Suppose
- $X_n$ is a submartingale w.r.t $\sigma$-field $\mathcal{F}_n$, for $n = 1,2,\dots$, i.e., $\mathbb{E}[X_{n+1}\mid \mathcal{F}_n] \ge X_{n}$.
- $X_n \in [0,1]$ for all $n$.
- There exists a function $\phi : [0,1]\to[0,1]$, such that $\phi$ is strictly concave, infinitely differentiable, $\phi(0) = 0$ and $\phi(1) = 1$.
- $\mathbb{E}[X_{n+1}\mid \mathcal{F}_n] \ge \phi(X_{n})$ (note that $\phi(X_n) \ge X_n$ by our assumptions on $\phi$)
- $X_1 > 0$ almost surely and $\Pr(X_1 < t) \le \exp(-c/t)$ for some $c>0$, for all $t > 0$
I have two questions:
- If $X_1 > 0$ almost surely, then is it true that $\lim_{n\to \infty} X_n = 1$ almost surely?
- If the above is true, can we say anything about the rate of convergence, e.g., an inequality of the form: for all $\epsilon > 0$ there exists a $C > 0$ such that for all $n$ $$ \Pr(1 - X_n > \epsilon)\le \exp(-n\epsilon/C) $$ (Of course, inequalities of other forms are welcome as well. I'm unfamiliar with the area.)