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1、PrefaceBooksareindividualandidiosyncratic.Intryingtounderstandwhatmakesagoodbook,thereisalimitedamountthatonecanlearnfromotherbooks;butatleastonecanreadtheirprefaces,inhopeofhelp.Ourownresearchshowsthatauthorsuseprefacesformanydi?erentreasons.Prefacescanbeexplanationsoftheroleandthecontentsof
2、thebook,asinChung[49]orRevuz[223]orNummelin[202];thiscanbecombinedwithwhatisalmostanapologyforbotheringthereader,asinBillingsley[25]orC?inlar[40];prefacescandescribethemathematics,asinOrey[208],ortheimportanceoftheapplications,asinTong[267]orAsmussen[10],orthewayinwhichthebookworksasatext,asi
3、nBrockwellandDavis[32]orRevuz[223];theycanbetheonlyavailableoutletforthankingthosewhomadethetaskofwritingpossible,asinalmostalloftheabove(althoughweparticularlylikethefamilialgratitudeofResnick[222]andthededicationofSimmons[240]);theycancombinealltheseroles,andmanymore.Thisprefaceisnodi?erent
4、.Letusbeginwiththosewehopewillusethebook.Whowantsthisstu?anyway?ThisbookisaboutMarkovchainsongeneralstatespaces:sequencesΦnevolvingrandomlyintimewhichremembertheirpasttrajectoryonlythroughitsmostrecentvalue.Wedeveloptheirtheoreticalstructureandwedescribetheirapplication.Thetheoryofgeneralstat
5、espacechainshasmaturedoverthepasttwentyyearsinwayswhichmakeitverymuchmoreaccessible,verymuchmorecomplete,and(weatleastthink)ratherbeautifultolearnanduse.Wehavetriedtoconveyallofthis,andtoconveyitatalevelthatisnomoredi?cultthanthecorrespondingcountablespacetheory.Theeasiestreaderforustoenvisag
6、eisthelong-su?eringgraduatestudent,whoisexpected,inmanydisciplines,totakeacourseoncountablespaceMarkovchains.Suchagraduatestudentshouldbeabletoreadalmostallofthegeneralspacetheoryinthisbookwithoutanymathematicalbackgrounddeeperthanthatneededforstudyingchainsoncountablespaces,providedonlythatt
7、hefearofseeinganin-tegralratherthanasummationsigncanbeovercome.Verylittlemeasuretheoryoranalysisisrequired:virtuallynomoreinmostplacesthanmustbeusedtode?netransitionprobabilities.TheremarkableNummelin-Athreya-Neyregenerationtech-nique,togethe
