## The Annals of Probability

### Prediction of weakly stationary sequences on polynomial hypergroups

#### Abstract

We investigate random sequences $(X_n)_{\nin_0}$ with spectral representation based on certain orthogonal polynomials, that is, random sequences that are weakly stationary with respect to polynomial hypergroups. We present various situations where one meets this kind of sequence. The main topic is on the one-step prediction. In particular, it is examined when the mean-squared error tends to zero. For many cases we present a complete solution for the problem of $(X_n)_{\nin_0}$ being asymptotically deterministic.

#### Article information

Source
Ann. Probab., Volume 31, Number 1 (2003), 93-114.

Dates
First available in Project Euclid: 26 February 2003

https://projecteuclid.org/euclid.aop/1046294305

Digital Object Identifier
doi:10.1214/aop/1046294305

Mathematical Reviews number (MathSciNet)
MR1959787

Zentralblatt MATH identifier
1014.60039

#### Citation

Hösel, Volker; Lasser, Rupert. Prediction of weakly stationary sequences on polynomial hypergroups. Ann. Probab. 31 (2003), no. 1, 93--114. doi:10.1214/aop/1046294305. https://projecteuclid.org/euclid.aop/1046294305

#### References

• ARNAUD, J. P. (1994). Stationary processes indexed by a homogeneous tree. Ann. Probab. 22 195- 218.
• BLOOM, W. R. and HEy ER, H. (1995). Harmonic Analy sis of Probability Measures on Hy pergroups. de Gruy ter, Berlin.
• BLOWER, G. (1996). Stationary processes for translation operators. Proc. London Math. Soc. 72 697-720.
• BRESSOUD, D. M. (1981). Linearization and related formulas for q-ultraspherical poly nomials. SIAM J. Math. Anal. 12 161-168.
• BROCKWELL, P. J. and DAVIES, R. A. (1991). Time Series: Theory and Methods. Springer, New York.
• CHIHARA, T. S. (1978). An Introduction to Orthogonal Poly nomials. Gordon and Breach, New York.
• COWLING, M., MEDA, ST. and SETTI, A. G. (1998). An overview of harmonic analysis on the group of isometries of a homogeneous tree. Expo. Math. 16 385-424.
• DEVROy E, L. and GYÖRFI, L. (1985). Nonparametric Density Estimation: The L1-View. Wiley, New York.
• GERONIMUS, Y. L. (1960). Poly nomials Orthogonal on a Circle and Interval. Pergamon, Oxford.
• HEy ER, H. (1991). Stationary random fields over hy pergroups. In Gaussian Random Fields 197- 213. World Scientific, Singapore.
• HEy ER, H. (2000). The covariance distribution of a generalized random field over a commutative hy pergroup. Contemp. Math. 261 73-82.
• HÖSEL, V. (1998). On the estimation of covariance functions on Pn-weakly stationary processes. Stochastic Anal. Appl. 16 607-629.
• HÖSEL, V. and LASSER, R. (1992). One-step prediction for Pn-weakly stationary processes. Monatsh. Math. 113 199-212.
• KAKIHARA, Y. (1997). Multidimensional Second Order Stochastic Processes. World Scientific, Singapore.
• KNOPP, K. (1922). Theorie und Anwendung der unendlichen Reihen. Springer, Berlin.
• LASSER, R. (1983). Orthogonal poly nomials and hy pergroups. Rend. Mat. 3 185-209.
• LASSER, R. (1994). Orthogonal poly nomials and hy pergroups II-The sy mmetric case. Trans. Amer. Math. Soc. 341 749-770.
• LASSER, R. and LEITNER, M. (1989). Stochastic processes indexed by hy pergroups I. J. Theoret. Probab. 2 301-311.
• LASSER, R. and LEITNER, M. (1990). On the estimation of the mean of weakly stationary and poly nomial weakly stationary sequences. J. Multivariate Anal. 35 31-47.
• LASSER, R., OBERMAIER, J. and STRASSER, W. (1993). On the consistency of weighted orthogonal series density estimators with respect to L1-norm. Nonparametric Statist. 3 71-80.
• LEITNER, M. (1991). Stochastic processes indexed by hy pergroups II. J. Theoret. Probab. 4 321- 332.
• LUBINSKY, D. S. (1987). A survey of general orthogonal poly nomials for weights on finite and infinite intervals. Acta Appl. Math. 10 237-296.
• MITRINOVI ´C, D. S. (1970). Analy tic Inequalities. Springer, Berlin.
• RAO, M. M. (1989). Bimeasures and harmonizable processes. Probability Measures on Groups IX. Lecture Notes in Math. 1379 254-298. Springer, Berlin.
• SHIRy AYEV, A. N. (1984). Probability. Springer, New York.
• SZEGÖ, G. (1975). Orthogonal Poly nomials. Amer. Math. Soc., Providence, RI.
• VOIT, M. (1990). Central limit theorems for random walks on N0 that are associated with orthogonal poly nomials. J. Multivariate Anal. 34 290-322.
• VOIT, M. (1991). Factorization of probability measures on sy mmetric hy pergroups. J. Austral. Math. Soc. Ser. A 50 417-467.
• YAGLOM, A. M. (1987). Correlation Theory of Stationary and Related Random Functions I. Springer, New York.