9.4 Proof of the Main Theorem 173and we therefore obtain
Q
[
(Rn·A)Tn,i−(Rn·A)Tn,i− 1 ≥α−2 λ
n]
>β−n−^2fori=1,...,knand alln≥N.
Unfortunately we do not know thatRnis 1-admissible or even admissible.
A final stopping time argument and some estimates will allow us to control
the “admissibility” ofRn.ThejumpsofRn·Sare part of the jumps ofLn·S
and hence
∆(Rn·S)≥∆(Ln·S)≥−n+1
n^2≥−
2
n.
An upper bound for (Rn·M) is obtained by
∥
∥(Rn·M)Tn,k
n∥
∥^2
L^2 (Q)≤
∥
∥(Ln·M)Tn,k
n∥
∥^2
L^2 (Q)≤∑kni=1‖fn,i‖^2 L (^2) (Q).
Forn≥Nthis is smaller than 4kn. Doob’s maximal inequality applied on
theL^2 (Q)-martingale (Rn·M)Tn,knyields
∥
∥
∥
∥supt≥ 0 |(R
n·M)t|
∥
∥
∥
∥
L^2 (Q)≤ 4
√
kn.This inequality will show thatRn·Swill not become too negative on big
sets. First note that we may estimate (Rn·S)frombelowbyRn·M. Indeed,
Rn·S=Rn·M+Rn·A≥Rn·MsinceRn·Ais increasing and hence positive.
The following estimates hold
Q
[
inf
t≥ 0
(Rn·S)t≤−knn−(^14)
]
≤Q
[
sup
t≥ 0|(Rn·M)t|≥knn−(^14)
]
≤ 16
√
n
knby Tchebycheff’s inequality and the above estimate≤ 64 α1
√
n.
Let nowUn =inf{t|(Rn·S)t<−knn−(^14)
}. The preceding inequality
says thatQ[Un<∞]≤ 64 α√^1 n. We define yet another integrand: letVn=
1
knR
n 1
[[ 0,Un]].ThejumpsofV
n·Sare then bounded from below by − 2
nknand
the process (Vn·S) is therefore bounded below by−n−
(^14)
−nk^2 n. The integrands
Vnare therefore admissible and their uniform lower bound tends to zero. We
now claim that (Vn·S)∞is positive with high probability.