# Proof for an extension of Azuma's inequality

I am trying to understand a part of the proof of an extension of Azuma's inequality, where there is a small failure probability, as it appears in proposition 34 in "Random matrices: universality of local spectral statistics of non-hermitian matrices" by Terence Tao and Van Vu.

Here's the url for Arxiv: https://arxiv.org/pdf/1206.1893.pdf

To my understanding, the basic idea for the proof is that from the original function $$Y$$ and a martingale sequence ($$E[Y|\xi_1,\dots,\xi_{i}])_{i=1}^n$$, a modified function $$Y'$$ and martingale sequence ($$E[Y'|\xi_1,\dots,\xi_{i}])_{i=1}^n$$ are constructed, by alternating $$Y$$ on "bad" sets where martingale difference terms are big.

I am having a hard time understanding why the new martingale sequence based on $$Y'$$ should have the bounded difference, i.e. $$|E[Y'|\xi_1,\dots,\xi_{i}]-E[Y'|\xi_1,\dots,\xi_{i-1}]| \leq \alpha_i$$ a.s.

Any input will be very much appreciated. Thanks!

The statement "$$Y'$$ satisfies the condition of Azuma’s inequality" is incorrect in general if it is supposed to mean, for instance, that $$C'_i\le C\alpha_i$$ for some real constant $$C$$ not depending on the distribution of $$Y$$, where $$C'_i=C'_i(\xi)$$ is defined similarly to $$C_i(\xi)$$ in definition (4.1) on page 28 in the linked paper but with $$Y'$$ in place of $$Y$$: $$C'_i(\xi):=|E(Y'|\xi_1,\dots,\xi_i)-E(Y'|\xi_1,\dots,\xi_{i-1})|.$$
Indeed, suppose that $$n=1$$, $$\xi_1=Y$$, $$P(Y=n)=\frac1{n+1}=1-P(Y=-1)$$ for natural $$n\ge2$$, so that $$EY=0$$. Let $$\alpha_1=2$$. Then $$C_1=|E(Y|\xi_1)-EY|=|Y|$$, $$B_1=\{C_1\ge2\}=\{Y=n\}$$, $$Y'=1_{B_1}\,E_{B_1}Y+1_{B_1^c}\,Y=1_{Y=n}\,n-1_{Y=-1}$$, $$C'_1=|Y'|=1_{Y=n}\,n+1_{Y=-1}$$, which is not bounded by any constant not depending on the distribution of $$Y$$.