Course in Probability Theory

9.3 BASIC PROPERTIES OF SMARTINGALES 1 33 7PROOF . For a martingale we have equality in (5) for any convex cp, hencewe may take cp(x) = Ix 1, Ix I P or Ix I log' Ix I in the proof above .Thus for a martingale IX,), all three transmutations : {X; }, {X ;, } and{ 1X„ I } are submartingales . For a submartingale {X n }, nothing is said about thelast two .Corollary 3 . If {Xn , 3 n } is a supermartingale, then so is {X, A A, ~, n } whereA is any constant .PROOF . We leave it to the reader to deduce this from the theorem, buthere is a quick direct proof:(6)X„ AA > f(Xn+1 i 7n) A e` (A 1 Jan) > (r (Xn+1 AA I ~T n)It is possible to represent any smartingale as a martingale plus or minussomething special . Let us call a sequence of r .v .'s {Z,,, n E N) an increasingprocess iff it satisfies the conditions :( i) Z1 = 0 ; Z n < Zn+1 for n > 1 ;(ii) (-'C- (Z„) < oo for each n .It follows that Z, = T Z n exists but may take the value +oo ; Z,,.is integrable if and only if {Z„ } is L 1 -bounded as defined above, whichmeans here f (Z,) < oo . This is also equivalent to the uniformintegrability of {Z,,) because of (i) . We can now state the result as follows .Theorem 9 .3 .2 . Any submartingale {X n , Jan } can be written asXn = Yn +Zn,where {Y n , ~, } is a martingale, and {Z„ } is an increasing process .PROOF . From {X„ } we define its difference sequence as follows :(7) x1 = X1, x,, = X„ - Xn-1, n > 2,so that X,, = E~=1 x j , n > 1 (cf. the notation in the first paragraph of thissection) . The defining relation for a submartingale then becomesc" {Xn I ~ - 1) >- 0 ,with equality for a martingale . Furthermore, we putnY1 = X1, yn = x n - ~, {xn Y„ = yj'j=1nZ1 = 0, Zn Zn = Zj .j=1