Concentration bound for a martingale-like setting (the expected difference decreases as the sequence increases)

Question

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.

Answer 1

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.