The law of the logarithm for weighted sums of independent random variables (Q1411343)

Let \(\{X,X_n;n\geq 1\}\) 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 \(1/2\leq \alpha < 1\). Set \(b_n=B(n)\), \(n\geq 1\). If \[ E(B^{-1} (|X|)) < \infty,\quad B^{-1}(x \log x) P(|X|> x)=o(\log x)\quad \text{as }\chi\rightarrow \infty, \] and \[ \limsup_{n\rightarrow\infty} {{n\log nE(X^2I_{\{x^2 < b^2_n (\log n)^2\}})}\over{b^2_n}} = \lambda^2 \] for some \(\lambda \in [0,\infty)\), then it is shown that \[ \limsup_{n\rightarrow\infty} {{|\sum^n_{k=1} a_{n,k} X_k|}\over{b_n}}\leq 6\sqrt{2} \lambda \sup_{1\leq k\leq n,n\geq 1} |a_{n,k}|\quad \text{a.s.} \] and \[ \limsup_{n\rightarrow\infty} \underset {n\rightarrow\infty}{(\liminf)}{{\sum^n_{k=1} W_{n,k} X_k}\over{b_n}}=+(-)\sqrt{2} \lambda\sigma(W)\quad\text{a.s.} \] for every array of bounded real-valued i.i.d. random variables \(\{W,W_{n,k}\); \(1\leq k\leq n, n\geq 1\}\) independent of \(\{X,X_n\); \(n\geq 1\}\), where \(\sigma(W)=(E(W-E(W))^2)^{1/2}\). An analogous law of the iterated logarithm for the unweighted sums \(\{\sum^n_{k=1} X_k\); \(n\geq 1\}\) is also given, along with some illustrative examples.