Azuma's Inequality When $| \triangle_{n} | \leq k $

Question

I saw Azuma's Inequality on my textbook, which is stated below:

Let {$W_{n}$} be a martingale with $|\triangle_{n}| \leq 1$ for all n. Let u > 0 by any constant. Then $\mathbb{P}(W_{N} \geq u) \leq e^{-u^2 / 2N}$

and there is a follow up question asking what would be different if instead we have: $|\triangle_{n}| \leq k$. I think the constraint $|\triangle_{n}| \leq 1$ is used in a middle step of the proof:

Let Z have the conditional law of $\triangle_{n}$ given $F_{n-1}$. The supermartingale property follows from $\mathbb{E} e ^{\lambda z} \leq cosh(\lambda)$, which holds for any random variable with mean 0 in [-1,1].

But I am not sure what difference would this incur. Any hints or comments are welcome. Thanks a lot.

Answer 1

We can use a rescaling: apply the previous result to $W'_n=W_n/k$ instead of $W_n$ and $u'=u/k$ instead of $u$.