A general result on complete convergence for weighted sums of linear processes and its statistical applications

ABSTRACT

Consider the linear process $X_i=\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{\psi }_j{e}_{i-j}$, where $\left\{e_i\right\}$ is a sequence of identically distributed, negatively associated random variables with $E{e}_0=0$, and $\left\{{\psi }_j\right\}$ is a sequence of real numbers with $\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}|{\psi }_j|<\mathrm{\infty }$. Under some mild conditions, we first establish a general result on complete convergence for weighted sums of such linear process, and then describe its statistical properties and interpretations in both semiparametric and nonparametric regression models. We also prove the complete consistency for the parameter estimators. In addition, we have conducted comprehensive simulation studies to demonstrate the validity of obtained theoretical results.

KEYWORDS:

• Linear process
• complete convergence
• semiparametric regression model
• nonparametric regression model
• complete consistency

MATHEMATICAL SUBJECT CLASSIFICATIONS:

• 62F12
• 62G05
• 60F15

Acknowledgments

The authors are most grateful to the Editor and three anonymous reviewers for carefully reading the manuscript and for valuable suggestions which helped in improving an earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by the National Natural Science Foundation of China (11671012, 11871072, 11701004, 11701005), the Natural Science Foundation of Anhui Province (1808085QA03, 1908085QA01) and the Project on Reserve Candidates for Academic and Technical Leaders of Anhui Province (2017H123).