On the functional law of the iterated logarithm for partially observed sums of random variables

We consider partial-sum process generated by a sequence of nonidentically distributed independent random variables. Assuming that this process is available for observations along an arbitrary time sequence, we fill the gaps by linear interpolation and prove the functional law of the iterated logarithm (FLIL) for sample paths obtained in this way. Assuming that condition of V. A. Egorov holds, we show that FLIL is valid, while under other conditions sufficient for usual law of the iterated logarithm FLIL may fail.