Almost periodic stochastic processes with applications to analytic number theory

A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if f f is a Besicovitch almost periodic function and V V is a random variable uniformly distributed on [ − 1 , 1 ] [-1,1] , then the random variables f ( L ⋅ V ) f(L· V) converge in distribution, as L → ∞ L→ ∞ , to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes ( f ( L ⋅ V + t ) ) t ∈ R (f(L· V+t))t∈ R in the space of Besicovitch almost periodic functions, and also in the sense of weak convergence of finite-dimensional distributions. Further we investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of Akbary, Ng and Shahabi (2014) and earlier works.