General analogues to the law of the iterated logarithm

  • William Feller


Let X1, X2,⋯. be independent and Sn=X1+⋯.+Xn. Suppose E(Xk)=0 and sn2= E(Sn2)<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 xnn) 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 Xn
$$P\{ S_n > s_n x_n (\zeta _n )\} = \exp [ - \log \log s_n + 0(\log \log s_n )].$$
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 }snxnn)}.


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  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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