I have a series of $n$ independent random variables $X\_1,\ldots, X\_n$, each with the support $[0,1]$, and a monotone convex function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ that is 1-Lipshitz in L1 norm, i.e., for every $x,y \in \mathbb{R}^n$, it holds that $f(x)-f(y) \leq \sum\_{i=1}^{n} |x\_i-y\_i|$.
I want to have a concentration bound like Talagrand.


Is it true that $$Pr[ \mid f(X\_1,\ldots,X\_n) - E[f(X\_1,\ldots,X\_n)] \mid > t] \leq c\_1 \cdot e^{-\frac{t^2}{c\_2}} $$
for some constants $c\_1,c\_2>0$ (that are independent of $n$, and the distributions of $X\_i$, and the function $f$, as long as the conditions hold)?


Do I need more conditions for the inequality to hold?



Talagrand's inequality for L1 norm
Yes, you do need more conditions. For instance, if $f(x\_1,\dots,x\_n)\equiv x\_1+\dots+x\_n$ and the $X\_i$'s are (say) iid Bernoulli with parameter $1/2$, then $f$ is $1$-Lipschitz in the $L^1$-norm but, by the central limit theorem, your inequality will hold for all real $t>0$ only if $c\_2\gtrsim n/2$ as $n\to\infty$.


