Random Sampling of Almost Periodic Stochastic Processes

Abstract

We study the effect of random sampling on stochastic processes whose finite-dimensional distributions are Bohr almost periodic under time shifts. The sampling times are generated independently by a two-sided random walk with positive increments or by a two-sided subordinator. Under a non-degeneracy condition we show that the shifted finite-dimensional distributions of the sampled process converge to those of a strictly stationary process. Consequently, if the sampled process is itself almost periodic in distribution, then it must already be strictly stationary. We further prove an L²-law of large numbers for sampled averages under covariance decay and a uniform second-moment bound, and identify its limit with the mean of the stationary limit process. Finally, for m-dependent, L²-uniformly integrable almost periodic processes, we establish a central limit theorem for the centered sampled sums, whose asymptotic variance is given by the autocovariance series of the stationary limit.

Weyl Pseudo Almost Periodic Type Solutions to Semilinear Stochastic Evolution Equations Driven by Fractional Brownian Motion
Dimplekumar N. Chalishajar, Marko Kostic, Daniel Velinov
arXiv21 Sep 2026PreprintCC BY 4.0Full textAnalysis & PDEs ProbabilityarXiv
Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative α-stable processes: Existence, uniqueness and numerical approximation
Jiaquan Lu, Huaizhong Zhao
arXiv26 Oct 2026PreprintarXiv licenceFull textProbabilityarXiv
Random attractors and almost-sure stability under discretization of a stochastic autoparametric system
Chuchu Chen, Jialin Hong, Yibo Wang
arXiv29 Aug 2026PreprintarXiv licenceFull textDynamical systems Numerical analysis & scientific computingarXiv
Fast Almost-Uniform Sampling of Random k-SAT Solutions
Kun He, Zhidan Li, Kuan Yang
arXiv7 Oct 2026PreprintCC BY 4.0Full textProbabilityarXiv
Real-Space renormalization for the drawups and drawdowns of stochastic trajectories: universality classes for diffusion processes and random walks via the scale function
Cecile Monthus
arXiv3 Nov 2026PreprintarXiv licenceFull textProbability Statistical & nonlinear physicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{berger2026random,
  title = {{Random Sampling of Almost Periodic Stochastic Processes}},
  author = {David Berger and Maximilian Strobel},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23669},
  eprint = {2610.23669},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23669},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Random Sampling of Almost Periodic Stochastic Processes
AU  - David Berger
AU  - Maximilian Strobel
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23669
UR  - https://arxiv.org/abs/2610.23669
AB  - We study the effect of random sampling on stochastic processes whose finite-dimensional distributions are Bohr almost periodic under time shifts. The sampling times are generated independently by a two-sided random walk with positive increments or by a two-sided subordinator. Under a non-degeneracy condition we show that the shifted finite-dimensional distributions of the sampled process converge to those of a strictly stationary process. Consequently, if the sampled process is itself almost periodic in distribution, then it must already be strictly stationary. We further prove an L^2-law of large numbers for sampled averages under covariance decay and a uniform second-moment bound, and identify its limit with the mean of the stationary limit process. Finally, for m-dependent, L^2-uniformly integrable almost periodic processes, we establish a central limit theorem for the centered sampled sums, whose asymptotic variance is given by the autocovariance series of the stationary limit.
ER  -

Abstract

Related

Cite