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Title: On a simple estimate of correlations of stationary random sequences (English)
Author: Hurt, Jan
Language: English
Journal: Aplikace matematiky
ISSN: 0373-6725
Volume: 18
Issue: 3
Year: 1973
Pages: 176-187
Summary lang: English
Summary lang: Czech
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Category: math
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Summary: Suppose that $\{X_t\}_{t\in T}$ is a stationary Gaussian discrete random process where $T$ is the set of integers. Assume $EX_t=0,\ t\in T$, and denote $Z_t=sign X_t,\ T_{tj}=Z_tZ_{t+j}$ for $j$ natural. It is shown that $ET_{tj}=2\ arcsin\ \rho_j/\pi$ so that the quantities $T_{tj}$ may be used to estimate the correlation function $\{\rho_j\}_{j \in N}$. (Here $\rho_j$ denotes the correlation between $X_t$ and $X_{t+j}$.) Further, the formula for $cov(T_{0j},T_{kj})$ in terms of $\rho$'s is given. Asymptotic properties of the mean $\bar{T}_j=\sum^{N-j}_{t=1} T_{tj}/(N-j)$ are studied under the asumption that the spectral density of $\{X_t\}_{t\in T}$ is nonzero and possesses bounded second derivative. Particularly, the derived results hold for stationary autoregressive Gaussian random sequences which is the most important case in practice. It is proved that $\bar{T}_j}$ is asymptotically normally distributed and that the sequence $\{T_{tj}\}_{t\in T}$ satisfies the law of large numbers. Finally, some numerical examples and Monte-Carlo studies are given. ()
MSC: 62E20
MSC: 62M10
idZBL: Zbl 0265.62032
idMR: MR0317496
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Date available: 2008-05-20T17:56:07Z
Last updated: 2015-08-06
Stable URL: http://hdl.handle.net/10338.dmlcz/103468
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Reference: [1] Hájek J., Anděl J.: Stationary processes.(Lecture Notes). SPN, Praha, 1969.
Reference: [2] Hurt J.: Tests for the goodness of fit in stationary autoregressive sequences.(Unpublished thesis.)
Reference: [3] Ibragimov I. A., Linnik Yu. V.: Independent and stationary sequences of random variables.(Russian), Nauka, Moskva, 1965.
Reference: [4] Kendall M. G.: Tables of autoregressive series.Biometrika 36 (1949), p. 267. MR 0033506, 10.1093/biomet/36.3-4.267
Reference: [5] Loève M.: Probability theory.(3d ed.). D. Van Nostrand, New York, 1963. MR 0203748
Reference: [6] Plackett R. L.: A reduction formula for normal multivariate integrals.Biometrika 41 (1954), p. 351. Zbl 0056.35702, MR 0065047, 10.1093/biomet/41.3-4.351
Reference: [7] Rozanov Yu. V.: Stationary random processes.(Russian). GIFML, Moskva, 1963.
Reference: [8] Wold H.: Random normal deviates.Tracts for Computers, No. XXV, Cambridge, 1948. Zbl 0036.20603
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