The Annals of Mathematical Statistics

Feller's Parametric Equations for Laws of the Iterated Logarithm

Howard H. Stratton

Abstract

In this paper the author considers the two methods that Feller discusses in [3] and [4] which find a sequence $b_n$ so that $\lim \sup S_n(b_ns_n)^{-1} = 1$ a.s. where $S_n = \sum^n_{i=1}X_i$ and $X_i$ are independent random variables with $EX = 0, EX^2 < \infty$ and $E\lbrack\exp(hX_i)\rbrack < \infty$ for all $h < 0$. The more elementary and general method, which is not developed by Feller in [3], is used in a most elementary manner to derive a theorem general enough to include: $(l(n) \equiv (2lnlns_n)^{\frac{1}{2}})$. (A) Kolmogorov's classical law of the iterated logarithm and the result of Egorov [2]: $X_i$'s bounded and $\sup(X_i)l(n)s_n^{-1} = O(1)$ implies $0 < \lim \sup S_n(l(n)s_n)^{-1} < \infty$. (B) A slightly different version of a result of Feller [3]: $X_i$ bounded above, $\sup (X_i)l(n)/s_n = O(1)$ and two other conditions then $0 < \lim \sup S_n(l(n)s_n)^{-1} < \infty$ (the "slightly different version" is to replace one of the "two other conditions" with a different condition). (C) A generalization of a Thompson [5]: $X_i = a_i Y_i$, where $Y_i$'s are identically distributed with common negative exponential distribution, then $a_il(n)/s_n = O(1)$ implies $\lim \sup S_n(s_nl(n))^{-1} = 1$ (the generalization is to require only that $Y_i$'s be identically distributed with $E\lbrack\exp(hY_i)\rbrack < \infty$ for all $h > 0$). Also under these conditions the theorem includes: $a_1l(n)/S_n = O(1)\quad \text{implies}\quad 0 < \lim \sup S_n(s_nl(n))^{-1} < \infty.$

Article information

Source
Ann. Math. Statist., Volume 43, Number 6 (1972), 2104-2109.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177690893

Digital Object Identifier
doi:10.1214/aoms/1177690893

Mathematical Reviews number (MathSciNet)
MR362464

Zentralblatt MATH identifier
0253.60032

JSTOR