Question

Suppose $\xi_1,\ldots \xi_T$ is a martingale difference sequence. Then,

1) For any $a\in \mathbb{R}^{+}$, can we say something about the sequence $\xi_1^2\mathbb{1}(\xi_1\geq a),\ldots, \xi_T^2\mathbb{1}(\xi_T\geq a)$ ? Is it a (sub/sup) martingale difference sequence?

2) Suppose each $|\xi_t|\leq B_t$ a.s., and $B_1\leq B, B_T\leq B$ a.s. then can we provide an upper bound for $\mathbb{P}(\sum_{t=1}^T \xi_t\geq z)$?

I guess if we can prove that the sequence in (1) is a (sub/sup) martingale difference sequence then one can apply standard maximal inequalities to solve (2). However I am not able to resolve (1), and my intuition says that, for the sequence in (1) one cannot claim any (sub/sup) martingale difference behaviour. However I do not have a formal proof or a counterexample. Also if it turns out that the sequence in (2) is not a (sub/sup) martingale then how do we go about establishing maximal inequalities?

Answer 1

I must admit that I understand neither of your questions, or rather what sort of nontrivial answer you want.

1) $\xi^2 \mathsf{1} \lbrace \xi \ge a \rbrace \ge 0$, so it is a submartingale difference in an extremely boring way. I don't see what you can possibly want from it.

2) No, we can't bound it in a nontrivial way. As an example, consider independent random variables that equal $B$ with high probability and something very negative with small probability. If $|\xi_t| \le B$ instead, then see http://en.wikipedia.org/wiki/Azuma_inequality.