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A Probability Path.pdf

# Theorem 10101 krickeberg decomposition if xn bn n 0

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Theorem 10.10.1 (Krickeberg Decomposition) If {(Xn. Bn). n :::: 0} is a sub- martingale such that supE(X:) < oo , n then there exists a positive martingale {(Mn, Bn), n :::: 0} and a positive super- martingale {(Yn, Bn), n 2:: 0} and Xn = Mn- Yn. Proof. If {Xn} is a submartingale, then also {X:} is a submartingale. (See Exam- ple 10 .6.1.) Additionally, {E(XtiBn), p :::: n} is monotone non-decreasing in p. To check this, note that by smoothing, E(X;+ 1 ll3n) = E(E(X;+1 l l3p)ll3n) 2:: E(XtlBn) where the last inequality follows from the submartingale property . Monotonicity in p implies exists. We claim that {(Mn, Bn), n 2:: 0} is a positive martingale. To see this, observe that (a) Mn E Bn, and Mn 2:: 0. (b) The expectation of Mn is finite and constant inn since E(Mn) = E( lim t E(X;ll3n)) p-+00 = lim t £(£(X;Il3n)) p-+00 = lim t Ex+ p-+00 p =sup Ex;< oo, p?:O (monotone convergence) since expectations of submartingales increase. Thus E(Mn) < oo.

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10.10 Martingale and Submartingale Convergence 387 (c) The martingale property holds since E(Mn+IIBn) = E{ lim t E(XtiBn+I)iBn) p-+00 = lim t E(E<XtiBn+I)iBn) (monotone convergence) p-+00 = lim t E(XtiBn) = Mn· p-+00 (smoothing) We now show that is a positive supermartingale. Obviously, Yn E Bn. Why is Yn 2: 0? Since Mn = limp-+oo t E(XtiBn). if we take p = n, we get Mn ::: E(X:IBn) = x: ::: x: - x; = Xn . To verify the supermartingale property note that E(Yn+IIBn) = E(Mn+IIBn)- E(Xn+tiBn) :5 Mn- Xn = Yn 0 10.10.2 Doob's (Sub)martingale Convergence Theorem K.rickeberg's decomposition leads to the Doob submartingale convergence theo- rem. Theorem 10.10.2 (Submartingale Convergence) lf{(Xn. Bn). n ::: 0} is a (sub)- martingale satisfying supE(X:) < oo, neN then there exists X 00 e L 1 such that X a.s.x n oo· Remark. If {Xn} is a martingale supE(X:) < oo iff supE(IXnD < oo neN neN in which case the martingale is called L 1-bounded. To see this equivalence, ob- serve that if {(Xn, Bn). n eN} is a martingale then E<IXnD = E(X:) + E(X;) = 2E(X:)- E(Xn) = x: - const.
388 10. Martingales Proof. From the Krickberg decomposition, there exist a positive martingale {Mn} and a positive supermartingale {Yn} such that From Theorem 10 .8. 5, the following are true: M a.s.M n 00• y; a.s . y; n oo so and M 00 and Y 00 are integrable. Hence M 00 and Y 00 are finite almost surely, X 00 =Moo- Y 00 exists, and Xn X 00 0 10.11 Regularity and Closure We begin this section with two reminders and a recalled fact. Reminder 1. (See Subsection 10.9.2.) Let {Zn} be a simple branching process with P( extinction)= 1 =: q. Then with Zo = 1, E(Z 1) = m So the martingale { Wn} satisfies E(Wn) = 1 £(0) = 0 so { Wn} does NOT converge in L Also, there does NOT exist a random variable W 00 such that Wn = E(W 00 1Bn) and {Wn} is NOT uniformly integrable (ui). Reminder 2. Recall the definition of uniform integrability and its character- izations from Subsection 6.5.1 of Chapter 6. A family of random variables (X, , t e I} is ui if X, e L 1 for all t e I and lim sup] IX 1 IdP = 0. b-+OO IE/ IXri>b Review Subsection 6.5.1 for full discussion and characterizations and also re- view Theorem 6.6.1 on page 191 for the following FACT: If {Xn} converges a. s.

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