Abstract
Let the time series {X(t), t=1, 2, ...} satisfy φ(B)(1−B)d X(t)=ϑ(B)e(t), whereB is a backward shift operator, defined byBX(t)=X(t−1), and φ(z)=1+φ1 z+...+φ p z p, ϑ(z)=1+ϑ1 z+...+я q z q, and all the roots of φ(z) lie outside the unit circle; {e(t)} is a sequence of iid random variables with mean zero andE|e(t)|4+r<∞ (r>0). In this paper, the limit properties of\(\sum\limits_{t = 1}^n {X(t)^2 /t^{2d} \log n}\), where the integerd⩾1, have been considered.
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An, H., Gao, H. Two limit theorems on ARIMA models. Acta Mathematicae Applicatae Sinica 4, 154–164 (1988). https://doi.org/10.1007/BF02006064
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DOI: https://doi.org/10.1007/BF02006064