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1、Probability,RandomProcesses,andErgodicPropertiesNovember3,2001iiProbability,RandomProcesses,andErgodicPropertiesRobertM.GrayInformationSystemsLaboratoryDepartmentofElectricalEngineeringStanfordUniversityiv°c1987bySpringerVerlag,2001revisionbyRobertM.Gray.vThisbookisa®ecti
2、onatelydedicatedtoElizabethDuboisJordanGrayandtothememoryofR.Adm.AugustineHeardGray,U.S.N.1888-1981SaraJeanDuboisandWilliamBilly"Gray1750-1825viPrefaceHistoryandGoalsThisbookhasbeenwrittenforseveralreasons,notallofwhichareacademic.Thismaterialwasformanyyearsthe¯rsthalfof
3、abookinprogressoninformationandergodictheory.Theintentwasandistoprovideareasonablyself-containedadvancedtreatmentofmeasuretheory,probabilitytheory,andthetheoryofdiscretetimerandomprocesseswithanemphasisongeneralalphabetsandonergodicandstationarypropertiesofrandomprocesses
4、thatmightbeneitherergodicnorstationary.Theintendedaudiencewasmathematicallyinclinedengineeringgraduatestudentsandvisitingscholarswhohadnothadformalcoursesinmeasuretheoreticprobability.Muchofthematerialisfamiliarstu®formathematicians,butmanyofthetopicsandresultshavenotprev
5、iouslyappearedinbooks.Theoriginalprojectgrewtoolargeandthe¯rstpartcontainedmuchthatwouldlikelyboremathematiciansanddiscouragethemfromthesecondpart.HenceI¯nallyfollowedasuggestiontoseparatethematerialandsplittheprojectintwo.Theoriginaljusti¯cationforthepresentmanuscriptwas
6、thepragmaticonethatitwouldbeashametowasteallthee®ortthusfarexpended.Amoreidealisticmotivationwasthatthepresentationhadmeritas¯llingaunique,albeitsmall,holeintheliterature.Personalexperienceindicatesthattheintendedaudiencerarelyhasthetimetotakeacompletecourseinmeasureandpr
7、obabilitytheoryinamathematicsorstatisticsdepartment,atleastnotbeforetheyneedsomeofthematerialintheirresearch.Inaddition,manyoftheexistingmathematicaltextsonthesubjectarehardforthisaudiencetofollow,andtheemphasisisnotwellmatchedtoengineeringapplications.Anotableexceptionis
8、Ash'sexcellenttext[1],whichwaslikelyin°uencedbyhisoriginaltrainingasanelectricalengineer.Still,e