linear statistical problems for stationary isotropic ...

Theor. Probability and Math. Statist. No. 18, 1979

T eo p . BeposTHOCT. h MaT. CraTHCT. B u n . 18, 1978

M. P. MOKUACUK AND M. I. JADRENKO K ie v U niversity

LINEAR STATISTICAL PROBLEMS FOR STATIONARY ISOTROPIC RANDOM FIELDS ON A SPHERE. I. A b stra c t.

T he sp e ctral re p re se n ta tio n o f a sta tio n a ry iso tro p ic ra n d o m

field o n a sp h ere, as w ell as its co rre la tio n fu n c tio n , are describ ed ; ex am p les o f co rre la tio n fu n c tio n s o f such fields are p ro d u c e d .

T he p ro b lem o f estim atin g

th e regression co effic ie n ts an d th e u n k n o w n e x p e c ta tio n o f a ra n d o m field is discussed. B ib liography:

11 title s.

UDC 519.21

Various problems in the theory of automatic control, radiophysics, geo­ chemistry, astronomy and meteorology force us to consider random functions depending on time and on a point on a sphere. In this context it is sometimes natural to assume stationarity in time and isotropism in space of the random functions (random fields). In the present paper, which is divided into two parts, linear statistical problems on stationary isotropic random fields on a sphere are discussed. In the first part, the spectral representation o f a stationary isotropic field on a sphere is described, examples of correlation functions o f such fields are produced, and the problem of estimating the unknown expectation of a random field is discussed. The second part (to appear in the next issue of this journal) will be devoted to the problem o f linear extrapolation for a stationary isotropic random field on a sphere. The results contained in this paper were the subject of a communication at the Fourth International Symposium on Information Theory (LeningradRepino, June, 1976).

§1. Notation and auxiliary information about spherical harmonics Let S

be a unit sphere in n-dimensional Euclidean space, (d x, . . . , Q„_2 ,

the spherical coordinates o f the point x € S n , and S lm (d 1, . . . , 0n _ 2 , y>) the orthonorm al spherical harmonics o f degree m (I = 1, H m , n ) - ( 2 m + n - 2 ) ("

, h(m, «)), (1.1)

being the number o f linearly independent spherical harmonics o f degree m (con19 8 0 M a th e m a tic s S u b je c t C lassification. 33A 4S , 6 2 G 9 9 .

P rim ary 6 0 G 6 0 ; S eco n d ary 6 0 G 1 0 , 62M 10,

C o p y rig h t © 1 9 8 0 , A m erican M a th em atic al S o ciety

115

M. P. MOKLJACUK AND M. I. JADRENKO

116

cerning the properties o f spherical harmonics, see, for instance, [1], [2], or [3]). Let Cvm be the Gegenbauer polynomials, defined by their generating function

(1 -

2zt + ^ )~v = J

C l (z) t m.

(1.2)

m= 0 We shall take advantage o f the following two im portant propositions in the theory of spherical functions: T h e C o m po sit io n T h e o r e m

fo r

S p h e r ic a l H a r m o n ic s . For any two

points JCj, x 2 ^ S n , n—2 ~T~

him.n) y

a

=

(1.3) n

—

Cm

i=1

( 1)

where cos 8 = cosOCj, x 2) = (X j, x 2) is the “angular” distance between the points Xj and x 2, and con = 2nn^2 T(«/2). T h e F u n k -He c k e T h e o r e m . I f \p(v) is a continuous function on [ - 1 , 1 ] ,

then, fo r any point x G S n and any m and I such that m > 0 and 1 < I < h(m, ri),

| i|) (cos ( x, y )) S lm (y) mn (dy) = bmSlm (x),

(1.4)

where

=

CÙ p/x—1 c j

1

n—2

j

n—3

(t) ( I - F) 2 dt,

(1.5)

(1) - 1

and m n( ■) is Lebesgue measure on S n . We also note the following relation, which can be obtained from (1.2): n—2 °° C ( 2 £ h (m >n ) ^ 2- - - tm = -------------------- „ m -°

Cm2 (1)

( i n < 1 , « > 2 ) . (1.6)

( 1 - 2 tx + t*)‘‘

§2. The spectral expansion of a stationary isotropic random field on a sphere Let %(t, x ) be a random field on R x S n , where R = (—°°,

We assume

LINEAR STATISTICAL PROBLEMS FOR RANDOM FIELDS. I

117

that f(t, x ) is mean square continuous, and that M 'f2(t, x ) < °°.* We call the random field iff, x) a (wide-sense) stationary isotropic random field on a sphere if M%(t, x ) is a constant (in what follows, we shall assume M%(t, x ) = 0), and if

M l (t, x) I (s, y) — B (t — s, cos (x,