On the law of the iterated logarithm for uniformly bounded orthonormal systems
Mary Jane Weiss · Transactions of the American Mathematical Society · 1959
It is a familiar fact that certain systems of functions, although themselves not independent, exhibit properties of independent random variables.For example it is well known that if {w*} is a lacunary sequence of positive integers, that is m+i/nk >q>l, then the sequence (1) {pk cos (nk + ak))Jo JoIt is well known that the only condition needed on the sequence {cp"] in order to find such a weakly convergent subsequence is that it be uniformly bounded.From now on we will denote the subsequence {",.}so chosen bv Now the method of selection of the subsequence of Theorem 1 will involve, for values of x for which /(x)>0, the use of cp"kix)/fix) and sums, products