On the Equidistribution of Sums of Independent Random Variables

where the constant M (A) is the mean value of A(x) as defined in the theory of almost periodic functions. Let it) —feitxdFix) denote the characteristic function of the X's. Concerning the roots of the equation o5(¿) = 1 one of the three following cases must hold. Case 1 ("General" case), (pit) = 1 if and only if t = 0. Case 2 ("Lattice" case). <p(f) is not identically equal to 1 but there exists a value ¿of^O such that #(/0) = 1. All the mass of F(x) is then concentrated at one or more of the points x = 2kir/t0 (A = 0, ±1, • • • ). It can be shown that there exists a largest positive number ß, 0 <ß < o», such that all the mass of F(x) is concentrated at one or more of the points x = A/3, and the number ß has the property that <Pit) = l if and only if t = 2kir/ß (A = 0, ±1, • • • ). Case 3 (Trivial case). <p(i) = l for all t. All the mass of F(x) is then concentrated at x = 0 and all the 5„ are identically 0. The distinction among these three cases is fundamental in what follows. Let F(x) be arbitrary but fixed. We define the mean value operator M (A) for certain complex-valued functions A(x) of a real variable x, — «j <x < oo, as follows. Case 1.