A weighted discrete limit theorem

A weighted discrete limit theorem Summary Let the function w(x) be defined for all x>2, WN=N∑k=2w(k), and limN→∞WN=+∞. Moreover, we suppose that the function w(x) has a continuous derivative w′(x) satisfying ∫N2u|w′(u)|du=O(WN). Let Ω=∏pγp, where γp={s∈C:|s|= 1} for all primes p. Denote by B(Ω) the Borel σ algebra of Ω. The torus Ω is a compact topological group. Therefore, on(Ω,B(Ω)), the probability Haar measure mH exists. For A∈B(Ω), define PN(A)=1WNN∑k=2w(k)I{l:(pillog2l:p∈P)∈A}(k), where P is the set of all prime numbers. Then we prove the following theorem. Suppose that the function w(x) satisfies all above hypothesis. Then PN converges weakly to the Haar measure mH as N→∞