An invariance principle for the law of the iterated logarithm

  • V. Strassen
Article

Summary

Let Sn be the sum of the first n of a sequence of independent identically distributed r. v. s. having mean 0 and variance 1. One version of the law of the iterated logarithm asserts that with probability one the set of limit points of the sequence
$$((2n{\text{ log }}\log n)^{ - 1/2} S_n )_{n \geqq 3} $$
coincides with «-1, 1» = {x:x real and ¦x¦≦ 1} (see Hartman-Wintner [6]). Now consider the continuous function ηn on «0, 1» obtained by linearly interpolating (2 n log log n)−1/2Si at i/n. Then we prove (theorem 3) that with probability one the set of limit points of the sequence (ηn)n≧3 with respect to the uniform topology coincides with the set of absolutely continuous functions x on «0, 1» such that
$$x(0) = 0$$
and
$$\int {\dot x^2 dt \leqq 1} $$
As applications we obtain, e. g.,
$$Pr\left\{ {\mathop {{\text{lim sup}}}\limits_{n \to \infty } n^{ - 1 - (a/2)} (2{\text{ log log }}n)^{ - (a/2)} \sum\limits_{i = 1}^n {|S_i |a} = \frac{{2(a + 2)^{(a/2) - 1} }}{{\left( {\int\limits_0^1 {\frac{{dt}}{{\sqrt {1 - t^a } }}} } \right)^a a^{a/2} }}} \right\} = 1$$
for any a ≧ 1, and
$$Pr\left\{ {\mathop {{\text{lim sup}}}\limits_{n \to \infty } v_n = 1 - {\text{exp}}\left\{ { - 4\left( {\frac{1}{{c^2 }} - 1} \right)} \right\}} \right\} = 1$$
Where vn is the frequency of the events
$$S_i > c(2i\log \log i)^{1/2} $$
among the first n integers i (0 ≦ c≦ 1).

Keywords

Continuous Function Stochastic Process Probability Theory Mathematical Biology Limit Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Chung, K. L.: On the maximum partial sum of sequences of independent random variables. Trans. Amer. Math. Soc., 64, 205–233 (1948).Google Scholar
  2. [2]
    Diedonné, I.: Foundations of Modern Analysis, Pure and Applied Mathematics. New York-London: 1960.Google Scholar
  3. [3]
    Doob, J. L.: Stochastic Processes, Wiley-Publications in Statistics. New York-London: Wiley 1959.Google Scholar
  4. [4]
    Erdös, P., and M. Kac: On certain limit theorems of the theory of probability. Bull. Amer. Math. Soc. 52, 292–302 (1946).Google Scholar
  5. [5]
    Feller, W.: The general form of the so-called law of the iterated logarithm. Trans. Amer. Math. Soc., 54, 373–402 (1943).Google Scholar
  6. [6]
    Hartman, P., and A. Wintner: On the law of the iterated logarithm. Amer. J. Math., 63, 169–176 (1941).Google Scholar
  7. [7]
    Kolmogorov, A.: Das Gesetz des iterierten Logarithmus. Math. Annalen 101, 126–135 (1929).Google Scholar
  8. [8]
    Lamperti, J.: On convergence of stochastic processes, Trans. Amer. Math. Soc. 104, 430–435 (1962).Google Scholar
  9. [9]
    Loève, M.: Probability Theory, The University series in higher Mathem., Princeton (1960).Google Scholar
  10. [10]
    Riesz, F., and B. Sz. Nagy: Vorlesungen über Funktionalanalysis. Hochschulbücher fur Mathematik, Berlin (1956).Google Scholar
  11. [11]
    Skorokhod, A.B.: Research on the Theory of Random Processes. Kiew (1961) (in Russian).Google Scholar

Copyright information

© Springer-Verlag 1964

Authors and Affiliations

  • V. Strassen
    • 1
    • 2
  1. 1.University of CaliforniaBerkeley
  2. 2.UniversitÄt GöttingenDeutschland

Personalised recommendations