Consider a martingale $M_t$ on $\mathbb{Z}$ starting from $M_0=0$ and such that $Var[M_t] \leq C \, t$, where $C>0$ is some constant. For a given $n \in \mathbb{N}$ and $t \leq n$, define a process $Q^n_t$ on $\mathbb{Z}$, which follows the law of $M_t$ conditioning on $M_n=0$ (i.e., conditioning on returning to 0 at time $n \geq t \geq 0$.).
Is the following statement true?
$$\forall \epsilon >0, \, \, \, \lim\limits_{n \rightarrow \infty} P( \exists 1 \leq t \leq n \, \, \mbox{s.t.} \, \, |Q^n_t|>\epsilon \cdot n \, \, ) =0.$$
Comments:
- If ``typical'' fluctuations around the origin are of order $\sqrt{t}$, then the statement should be true as the martingale will have few chances to reach a distance $ \epsilon \cdot n$ and then come back to $0$ at time $n$. How to prove it?
- Perhaps it is true that $|Q^n_t|$ is a super martingale. This could help proving the statement.