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General analogues to the law of the iterated logarithm

  • William Feller
Article

Summary

Let X1, X2,⋯. be independent and Sn=X1+⋯.+Xn. Suppose E(Xk)=0 and s n 2 = E(S n 2 )<t8 and put \(\psi _\mathfrak{n} (h) = \log E(e^{hS_n /\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{s} _n } )\) for h>0. To avoid trivialities we suppose that Sn+1<sn(log sn)p for some p>0. We assume that there exists a number q<1 such that if
$$q\log \log s_n < \zeta _n < q^{ - 1} \log \log s_n$$
there exists a (necessarily unique) number x n n ) determined by the parametric equations
$$\zeta _n = h\psi '_n (h) - \psi _n (h), x_n \zeta _n = \psi '_n (h).$$
It was shown in the preceding paper that under mild restrictions on the X n
$$P\{ S_n > s_n x_n (\zeta _n )\} = \exp [ - \log \log s_n + 0(\log \log s_n )].$$
It is now shown that under these conditions with probability one
$$\lim \sup \frac{{S_n }}{{s_n x_n (\zeta _n )}} = 1.$$
It is easy to give examples where
$$x_n \zeta _n \sim C(\log \log x_n )^{\tfrac{1}{2}}$$
with an arbitrary C>0. Other examples, however, exhibit an entirely different behavior of the sequence }s n x n n )}.

Keywords

Stochastic Process Probability Theory Mathematical Biology Parametric Equation Iterate Logarithm 
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.

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References

  1. 1.
    Feller, W.: Limit theorems for probabilities of large deviations. Z. Wahrscheinlichkeitsrechnung verw. Geb. 14, 1–20 (1969).Google Scholar
  2. 2.
    - On fluctuations of sums of independent random variables. To appear Proc. Nat. Acad. Sci., USA 62 (1969).Google Scholar

Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • William Feller
    • 1
  1. 1.Fine HallPrinceton UniversityPrincetonUSA

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