The Law of the Logarithm for Weighted Sums of Independent Random Variables

Abstract

Let X,X _n ;n≥1 be a sequence of real-valued i.i.d. random variables with E(X)=0. Assume B(u) is positive, strictly increasing and regularly-varying at infinity with index 1/2≤α<1. Set b _n =B(n),n≥1. If

$$E(B^{ - 1} (\left| X \right|)) < \infty ,{\text{ }}B^{ - 1} (x\log x)P(\left| X \right| >x) = o(\log x){\text{ }}as{\text{ x}} \to \infty $$

and

$$\mathop {\lim }\limits_{x \to \infty } \sup \frac{{n\log nE(X^2 I_{\left\{ {X^2 < b_n^2 /(\log n)^2 } \right\}} )}}{{b_n^2 }} = \lambda ^2$$

for some λ∈[0,∞), then it is shown that

$$\mathop {\lim }\limits_{x \to \infty } \sup \frac{{\left| {\sum\nolimits_{k = 1}^n {a_{n,k} X_k } } \right|}}{{b_n }} \leqslant 6\sqrt 2 \lambda {\text{ }}\mathop {\sup }\limits_{1 \leqslant k \leqslant n,n \geqslant 1} {\text{ }}\left| {a_{n,k} } \right|{\text{ a}}{\text{.s}}{\text{.}}$$

and

$$\mathop {\lim }\limits_{x \to \infty } \sup {\text{ }}(\mathop {\lim \inf }\limits_{x \to \infty } )\frac{{\left| {\sum\nolimits_{k = 1}^n {W_{n,k} X_k } } \right|}}{{b_n }} = + \left( - \right)\sqrt 2 {\text{ }}\lambda \sigma (W){\text{ a}}{\text{.s}}{\text{.}}$$

for every real triangular array (a _n,k ;1≤k≤n,n≥1) and every array of bounded real-valued i.i.d. random variables W,W _n,k ;1≤k≤n,n≥1`` independent of {X,X _n ;n≥1}, where σ(W)=(E(W−E(W))²)^1/2. An analogous law of the iterated logarithm for the unweighted sums ∑ⁿ _k=1 X _k ;n≥1} is also given, along with some illustrative examples.