Course in Probability Theory

8 .2 BASIC NOTIONS 1 275PROOF . The independence follows from Theorem 8 .2.2 by our showingthat 1 ,11, . . . , Vk_ 1 belong to the pre-,Bk_ 1 field, while V k belongs to the post-$k_1 field . The details are left to the reader ; cf. Exercise 6 below .To prove that Vk and Vk+1 have the same distribution, we may supposethat k = 1 . Then for each n E N, and each n-dimensional Borel set A, we have{w : a 2 (w) = n ; (Xaj+1(w), . . ., Xai+a 2 ((O)) E A}= {co : a l (t"co) = n ; (X 1 (r"CO), . . ., Xa i (r" (0) E A),sinceXaI (r " CJ) = X a i(~W)(t " co) = X a 2(w )(taw)= XaI (m)+a2((0) (co) = X,1 +,2 (w)by the quirk of notation that denotes by X,,( .) the function whose value atw is given by Xa( ,, ) (co) and by (7) with n = a2 (co) . By (5) of Sec . 8 .1, thepreceding set is the r-"-image (inverse image under r') of the set{w : a l (w) = n ; (X 1(w), . . ., Xal (w)) E A),and so by (10) has the same probability as the latter . This proves our assertion .Corollary. The r .v .'s {Y k , k E N}, whereYk(W) =f+1(P(Xn(w))and co is a Borel measurable function, are independent and identicallydistributed .For cP - 1, Y k reduces to ak . For co(x) - x, Yk = S& - S $A _, . The readeris advised to get a clear picture of the quantities ak, 13k' and Y k beforeproceeding further, perhaps by considering a special case such as (5) .We shall now apply these considerations to obtain results on the "globalbehavior" of the random walk . These will be broad qualitative statementsdistinguished by their generality without any additional assumptions .The optional r .v . to be considered is the first entrance time into the strictlypositive half of the real line, namely A = (0, oc) in (5) above . Similar resultshold for [0, oc) ; and then by taking the negative of each X n , we may deducethe corresponding result for (-oc, 0] or (-oo, 0) . Results obtained in this waywill be labeled as "dual" below . Thus, omitting A from the notation :00min{n E N : Sn > 0} on U {w : S n (w) > 0} ;(1 1) a(co)n=1+cc elsewhere ;