802 CHAPTER 29. MARTINGALES

Theorem 29.4.13 Let {X (k)} be a real sub-martingale and let λ > 0 be given.Then

P([max{|Xk| ,k = 1, ...,n}> λ ])≤ 2λ

∫|X (1)|+ |X (n)|dP

Proof: [max{|Xk| ,k = 1, ...,n}> λ ]⊆ [X∗n > λ ]∪ [Xn∗ <−λ ] and so

P([max{|Xk| ,k = 1, ...,n}> λ ]) ≤ 1λ

∫X+

n dP+1λ

∫(|X (1)|+ |X (n)|)dP

≤ 2λ

∫|X (1)|+ |X (n)|dP.■

29.5 Reverse Sub-martingale Convergence TheoremSub-martingale: E (Xn+1|Fn)≥ Xn. Reverse sub-martingale: E (Xn|Fn+1)≥ Xn+1 and herethe Fn are decreasing.

Definition 29.5.1 Let {Xn}∞

n=0 be a sequence of real random variables such thatE (|Xn|)< ∞ for all n and let {Fn} be a sequence of σ algebras such that Fn ⊇Fn+1 forall n. Then {Xn} is called a reverse sub-martingale if for all n,

E (Xn|Fn+1)≥ Xn+1.

Note it is just like a sub-martingale only the indices and σ algebras are going the otherway. Here is an interesting lemma. This lemma gives uniform integrability for a reversesub-martingale. The application I have in mind in the next lemma is that supn E (|Xn|)< ∞

but it is stated more generally and this condition appears to be obtained for free given theexistence of X∞ in the following lemma.

Lemma 29.5.2 Suppose for each n, E (|Xn|) < ∞, Xn is Fn measurable, Fn+1 ⊆Fnfor all 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 equi-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 equi-integrable. Let k be large enoughthat ∣∣∣E (Xk)− lim

m→∞E (Xm)

∣∣∣< ε (29.11)

and suppose n > k. Then if λ > 0,∫[|Xn|≥λ ]

|Xn|dP =∫[Xn≥λ ]

XndP+∫[Xn≤−λ ]

(−Xn)dP