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1、Probability,RandomProcesses,andErgodicPropertiesNovember3,2001iiProbability,RandomProcesses,andErgodicPropertiesRobertM.GrayInformationSystemsLaboratoryDepartmentofElectricalEngineeringStanfordUniversityiv°c1987bySpringerVerlag,2001revisionbyRobertM.Gray.vThisbookisa®e
2、ctionatelydedicatedtoElizabethDuboisJordanGrayandtothememoryofR.Adm.AugustineHeardGray,U.S.N.1888-1981SaraJeanDuboisandWilliamBilly"Gray1750-1825viPrefaceHistoryandGoalsThisbookhasbeenwrittenforseveralreasons,notallofwhichareacademic.Thismaterialwasformanyyearsthe¯rst
3、halfofabookinprogressoninformationandergodictheory.Theintentwasandistoprovideareasonablyself-containedadvancedtreatmentofmeasuretheory,probabilitytheory,andthetheoryofdiscretetimerandomprocesseswithanemphasisongeneralalphabetsandonergodicandstationarypropertiesofrandom
4、processesthatmightbeneitherergodicnorstationary.Theintendedaudiencewasmathematicallyinclinedengineeringgraduatestudentsandvisitingscholarswhohadnothadformalcoursesinmeasuretheoreticprobability.Muchofthematerialisfamiliarstu®formathematicians,butmanyofthetopicsandresult
5、shavenotpreviouslyappearedinbooks.Theoriginalprojectgrewtoolargeandthe¯rstpartcontainedmuchthatwouldlikelyboremathematiciansanddiscouragethemfromthesecondpart.HenceI¯nallyfollowedasuggestiontoseparatethematerialandsplittheprojectintwo.Theoriginaljusti¯cationfortheprese
6、ntmanuscriptwasthepragmaticonethatitwouldbeashametowasteallthee®ortthusfarexpended.Amoreidealisticmotivationwasthatthepresentationhadmeritas¯llingaunique,albeitsmall,holeintheliterature.Personalexperienceindicatesthattheintendedaudiencerarelyhasthetimetotakeacompleteco
7、urseinmeasureandprobabilitytheoryinamathematicsorstatisticsdepartment,atleastnotbeforetheyneedsomeofthematerialintheirresearch.Inaddition,manyoftheexistingmathematicaltextsonthesubjectarehardforthisaudiencetofollow,andtheemphasisisnotwellmatchedtoengineeringapplication
8、s.AnotableexceptionisAsh'sexcellenttext[1],whichwaslikelyin°uencedbyhisoriginaltrainingasanelectricalengineer.Still,e
