The functional law of the iterated logarithm for truncated sums

The functional law of the iterated logarithm (FLIL) is obtained for truncated sums $S_n = sum _{j = l}^n X_j I{ X_j^{text{2}} leqslant b_n } $ of independent symmetric random variables X_j, 1<-j≤n, b_n≤∞. Considering the random normalization $T_n^{1/{text{2}}} = left( {sumlimits_{j = 1}^n {X_j^{text{2}} } I{ X_j^{text{2}} leqslant b_n } } right)^{1/{text{2}}} ,$ we obtain an upper estimate in the FLIL, using only the condition that T_n→∞ almost surely. These results are useful in studying trimmed sums. Bibliography: 9 titles.