I went through several martingales concentration bounds, but none of them fit the settings I am interested in, which is the following. Suppose I have a sequence of nonnegative random variables $0=Y\_{0},\ldots,Y\_{n}$, where every $Y\_{i}$ is a function of some real random variables $X\_{1},\ldots,X\_{i}$ and I know that 
$$
\mathbb{E}[Y\_{i+1}-Y\_{i} | X\_{1},\ldots,X\_{i}] \le c\_{i}(1-i\sqrt{Y\_{i}})
$$
for some small $c\_{i}$-s. 
I believe that in such cases we can say, informally, the sequence is "almost" a supermartingale from some $i$ onwards.
Can I bound $\Pr[Y\_{n} \ge \alpha]$, as a function of the $c\_{i}$-s? It seems plausible that this sequence cannot grow rapidly, with high probability. However, I failed to deduce such a bound.


Thank you very much.



Concentration bound for a martingale-like setting (the expected difference decreases as the sequence increases)
We can use the decomposition 
$$Y\_n=\sum\_{i=0}^{n-1} Y\_{i+1}-Y\_i -\mathbb E[Y\_{i+1}-Y\_i\mid X\_1,\dots, X\_i ] + \sum\_{i=0}^{n-1}\mathbb E[Y\_{i+1}-Y\_i\mid X\_1,\dots, X\_i ],$$
hence by assumption, the bound 
$$Y\_n\leqslant \sum\_{i=0}^{n-1} Y\_{i+1}-Y\_i -\mathbb E[Y\_{i+1}-Y\_i\mid X\_1,\dots, X\_i ] + \sum\_{i=0}^{n-1} c\_i(1-i\sqrt{Y\_i}) $$
holds. We thus have for a fixed $\alpha$, 
$$\mathbb P(Y\_n\geqslant \alpha)\leqslant \mathbb P\left( \sum\_{i=0}^{n-1} Y\_{i+1}-Y\_i -\mathbb E[Y\_{i+1}-Y\_i\mid X\_1,\dots, X\_i ]\geqslant\frac\alpha 2 \right)+\\
+\mathbb P\left(\sum\_{i=0}^{n-1} c\_i(1-i\sqrt{Y\_i})\geqslant\frac\alpha 2 \right) =:P\_1+ P\_2.$$
An upper bound for $P\_1$ can be found using for example Azuma's inequality if the increments are bounded, or Burkholder's inequality in general. And $P\_2$ admits an upper bound which may be expressed with the $c\_i$-s. For example, if $\alpha\geqslant 2\sum\limits\_{i=0}^{n-1}c\_i$, then $P\_2=0$, hence for such $\alpha$-s, the bound reduces to concentrations inequalities for martingales, which has been extensively studied. 


