Summary
This work extends to symmetric α-stable (SαS) processes, 1<α<2, which are Fourier transforms of independently scattered random measures on locally compact Abelian groups, some of the basic results known for processes with finite second moments and for Gaussian processes. Analytic conditions for subordination of left (right) stationarily related processes and a weak law of large numbers are obtained. The main results deal with the interpolation problem. Characterization of minimal and interpolable processes on discrete groups are derived. Also formulas for the interpolator and the corresponding interpolation error are given. This yields a solution of the interpolation problem for the considered class of stable processes in this general setting.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Blum, J.R., Eisenberg, B.: Conditions for metric transitivity for stationary Gaussian processes on groups. Ann. Math. Statist. 43, 1737–1741 (1972)
Bruckner, L.: Interpolation of homogeneous random fields on discrete groups. Ann. Math. Statist. 40, 251–258 (1969)
Cambanis, S.: Complex symmetric stable variables and processes. In: Contributions to statistics: Essays in Honour of Norman L. Johnson, pp. 63–79. P.K. Sen (ed.). New York: North Holland 1983
Cambanis, S., Hardin, C.D., Jr., Weron, A.: Ergodic properties of stationary stable processes. Center for Stochastic Processes Tech. Rept. No. 59, Statistics Dept. Univ. of North Carolina, 1984
Cambanis, S., Miller, G.: Linear problems in pth order and stable processes. SIAM J. Appl. Math. 41, 43–69 (1981)
Cambanis, S., Soltani, R.: Prediction of stable processes: Spectral and moving average representations. Z. Wahrscheinlichkeitstheorie verw. Gebiete 66, 593–612 (1984)
Driscoll, M.F., McDonald, J.N., Weiss, N.A.: LLN for weakly stationary processes on locally compact Abelian groups. Ann. Probability 2, 1168–1171 (1974)
Hosoya, Y.: Harmonizable stable processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 60, 517–533 (1982)
Kolmogorov, A.N.: Stationary sequences in Hilbert space. Bull. Math. Moscow 2, 1–40 (1941)
Makagon, A., Weron, A.: q-variate minimal stationary processes. Studia Math. 59, 41–52 (1976)
Marcus, M.B., Pisier, G.: Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. Preprint (1982)
Masry, E., Cambanis, S.: Spectral density estimation for stationary stable processes. Stochastic Proc. Appl. 18, 1–31 (1984)
Morettin, P.A.: Walsh spectral analysis. SIAM Rev. 23, 279–291 (1981)
Pourahmadi, M.: Prediction and interpolation of stable processes. Preprint (1982)
Roy, R.: Spectral analysis for a random process on the sphere. Ann. Inst. Statist. Math. 28, 91–97 (1976)
Rudin, W.: Fourier Analysis on Groups. Interscience, Wiley, New York 1962.
Salehi, H.: Algorithms for linear interpolator and interpolation error for minimal stationary stochastic processes. Ann. Probability 7, 840–846 (1979)
Urbanik, K.: Lectures on Prediction Theory, Lecture Notes in Math. 44. Berlin-Heidelberg-New York: Springer 1967
Urbanik, K.: Harmonizable sequences of random variables. Colloques internationaux du C.N.R.S. 186, 345–361 (1970)
Weron, A.: On characterizations of interpolable and minimal stationary processes. Studia Math. 49, 165–183 (1974)
Weron, A.: Stable processes and measures: A survey. Lecture Notes in Math. 1080, pp. 306–364. Berlin-Heidelberg-New York: Springer 1984
Yaglom, A.M.: On a problem of linear interpolation of stationary random sequences and processes. Amer. Math. Soc. Sel. Transl. Math. Statist. 4, 330–344 (1963)
Author information
Authors and Affiliations
Additional information
Research supported by AFOSR contract F49620 82 C0009
On leave from the Institute of Mathematics, Technical University 50-370 Wrocław, Poland
Rights and permissions
About this article
Cite this article
Weron, A. Harmonizable stable processes on groups: spectral, ergodic and interpolation properties. Z. Wahrscheinlichkeitstheorie verw Gebiete 68, 473–491 (1985). https://doi.org/10.1007/BF00535340
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00535340