The Annals of Probability

Stationary random fields with linear regressions

Włodzimierz Bryc

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We analyze and identify stationary fields with linear regressions and quadratic conditional variances. We give sufficient conditions to determine one-dimensional distributions uniquely as normal and as certain compactly supported distributions. Our technique relies on orthogonal polynomials, which under our assumptions turn out to be a version of the so-called continuous q-Hermite polynomials.

Article information

Ann. Probab., Volume 29, Number 1 (2001), 504-519.

First available in Project Euclid: 21 December 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E99: None of the above, but in this section
Secondary: 60G10: Stationary processes

Conditional moments hypergeometric orthogonal polynomials q-Hermite polynomials q -Gaussian processes linear regression


Bryc, Włodzimierz. Stationary random fields with linear regressions. Ann. Probab. 29 (2001), no. 1, 504--519. doi:10.1214/aop/1008956342.

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  • [1] Akhiezer, N. I. (1965). The classical moment problem. Oliver and Boyd, Edinburgh.
  • [2] Allaway, W. (1980). Some properties of the q-Hermite polynomials. Canad. J. Math. 32 686-694.
  • [3] Askey, R. (1989). Continuous q-Hermite polynomials when q > 1. In q-Series and Partitions (D. Stanton, ed.) Springer, New York.
  • [4] Bryc, W. (1985). Some remarks on random vectors with nice enough behaviour of conditional moments. Bull. Polish Acad. Sci. 33 677-683.
  • [5] Bryc, W. (1995). Normal distribution: characterizations with applications. Lecture Notes in Statist. 100. Springer, Berlin.
  • [6] Bryc, W. (2000). Stationary Markov chains with linear regressions. Stoch. Process. Appl. To appear.
  • [7] Bryc, W. and Pluci ´nska, A. (1985). A characterization of infinite Gaussian sequences by conditional moments. Sankhy ¯a Ser. A 47 166-173.
  • [8] Carlitz, L. (1955). Some polynomials related to theta functions. Ann. Mat. Pura Appl. 41 359-373.
  • [9] Ismai, M. E. H., Stanton, D. and Viennot, G. (1987). The combinatorics of q-Hermite polynomials and the Askey-Wilson integral. European J. Combin. 8 379-392.
  • [10] Koekoek, R. and Swarttouw, R. F. (1994). The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 94-05, Technische Univ., Delft.
  • [11] Matysiak, W. (1999). Private communication.
  • [12] Pluci ´nska, A. (1983). On a stochastic process determined by the conditional expectation and the conditional variance. Stochastics 10 115-129.
  • [13] Shohat, J. A. and Tamarkin, J. D. (1943). The Problem of Moments. Amer. Math. Soc., New York.
  • [14] Szablowski, P. (1989). Can the first two conditional moments identify a mean square differentiable process? Comput. Math. Appl. 18 329-348.
  • [15] Szablowski, P. (1990). Expansions of E X Y + X and their applications to the analysis of elliptically contoured measures. Comput. Math. Appl. 19 75-83.
  • [16] Wesolowski, J. (1993). Stochastic process with linear conditional expectation and quadratic conditional variance. Probab. Math. Statist. 14 33-44.