreibergFreibergGerman Democratic Republic

Personalised recommendations