Feller's Parametric Equations for Laws of the Iterated Logarithm

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: $a_1l(n)/S_n = O(1)\quad \text{implies}\quad 0 < \lim \sup S_n(s_nl(n))^{-1} < \infty.$