| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| DOI:
|
10.21136/AM.1973.103468 |
| . |
| Date available:
|
2008-05-20T17:56:07Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103468 |
| . |
| 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[2] Hurt J.: Tests for the goodness of fit in stationary autoregressive sequences.(Unpublished thesis.)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 |
| . |
