1978 CHAPTER 60. CONDITIONAL, MARTINGALES
60.8 A Reverse Submartingale Convergence TheoremDefinition 60.8.1 Let {Xn}∞
n=0 be a sequence of real random variables such that E (|Xn|)<∞ for all n and let {Fn} be a sequence of σ algebras such that Fn ⊇Fn+1 for all n. Then{Xn} is called a reverse submartingale if for all n,
E (Xn|Fn+1)≥ Xn+1.
Note it is just like a submartingale only the indices are going the other way. Here is aninteresting lemma. This lemma gives uniform integrability for a reverse submartingale.
Lemma 60.8.2 Suppose E (|Xn|) < ∞ for all n, Xn is Fn measurable, Fn+1 ⊆ Fn forall n ∈ N, and there exist X∞ F∞ measurable such that F∞ ⊆ Fn for all n and X0 F0measurable such that F0 ⊇Fn for all n such that for all n ∈ {0,1, · · ·} ,
E (Xn|Fn+1)≥ Xn+1, E (Xn|F∞)≥ X∞,
where E (|X∞|)< ∞. Then {Xn : n ∈ N} is uniformly integrable.
Proof:E (Xn+1)≤ E (E (Xn|Fn+1)) = E (Xn)
Therefore, the sequence {E (Xn)} is a decreasing sequence bounded below by E (X∞) so ithas a limit. I am going to show the functions are equiintegrable. Let k be large enough that∣∣∣E (Xk)− lim
m→∞E (Xm)
∣∣∣< ε (60.8.20)
and suppose n > k. Then if λ > 0,∫[|Xn|≥λ ]
|Xn|dP =∫[Xn≥λ ]
XndP+∫[Xn≤−λ ]
(−Xn)dP
=∫[Xn≥λ ]
XndP+∫
Ω
(−Xn)dP−∫[−Xn<λ ]
(−Xn)dP
=∫[Xn≥λ ]
XndP−∫
Ω
XndP+∫[−Xn<λ ]
XndP
From 60.8.20,
≤∫[Xn≥λ ]
XndP−∫
Ω
XkdP+ ε +∫[−Xn<λ ]
XndP
By assumption,
E (Xk|Fn)≥ Xn