平稳随机场的预测理论(Ⅰ)——半平面预测

The Prediction Theory of Stationary Random Fields(Ⅰ) Half-Plane Prediction

本文将对离散参数的平稳随机场的半平面预测理论做比较系统的、严格的论证。内容有四个部分:§1.半平面正则性,奇异性的谱鉴别。§2.半平面正则随机场的典型表示。§3.具有谱密度的情形。§4.正则随机场的半平面滑动和表示。

This paper develops the half-plane predictiou theory of discrete stationary random fields. It consists of four parts. In §1, spectral characterizations of half-plane regularity and singularity of random fields are completely proved. (These characterization theorems have been stated in author's early paper [1] without proof, although their continuous counterparts heve been studied and proved in author's another paper [2]) In §2, We prove the following canonical representation theorem for half-plane regular stationary random field {x(s, t)}x ( s , t )=wheredZy0(λ,μ)=Moreover,1) ProjHx(t-k)6Hx(t-k-1)X(s,t)=2) dFy0(λ,μ)=dFx(λπ)μ 3) ProjHx(-1)X(s,t) = (t>0)4) for t>0,σ2(-t-1)=E|x(s,t)-ProjHx(-1)X(s,t)|2In particular,σ2(-1)=(2π)In §3, we study the particular important case when the random field possesses spectral density. In §4, we prove two theorems about the half-plane moving average representation for the half-plane regular stationary random fields.1) Half-plane regular stationary random field {x(s, t)} possesses the following half-plane moving average representation:x(s ,t)=(Where {u(s, t)} is 2-dim. white noise field.) iff {x(s, t)} possesses spectral density.2 ) the above moving average representation (*) is canonical, (i. e. {u(s, t)} is stationarily related with {x(s, t)} and H11(t) = Hx(t)) iff {x(s, t)} possesses positive spectral density: fr( λ,μ)>0, p·p·(dλdμ)When the condition is satisfied, the coefficients Cmn in (*) can be determined as follows.Cmn= (m = 0, ±1,±2, … n=0,1, 2, … )where(k = 0, 1, 2, …)?moreover,u(s,t)=