Almost periodic stochastic processes with applications to analytic number theory

2025/02/07 by Iksanov, Alexander, Kabluchko, Zakhar, Marynych, Alexander
#11M26 #60F17 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11N37 #Probability (math.PR) #secondary: 60G10

paper · doi:10.48550/arxiv.2502.04969

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. We further investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of [Limiting distributions of the classical error terms of prime number theory, Quart. J. Math. 65 (2014), 743--780] and earlier works.

Related

• Almost periodic stationary processes
• Almost periodicity and periodicity for nonautonomous random dynamical systems
• Sampling Almost Periodic and related Functions
• Generalized vector valued almost periodic and ergodic distributions
• SAMPLING ALMOST PERIODIC FUNCTIONS WITH RANDOM PROBES OF FINITE DENSITY