Springer.Markov Chains and Stochastic Stability.2003.pdf

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1、PrefaceBooksareindividualandidiosyncratic.Intryingtounderstandwhatmakesagoodbook,thereisalimitedamountthatonecanlearnfromotherbooks;butatleastonecanreadtheirprefaces,inhopeofhelp.Ourownresearchshowsthatauthorsuseprefacesformanydifferentreasons.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.Thisprefaceisnodifferent

4、.Letusbeginwiththosewehopewillusethebook.Whowantsthisstuffanyway?ThisbookisaboutMarkovchainsongeneralstatespaces:sequencesΦnevolvingrandomlyintimewhichremembertheirpasttrajectoryonlythroughitsmostrecentvalue.Wedeveloptheirtheoreticalstructureandwedescribetheirapplication.Thetheoryofgeneralstat

5、espacechainshasmaturedoverthepasttwentyyearsinwayswhichmakeitverymuchmoreaccessible,verymuchmorecomplete,and(weatleastthink)ratherbeautifultolearnanduse.Wehavetriedtoconveyallofthis,andtoconveyitatalevelthatisnomoredifficultthanthecorrespondingcountablespacetheory.Theeasiestreaderforustoenvisag

6、eisthelong-sufferinggraduatestudent,whoisexpected,inmanydisciplines,totakeacourseoncountablespaceMarkovchains.Suchagraduatestudentshouldbeabletoreadalmostallofthegeneralspacetheoryinthisbookwithoutanymathematicalbackgrounddeeperthanthatneededforstudyingchainsoncountablespaces,providedonlythatt

7、hefearofseeinganin-tegralratherthanasummationsigncanbeovercome.Verylittlemeasuretheoryoranalysisisrequired:virtuallynomoreinmostplacesthanmustbeusedtodefinetransitionprobabilities.TheremarkableNummelin-Athreya-Neyregenerationtech-nique,togethe