## The Annals of Statistics

- Ann. Statist.
- Volume 1, Number 4 (1973), 780-785.

### Estimation of the Covariance Function of a Homogeneous Process on the Sphere

#### Abstract

A homogeneous random process on the sphere $\{X(P): P \in S_2\}$ is a process whose mean is zero and whose covariance function depends only on the angular distance $\theta$ between the two points, i.e. $E\lbrack X(P)\rbrack \equiv 0$ and $E\lbrack X(P)X(Q)\rbrack = R(\theta)$. Given $T$ independent realizations of a Gaussian homogeneous process $X(P)$, we first derive the exact distribution of the spectral estimates introduced by Jones (1963 b). Further, an estimate $R^{(T)}(\theta)$ of the covariance function $R(\theta)$ is proposed. Exact expressions for its first- and second-order moments are derived and it is shown that the sequence of processes $\{T^{\frac{1}{2}}\lbrack R^{(T)}(\theta) - R(\theta)\rbrack\}^\infty_{T=1}$ converges weakly in $C\lbrack 0, \pi\rbrack$ to a given Gaussian process.

#### Article information

**Source**

Ann. Statist., Volume 1, Number 4 (1973), 780-785.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342475

**Digital Object Identifier**

doi:10.1214/aos/1176342475

**Mathematical Reviews number (MathSciNet)**

MR334443

**Zentralblatt MATH identifier**

0263.62052

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62M15: Spectral analysis

Secondary: 60G10: Stationary processes 60G15: Gaussian processes

**Keywords**

Homogeneous process on the sphere spectral estimates estimate of the covariance function weak convergence

#### Citation

Roy, Roch. Estimation of the Covariance Function of a Homogeneous Process on the Sphere. Ann. Statist. 1 (1973), no. 4, 780--785. doi:10.1214/aos/1176342475. https://projecteuclid.org/euclid.aos/1176342475