Second-order Homogeneous Random Fields
Second-order Homogeneous Random Fields
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二阶齐次随机场
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发表时间：
1961
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通讯作者：
A. Yaglom
中科院分区：
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作者：
A. Yaglom
(1.3)F(A)=Eiz(A)12.这里我们假设过程的时间参数t取所有实值。对于离散参数随机过程，(1.1)和(1.2)中的积分极限必须用-r到+7r来代替。对于具有多维参数t=(t1，t2，*--，tn)的平稳过程，即对于n维空间Rn中的齐次随机场t(T)，以及对于任意局部紧交换群G上的一类更一般的齐次场，存在类似的谱表示[参见下面的公式(2.21)到(2.23)]。此外，对于具有tErn的齐次场t(T)，任何关于其对称性的附加假设都会对协方差函数B(T)以及谱测量F(A)和Z(A)施加特殊限制。从应用的角度来看，最有趣的是均匀各向同性随机场的情况，即具有球对称性的齐次场t(T)。R中这种场的协方差函数B(R)的一般形式，其中T=jtj，由I.J.勋伯格[1]的著名公式给出，即
(1.3) F(A) = EIZ(A)12. We assume here that the time parameter t of the process takes on all real values. For discrete parameter random processes the limits of integration in (1.1) and (1.2) must be replaced by -r to +7r. Analogous spectral representations exist for stationary processes with a multidimensional parameter t = (tl, t2, *--, tn), that is, for homogeneous random fields t(t) in an n-dimensional space Rn, and for a more general class of homogeneous fields on an arbitrary locally compact commutative group G [see formulas (2.21) to (2.23) below]. Moreover, in the case of a homogeneous field t(t) with t E Rn any additional assumptions about its symmetry impose special restrictions on the covariance function B(T) and on the spectral measures F(A) and Z(A). From the point of view of applications the most interesting is the case of a homogeneous and isotropic random field, that is, the homogeneous field t(t) which possesses spherical symmetry. The general form of the covariance function B(r), with T = JTj, of such a field in R. is given by the well-known formula of I. J. Schoenberg [1], namely