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.
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.
@article{10_21136_AM_1973_103468,
author = {Hurt, Jan},
title = {On a simple estimate of correlations of stationary random sequences},
journal = {Applications of Mathematics},
pages = {176--187},
year = {1973},
volume = {18},
number = {3},
doi = {10.21136/AM.1973.103468},
mrnumber = {0317496},
zbl = {0265.62032},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1973.103468/}
}
TY - JOUR
AU - Hurt, Jan
TI - On a simple estimate of correlations of stationary random sequences
JO - Applications of Mathematics
PY - 1973
SP - 176
EP - 187
VL - 18
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.21136/AM.1973.103468/
DO - 10.21136/AM.1973.103468
LA - en
ID - 10_21136_AM_1973_103468
ER -
%0 Journal Article
%A Hurt, Jan
%T On a simple estimate of correlations of stationary random sequences
%J Applications of Mathematics
%D 1973
%P 176-187
%V 18
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/AM.1973.103468/
%R 10.21136/AM.1973.103468
%G en
%F 10_21136_AM_1973_103468
Hurt, Jan. On a simple estimate of correlations of stationary random sequences. Applications of Mathematics, Tome 18 (1973) no. 3, pp. 176-187. doi: 10.21136/AM.1973.103468
