On the law of the iterated logarithm for continued fractions with sequentially restricted partial quotients

We establish a law of the iterated logarithm (LIL) for the set of real numbers whose nth partial quotient is bigger than alpha(n), where (alpha(n)) is a sequence such that n-ary sumation 1/alpha(n) is finite. This set is shown to have Hausdorff dimension 1/2 in many cases and the measure in LIL is absolutely continuous to the Hausdorff measure. The result is obtained as an application of a strong invariance principle for unbounded observables on the limit set of a sequential iterated function system.