A finitely additive version of the law of the iterated logarithm

A finitely additive version of the law of the iterated logarithm (LIL) is proposed. The formulation involves only finite-dimensional distributions of a sequence of independent random variables $(X_n)_{n\ge 1}$. It is also proved that in the case where one deals with $\sigma$-additive probabilities, the given result is equivalent to the classical version of the LIL.