The functional law of the iterated logarithm for truncated sums (Q1575984)

The functional law of the iterated logarithm (FLIL) is obtained for truncated sums \(S_n= \sum^n_{j=1} X_jI\{X^2_j\leq b_n\}\) of independent symmetric random variables \(X_j\), \(1\leq j\leq n\), \(b_n\leq \infty\). Considering the random normalization \[ T^{1/2}_n= \Biggl(\sum^n_{j=1} X^2_j I\{X^2_j\leq b_n\}\Biggr)^{1/2}, \] an upper estimate in the FLIL is obtained, using only the condition that \(T_n\to \infty\) almost surely. These results are useful in studying trimmed sums.