• DOI:10.1007/s10474-016-0603-4
• Corpus ID: 119320307

Weak law of large numbers for linear processes

@article{Characiejus2016WeakLO,
  title={Weak law of large numbers for linear processes},
  author={Vaidotas Characiejus and Alfredas Ra{\vc}kauskas},
  journal={Acta Mathematica Hungarica},
  year={2016},
  volume={149},
  pages={215-232}
}

• V. Characiejus, A. Račkauskas
• Published 1 February 2016
• Mathematics
• Acta Mathematica Hungarica

We establish sufficient conditions for the Marcinkiewicz–Zygmund type weak law of large numbers for a linear process $${\{X_k:k\in\mathbb Z\}}$${Xk:k∈Z} defined by $${X_k=\sum_{j=0}^\infty\psi_j\varepsilon_{k-j}}$$Xk=∑j=0∞ψjεk-j for $${k\in\mathbb Z}$$k∈Z, where $${\{\psi_j:j\in\mathbb Z\}\subset\mathbb R}$${ψj:j∈Z}⊂R and $${\{\varepsilon_k:k\in\mathbb Z\}}$${εk:k∈Z} are independent and identically distributed random variables such that $${{x^p\Pr\{|\varepsilon_0| > x\}\to 0}}$$xpPr{|ε0|>x}→0 as $${{x\to \infty}}$$x→∞ with $${1 < p < 2}$$1<p<2 and $${E \varepsilon_0=0}$$Eε0=0. We use an abstract norming sequence that does not grow faster than $${n^{1/p}}$$n1/p if $${\sum|\psi_j| < \infty}$$∑|ψj|<∞. If $${\sum|\psi_j|=\infty}$$∑|ψj|=∞, the abstract norming sequence might grow faster than $${n^{1/p}}$$n1/p as we illustrate with an example. Also, we investigate the rate of convergence in the Marcinkiewicz–Zygmund type weak law of large numbers for the linear process.