Course in Probability Theory

3 60 1 CONDITIONING . MARKOV PROPERTY . MARTINGALE9.5 ApplicationsAlthough some of the major successes of martingale theory lie in the field ofcontinuous-parameter stochastic processes, which cannot be discussed here, ithas also made various important contributions within the scope of this work .We shall illustrate these below with a few that are related to our previoustopics, and indicate some others among the exercises .(I)The notions of "at least once" and "infinitely often"These have been a recurring theme in Chapters 4, 5, 8, and 9 and play importantroles in the theory of random walk and its generalization to Markovprocesses . Let {X,I , n E N ° ) be an arbitrary stochastic process ; the notationfor fields in Sec . 9 .2 will be used . For each n consider the events :00A n =U{Xj EB1 ),j=n00M= I I A n = {Xj E B j i .o .},n=1where B n are arbitrary Borel sets .Theorem 9 .5 .1 .We have(1) lim ,?I {An+1 13[0,n)} = 1 M a .e .,where [o .n] may be replaced by {n } or X n if the process is Markovian .PROOF .By Theorem 9 .4 .8, (14a), the limit is~'{M I ~io,00>} = 1 M .The next result is a "principle of recurrence" which is useful in Markovprocesses ; it is an extension of the idea in Theorem 9 .2 .3 (see also Exercises 15and 16 of Sec . 9 .2) .Theorem 9 .5 .2 . Let {Xn , n E N°} be a Markov process and A, Bn Borelsets . Suppose that there exists 8>0 such that for every n,00(2) ){ Uthen we have[X j E B j] I X n } > 8 a .e . on the set {X n E A n } ;j=n+1(3) J'{[X j E A j i .o.]\[Xj E Bj i .o .]} = 0 .