Summary
For sums of finite range potential functions of an iid random field we derive the validity of formal expansions of length two. Under standard conditions, formal expansions are valid if and only if the characteristic functions of the sum converge to zero for all nonzero frequency parameters. If this convergence fails, the distribution of the sum can be approximated by a mixture of lattice distributions. The result applies to m-dependent random fields generated by independent random variables.
Article PDF
Similar content being viewed by others
References
Bhattacharya, R.N., Rango Rao, R.: Normal approximation and asymptotic expansions. New York: Wiley 1976
Götze, F., Hipp, C.: Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrscheinlichkeitstheor. Verw. Geb. 64, 211–239 (1983)
Götze, F., Hipp, C.: Local limit theorems for sums of finite range potentials of a Gibbsian random field. Ann. Probab. (in press)
Guyon, K., Richardson, S.: Vitesse de convergence du théoréme de la limite centrale pour des champs faiblement dépendents. Z. Wahrscheinlichkeitstheor. Verw. Geb. 66, 297–314 (1984)
Heinrich, L.: A method for the derivation of limit theorems for random fields with finite range dependence. Z. Wahrscheinlichkeitstheor. Verw. Geb. 60, 501–515 (1982)
Heinrich, L.: Non-uniform estimates and asymptotic expansions of the remainder in the central limit theorem for m-dependent random variables. Math. Nachr. 115, 7–20 (1984)
Heinrich, L.: Stable limit theorems for sums of multiply indexed m-dependent random variables. Math. Nachr. 127, 193–210 (1986)
Heinrich, L.: Asymptotic expansions in the central limit theorem for a special class of m-dependent Random Fields I. Math. Nachr. 134, 83–106 (1987)
Petrov, V.V.: Sums of independent random variables. Berlin Heidelberg New York: Springer 1975
Prakasa Rao, B.L.S.: A non-uniform estimate of the rate of convergence in the central limit theorem for m-dependent random fields. Z. Wahrscheinlichkeitstheor. Verw. Geb. 58, 247–256 (1981)
Riauba, B.: Speed of convergence in the central limit theorem for m-dependent random fields. Litovsk. Mat. Sb. 20, 157–163 (1980) [English Translation: Lithuanian Math. J. 20, 71–75 (1980)]
Shergin, V.V.: On the convergence rate in the central limit theorem for m-dependent random variables. Teor. Verojatnost. i Primenen 24, 781–794 (1979) [English Translation: Theor. Probability Appl. 24, 782–796 (1979)]
Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Symp. Math. Stat. Probab. vol. II, pp. 583–602 1972
Takahata, H.: On the rates in the central limit theorem for weakly dependent random fields. Z. Wahrscheinlichkeitstheor. Verw. Geb. 64, 445–456 (1983)
Tikhomirov, A.N.: On the convergence rate in the central limit theorem for weakly dependent random variables. Teor. Verojatnost. i Primenen 25, 800–818 (1980) [English Translation: Theor. Probability Appl. 25, 790–809 (1980)]
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Götze, F., Hipp, C. Asymptotic expansions for potential functions of i.i.d. random fields. Probab. Th. Rel. Fields 82, 349–370 (1989). https://doi.org/10.1007/BF00339992
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00339992