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!



Proof for an extension of Azuma's inequality
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 $Y$. 


