My problem can be described easily: I have a sequence $(X_l)_{l \in \mathbb{N}}$ of r.v. adapted to some filtration $(\mathcal{F}_l)_{l \in \mathbb{N}}$, such that
- $\left|X_{l+1}-X_l\right|\le R$ a. s.
- $\mathbb{E}[X_{l+1}-X_l| \mathcal{F}_l]\le \delta$ a. s.
- $X_0 = x_0$ a.s. where $x_0 \in \mathbb{R}$ is fixed
- $\operatorname{Var}[X_{l+1}-X_l |\mathcal{F}_l] \ge v$ where $v>0$.
I want to prove $\mathbb{P}[X_l \le x_0-c\sqrt{vl}+\delta l]\ge p $ for some $p>0$ and some constant $c>0$.
By using 2) one can easily reduce the problem to a martingale $$\bar{X}_l:= \left(\sum_{k=1}^l D_k - \mathbb{E}[D_k| \mathcal{F}_{k-1}]\right)+x_0$$ where $D_k:=X_k-X_{k-1}$. The martingale again has bounded increments, $\bar{X}_l=x_0$ and $\operatorname{Var}[\bar{X}_l]\ge vl$. The task then is to prove $\mathbb{P}[\bar{X}_l \le x_0-c\sqrt{vl}]\ge p$.
Sadly, the proof in the original paper (Lemma 6.5) was wrong and as I've found out one can't adapt it after having corrected the wrong inequality (Jensen's inequality used in the wrong direction).
The only possibility I see so far is using the Martingale Central limit theorem and then try to control the speed of convergence for my original process using the inequalities described in the beginning of this paper. This seems pretty hard to check for my concrete considered process.
May I miss any obvious proof using simple martingale theory? Can anybody prove this result using only the above assumptions (or maybe some small additional assumptions, e.g. $L^{2+\delta}$ moments) ?
