Course in Probability Theory

9.5 APPLICATIONS 1 361PROOF. Let A _ {Xj E Aj i .o .} and use the notation An and M above .We may ignore the null sets in (1) and (2) . Then if w E A, our hypothesisimplies thatZII{An+l I X„}((o) > 8 i.o .In view of (1) this is possible only if w E M . Thus A C M, which implies (3) .The intuitive meaning of the preceding theorem has been given byDoeblin as follows : if the chance of a pedestrian's getting run over is greaterthan 8 > 0 each time he crosses a certain street, then he will not be crossingit indefinitely (since he will be killed first)! Here IX, E A,,) is the event ofthe nth crossing, {Xn E Bn} that of being run over at the nth crossing .(II) Harmonic and superharmonic functions for a Markov processLet {X,,, n E N°} be a homogeneous Markov process as discussed in Sec . 9 .2with the transition probability function P( ., . ). An extended-valued functionf on R1 is said to be harmonic (with respect to P) iff it is integrable withrespect to the measure P(x, .) for each x and satisfies the following "harmonicequation" ;(4) Vx E ~W1 : f (x) = f P(x, dy)f(y) .It is superharmonic (with respect to P) iff the "_" in (4) is replaced byin this case f may take the value +oo .Lemma . If f is [super]harmonic, then { f (X,), n e N° }, where X0 - x° forsome given x° in is a [super]martingale .PROOF . We have, recalling (14) of Sec . 9 .2,{f(Xn)} = f P" (xo,dy)f(y) < oc,as is easily seen by iterating (4) and applying an extended form of Fubini'stheorem (see, e .g ., Neveu [6]). Next we have, upon substituting X„ for x in (4) :f (X,) = f P(Xn, d y)f (y) = (`{ .f (Xn+l) I Xn} = ("If (Xn+1) 1 ~[°,nl},where the second equation follows by Exercise 8 of Sec . 9 .2 and the third byMarkov property . This proves the lemma in the harmonic case ; the other caseis similar . (Why not also the "sub" case?)The most important example of a harmonic function is the g( ., B) ofExercise 10 of Sec . 9 .2 for a given B ; that of a superharmonic function is the