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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!

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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 $Y$.

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