##
**A theorem of Feller revisited.**
*(English)*
Zbl 0669.60034

W. Feller [Ann. Math., II. Ser. 47, 631-638 (1946)] proposed and proved a law of the iterated logarithm. The proofs were later found to be incorrect. This paper provides a new proof of the main result. Let \(X_ i\) be i.i.d. centred real random variables with distribution function F, and set \(S_ n=\sum^{n}_{1}X_ i\). Feller claimed that if, as \(x\to \infty\),
\[
\int_{| t| >x}t^ 2 dF(t)=O(1/\log \log x),
\]
then the sequence \(\sqrt{n}\phi (n)\) belongs to the upper (lower) class if and only if
\[
(*)\quad \sum^{\infty}_{n=1}n^{-1}\phi (n)\exp \{-2^{-1}\phi^ 2(n)\}<+\infty \quad (=\infty).
\]
Here \(\phi\) (n) is an increasing sequence of positive numbers. This sequence belongs to the upper (lower) class if \(S_ n>\phi (n)\) finitely (infinitely) often in n. Feller’s proof was based on his Theorem 2. The essential difficulty was an (incorrect) decomposition of \(X_ i\) into three pieces.

This paper analyzes the upper and lower classes of \(S_ n\) through the behaviour of the sum of certain decompositions of \(X_ i\), i.e. \[ Y_ i=X_ iI\{| X_ i| <\sqrt{i \log \log i}\}-{\mathbb{E}}(X_ iI\{| X_ i| <\sqrt{i \log \log i}\}). \] The principal theorem (Theorem 1) states that if \(\sigma^ 2_ i={\mathbb{E}}y^ 2_ i\), \(B^ 2_ n=\sum^{n}_{1}\sigma^ 2_ i\), and \({\mathbb{E}}X^ 2_ i=1\), then \(\{B_ n\phi (n)\}\) belongs to the upper (lower) class if and only if (*) holds.

The proof of this theorem rests on four lemmas and a result of Feller. The lemmas permit the reduction of the problem to the \(Y_ i\), and a specific class of \(\phi\) (n). The behaviour of \(\sum^{n}_{1}Y_ i\) is described by a theorem of W. Feller [Ann. Math., II. Ser. 91, 402-418 (1970; Zbl 0252.60014)] with an appropriate choice of constants. Theorem 1 differs slightly from Feller’s Theorem 2, but yields the above claim on upper and lower class results for \(\sqrt{n}\phi (n)\). Theorem 2 refines Theorem 1 under Feller’s original condition. Theorem 3 expands Theorem 1 so that both Theorem 1 and 2 of Feller are corollaries.

This paper analyzes the upper and lower classes of \(S_ n\) through the behaviour of the sum of certain decompositions of \(X_ i\), i.e. \[ Y_ i=X_ iI\{| X_ i| <\sqrt{i \log \log i}\}-{\mathbb{E}}(X_ iI\{| X_ i| <\sqrt{i \log \log i}\}). \] The principal theorem (Theorem 1) states that if \(\sigma^ 2_ i={\mathbb{E}}y^ 2_ i\), \(B^ 2_ n=\sum^{n}_{1}\sigma^ 2_ i\), and \({\mathbb{E}}X^ 2_ i=1\), then \(\{B_ n\phi (n)\}\) belongs to the upper (lower) class if and only if (*) holds.

The proof of this theorem rests on four lemmas and a result of Feller. The lemmas permit the reduction of the problem to the \(Y_ i\), and a specific class of \(\phi\) (n). The behaviour of \(\sum^{n}_{1}Y_ i\) is described by a theorem of W. Feller [Ann. Math., II. Ser. 91, 402-418 (1970; Zbl 0252.60014)] with an appropriate choice of constants. Theorem 1 differs slightly from Feller’s Theorem 2, but yields the above claim on upper and lower class results for \(\sqrt{n}\phi (n)\). Theorem 2 refines Theorem 1 under Feller’s original condition. Theorem 3 expands Theorem 1 so that both Theorem 1 and 2 of Feller are corollaries.

### MSC:

60F15 | Strong limit theorems |