Two limit theorems on ARIMA models
Tóm tắt
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.
Tài liệu tham khảo
Box, G. E. P. and Jenkins, G. M., Time Series Analysis, Forecasting and Control, Holden-Day, San Francisco, 1970.
Chow, Y. S. and Teicher, H., Probability Theory, Springer-Verlag, New York, 1978.
Dickey, D. A., Estimation and Hypothesis Testing in Non-Stationary Time Series, Unpublished Ph. D Thesis, Iowa State University, 1976.
Dickey, D. A., Easza, D. A. and Fuller, W. A., Testing for Unit Roots in Seasonal Time Series,J. of the American Statistical Association,79 (1984), 355–367.
Hall, P. and Heyde, C. C., Martingale Limit Theory and Its Application, Academic Press, New York, 1980.
Lai, T. L. and Robbins, H., Adaptive Design and Stochastic Approximation,Ann. Statist.,7 (1979), 1196–1221.
Lai, T. L. and Wei, C. Z., Asymptotic Properties of Projections with Applications to Stochastic Regression Problem,J. Multivariate Anal.,12 (1982), 346–370.
Lai, T. L. and Wei, C. Z., A Law of the Iterated Logarithm for Double Arrays of Independent Random Variables with Applications to Regression and Time Series Models,Ann. Prob.,10 (1982), 320–335.
Lai, T. L. and Wei, C. Z., Least Squares Estimates in Stochastic Regression Models with Applications to Identification and Control of Dynamic Systems,Ann. Statist.,10 (1982), 154–166.
Said, S. E. and Dickey, D. A., Hypothesis Testing in ARIMA (p, 1,y) Models,L. of the American Statistical Association,80 (1985), 369–374.
Yaglom, A. M., The Correlation Theory of Processes Whosenth Differences Constitute a Stationary Process,Mat. Sb.,37 (79): 1(1955), 141–196.
Fuller, W. A., Nonstationary Autoregressive Time Series, In: E. J. Hannan, P. R. Krishnaiah, M. M. Rao, eds., Handbook of Statistics, Vol. 5, North-Holland, 1–23, 1985.