Theory of Probability and Mathematical Statistics
Almost periodic stochastic processes with applications to analytic number theory
Alexander Iksanov, Zakhar Kabluchko and Alexander Marynych
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Abstract: 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 is a Besicovitch almost periodic function and V is a random variable uniformly distributed on [-1,1], then the random variables f(L⋅V) converge in distribution, as 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 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.
Keywords: Almost periodic functions, ergodic stochastic process, functional limit theorem, Mertens conjecture, Möbius function, stationary process, von Mangoldt function, zeros of zeta-function, zeta-function
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