17.3 Stationary Handoni Processes 413 That means that for s t a t i o ~ random ~ r ~ processes, the first-order expected wfiies not ctepeitd oil time In partictilar, for the linear average and the va,riance: /&(tj = ps; 2 ( ) = o, z * (17.14) The auto-correlation filnction c m be expressed more simply with z = tl - tz: E{Z(tl)s(t2)) f= E(zftl)z(tl,.- c')) = E { ~ ( t + z ~ > ~ : f l: a )pXz(.) } (17.15) The a , u L o - c ~ ~ r ~ I afunction , ~ i ~ n pZ2(z); that we introduced in (17.15j ;ls R orie~ ~ i ~ e fuiiction n ~ ~ isonot ~ the a ~same fimctitm as pzz(tl,tz) in (17,101,which is a funcliori of two vttriables, t L and t z The two pE1: are linked for sta,tionaxyrandom sigmls (17.15) If the stationary coiidition only holds for (17.16) (17.17) however, tmd not for gewral funcLions f ( , ), then the random process ts called weak s ~ ~ ~ ~Also ~ for ~ ~the( linear ~ r ~ avex age, the vaxiaace iilid the ~ ~ u ~ o - ~ o r r e i a ~ i o ~ function of weak stationary proc PS (17.131, (17.14), arid (17.15) hold We can thus obttiin the following staternent: for a weak stationary process the linear average and the correlatioii properties coiitainecl in the auto-corrclation function are is iisually used in constant with time The concept of weak stationary proces the context of rnotlelling and analysis of random procesms, where it is often a, prclcondition when only the linear average and auLtw.orrelation fund ion itre coxisidered, weal