Course in Probability Theory

348 1 CONDITIONING . MARKOV PROPERTY . MARTINGALEWe now come to a new kind of inequality, which will be the tool forproving the main convergence theorem below . Given any sequence of r .v .'s{X j }, for each sample point w, the convergence properties of the numericalsequence {X j (w)} hinge on the oscillation of the finite segments {Xj(w), j EN„ } as n -* cc . In particular the sequence will have a limit, finite or infinite, ifand only if the number of its oscillations between any two [rational] numbers aand b is finite (depending on a, b and w) . This is a standard type of argumentused in measure and integration theory (cf. Exercise 10 of Sec . 4 .2) . Theinteresting thing is that for a smartingale, a sharp estimate of the expectednumber of oscillations is obtainable .Let a < b . The number v of "upcrossings" of the interval [a, b] by anumerical sequence {x 1 , . . . , x, } is defined as follows . Setal = min{ j : 1 < j < n, xj < a},a2 =min{ j : ai < j < n, x j > b) ;if either al or a2 is not defined because no such j exists, we define v = 0 . Ingeneral, for k > 2 we seta2k-1 = min{j :a2k-2 < j ::~ n, xj < a},a2k = min{j : a2k-1 < j < n, x> > b} ;if any one of these is undefined, then all the subsequent ones will be undefined .Let at be the last defined one, with £ = 0 if al is undefined, then v is definedto be [f/2] . Thus v is the actual number of successive times that the sequencecrosses from < a to > b . Although the exact number is not essential, since acouple of crossings more or less would make no difference, we must adhereto a rigid way of counting in order to be accurate below .Theorem 9 .4 .2 . Let {X j , .-Tj , j E N„} be a submartingale and -oc < a