1970 CHAPTER 60. CONDITIONAL, MARTINGALES

and so, subtracting the last term on the right from both sides,

E(X+

n)≥

∫[maxk Xk≥λ ]

XndP =∫[maxk Xk≥λ ]

XT dP

≥ λP([

maxk

Xk ≥ λ

])because XT (ω)≥ λ on [maxk Xk ≥ λ ] from the definition of T . This establishes 60.6.12.

Next let T (ω) be the first time Xk (ω) is≤−λ or if this does not happen for k≤ n, thenT (ω)≡ n. Then this is a stopping time by similar reasoning and 1≤ T (ω)≤ n are stoppingtimes and so by the optional stopping theorem, X1,XT ,Xn is a submartingale. Therefore, on[

mink

Xk ≤−λ

], XT (ω)≤−λ

and E (XT |F1)≥ X1 and so

E (X1)≤ E (E (XT |F1)) = E (XT )

which implies

E (X1) ≤ E (XT ) =∫[mink Xk≤−λ ]

XT dP+∫[mink Xk>−λ ]

XT dP

=∫[mink Xk≤−λ ]

XT dP+∫[mink Xk>−λ ]

XndP

and so

E (X1)−∫[mink Xk>−λ ]

XndP ≤∫[mink Xk≤−λ ]

XT dP

≤ −λP([

mink

Xk ≤−λ

])which implies

λP([

mink

Xk ≤−λ

])≤

∫[mink Xk>−λ ]

XndP−E (X1)

≤∫

Ω

(|Xn|+ |X1|)dP

and this proves 60.6.13.The last estimate follows from these. Here is why.[

max1≤k≤n

|Xk| ≥ λ

]⊆[

max1≤k≤n

Xk ≥ λ

]∪[

min1≤k≤n

Xk ≤−λ

]and so

λP([

max1≤k≤n

|Xk| ≥ λ

])≤ λP

([max

1≤k≤nXk ≥ λ

]∪[

min1≤k≤n

Xk ≤−λ

])