zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the spectral density and asymptotic normality of weakly dependent random fields. (English) Zbl 0787.60059

Let X=(X k ,k d ) be a centered complex weakly stationary random fields. For any disjoint 𝒮,𝒟 d define supEVW ¯ (V 2 ·W 2 =r(S,𝒟) where the supremum is taken over all pairs of random variables V= kS * a k X k , W= k𝒟 * b k X k , where 𝒮 * and 𝒟 * are finite subset of 𝒮 and 𝒟, and a k and b k are complex numbers. For every real number s1, define r * (s)=supr(𝒮,𝒟), where the supremum is taken over all pairs of nonempty disjoint subsets 𝒮,𝒟 d :dist(𝒮,𝒟)s.

Theorem 1. If r * (s)0 as s, then X has a continuous spectral density.

Theorem 2. The following two statements are equivalent: (1) r * (1)<1 and r * (s)0 as s; (2) X has a continuous spectral density.

The central limit theorem is proved. No mixing rate is assumed.


MSC:
60G60Random fields
60G10Stationary processes
60F05Central limit and other weak theorems
References:
[1]Bergh, J., and Löfström, J. (1976).Interpolation Spaces, Springer, New York.
[2]Bradley, R. C. (1987). The central limit question under ?-mixing.Rocky Mountain J. Math. 17, 95-114. · Zbl 0646.60027 · doi:10.1216/RMJ-1987-17-1-95
[3]Bradley, R. C. (1988). A central limit theorem for stationary ?-mixing sequences with infinite variance.Ann. Prob. 16, 313-332. · Zbl 0643.60018 · doi:10.1214/aop/1176991904
[4]Bradley, R. C., and Bryc, W. (1985). Multilinear forms and measures of dependence between random variables.J. Multivar. Anal. 16, 335-367. · Zbl 0586.62086 · doi:10.1016/0047-259X(85)90025-9
[5]Dvoretzky, A. (1972). Asymptotic normality for sums of dependent random variables.Sixth Berkeley Symp. Math. Stat. Prob. 2, 513-535.
[6]Goldie, C. M., and Greenwood, P. (1986). Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes.Ann. Prob. 14, 817-839. · Zbl 0604.60032 · doi:10.1214/aop/1176992440
[7]Goldie, C. M., and Morrow, G. J. (1986). Central limit questions for random fields. In Eberlein, E., and Taqqu, M. S., (eds.).Dependence in Probability and Statistics, Progress in Probability and Statistics, Vol. 11, Birkhäuser, Boston, pp. 275-289.
[8]Gorodetskii, V. V. (1984). The central limit theorem and an invariance principle for weakly dependent random fields.Soviet Math. Dokl. 29, 529-532.
[9]Ibragimov, I. A. (1975). A note on the central limit theorem for dependent random variables.Theor. Prob. Appl. 20, 135-141. · Zbl 0335.60023 · doi:10.1137/1120011
[10]Ibragimov, I. A., and Linnik, Yu. V. (1971).Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen.
[11]Ibragimov, I. A., and Rozanov, Y. A. (1978).Gaussian Random Processes, Springer, New York.
[12]Peligrad, M. (1986). Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables (a survey). In Eberlein, E., and Taqqu, M. S. (eds.),Dependence in Probability and Statistics, Progress in Probability and Statistics, Vol. 11, Birkhäuser, Boston, pp. 193-223.
[13]Rosenblatt, M. (1985).Stationary Sequences and Random Fields, Birkhäuser, Boston.
[14]Sarason, D. (1972). An addendum to ?Past and Future?.Math. Scand. 30, 62-64.
[15]Shao, Q. (1986).An Invariance Principle for Stationary ?-Mixing Sequences with Infinite Variance. Report, Department of Mathematics, Hangzhou University, Hangzhou, Peoples Republic of China.
[16]Withers, C. S. (1981). Central limit theorems for dependent random variables.Z. Wahrsch. verw. Gebiete 57, 509-534. · Zbl 0451.60027 · doi:10.1007/BF01025872
[17]Zhurbenko, I. G. (1986).The Spectral Analysis of Time Series, North-Holland, Amsterdam.