59.2.3 Doob Submartingale Estimate
Another very interesting result about submartingales is the Doob submartingale
estimate.
Theorem 59.2.8 Let
i=1∞ be a submartingale. Then for λ > 0
,
([ ]) ∫
P max Xk ≥ λ ≤ 1- X+ dP
1≤k≤n λ Ω n
|
|
Proof: Let
A1 ≡ [X1 ≥ λ],A2 ≡ [X2 ≥ λ]∖A1,
⋅⋅⋅,A ≡ [X ≥ λ]∖(∪k−1A )⋅⋅⋅
k k i=1 i
Thus each
Ak is
Sk measurable, the
Ak are disjoint, and their union equals
. Therefore from the definition of a submartingale and Jensen’s
inequality,
([ ]) ∑n ∑n ∫
P max Xk ≥ λ = P (Ak) ≤ 1 XkdP
1≤k≤n k=1 λ k=1 Ak
1∑n ∫
≤ -- E (Xn |Sk)dP
λk=1 Ak
1∑n ∫ +
≤ λ- E (Xn |Sk) dP
k=1∫Ak
1∑n ( + )
≤ λ Ak E Xn |Sk dP
k=n1∫ ∫
= 1∑ X+ dP ≤ 1- X+ dP. ■
λk=1 Ak n λ Ω n