On connection berween the law of large numbers for squares and the law of iterated logarithm

Let $\{X_i\}$ be a sequence of independent random variables with $EX_i=0, i=1,2,\dots,\quad b_n\uparrow\infty$ be a sequence of real numbers. Under some conditions it is proved that if $\sum_{i=1}^nX_i^2\stackrel{p}{=}O(b_n)$ $(\sum_{i=1}^nX_i^2=O(b_n) \text{ a.s.})$, then $\sum_{i=1}^nX_i\stackrel{p}{=}O(\varphi(b_n))$ $(\sum_{i=1}^nX_i=O(\varphi(b_n)) \text{ a.s.})$, where $\varphi(x)=\sqrt x\quad(\varphi(x)=\sqrt{x\ln\ln x})$.