Laws of the iterated logarithm for weighted sums of independent random variables
Necessary and sufficient conditions for LIL for weighted sums of the form \[ S_n= \sum^n_{k=1} f(k/n)X_k/(2n\log\log n)^{1/2} \] are given. Here \(f\) is a continuous function on \([0,1]\) which can be approximated by polynomials in total variation norm and \(\{X,X_n\}\) is a sequence of i.i.d. mean zero random variables with \(EX^2=1\). The main result states that \(\limsup_{n\to\infty}S_n\) is equal to \(L_2\)-norm of \(f\). Conversely, if this \(\limsup\) is finite for some considered \(f\neq 0\), then \(EX^2<\infty\). Conditions for compact LIL are also given when \(X\) takes values in a separable Banach space, and a LIL for weighted sums is obtained in a Banach space setting.
scientific article; zbMATH DE number 896085