Let $X\_1, \ldots, X\_n$ be independent and identically distributed random variables. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a bounded difference function, i.e., for any $x,y \in \mathbb{R}^n$ that differ only in coordinate $i$, we have $|f(x) - f(y)| \leq c\_i$. Under this setting, [McDiarmid’s inequality](https://en.wikipedia.org/wiki/Doob_martingale#McDiarmid%27s_inequality) provides a well-known concentration result.


Under the same setting, do we also have a central limit theorem for an appropriately scaled version of $f(X\_1, \ldots, X\_n) - E[f(X\_1, \ldots, X\_n)]$ ? I am ok with making additional assumptions on $(X\_i)\_i$ sequence.



CLT for bounded difference functions
In the general setting you propose, the answer is negative. Suppose that $X\_i$ take the values $\pm 1$ with equal probability. Let $h: {\mathbb R} \to {\mathbb R}$ be a Lipschitz function with constant 1, e.g., $$ (\*) \quad h(x)=\max\{x,0\} \, ,$$ and define
$$f\_n(x\_1,\dots,x\_n):=\sqrt{n} \cdot h \left(\frac{x\_1+\dots +x\_n}{\sqrt{n}}\right) \,.$$
(In the special case (\*), one can skip dividing and multiplying by $\sqrt{n}$.)
Then $f\_n$ satisfy the bounded difference condition with constants $c\_i=2$ on the support of the distribution.


(If you want the bounded difference condition as stated to hold everywhere, consider the functions
$\widetilde{f}\_n(x\_1,\dots,x\_n):=f\_n({\rm sign}(x\_1),\dots,{\rm sign}(x\_n))$ instead.)


Let $Z$ be a standard normal variable. Then we have the convergence in law
$$\frac{f(X\_1, \ldots, X\_n) - E[f(X\_1, \ldots, X\_n)]}{\sqrt{{\rm Var} [f(X\_1, \ldots, X\_n)]}} 
\Rightarrow \frac{h(Z)-E[h(Z)]}{\sqrt{{\rm Var}[h(Z)] }}\,. $$
The distribution of the right-hand side is not Gaussian for most functions $h$.
In the special case (\*), this distribution will have a density which is the right half of the Gaussian density, shifted and scaled to have mean zero and variance 1.


**Postscript:** Perhaps you can post another question with the particular bounded difference functions $f\_n$ you have in mind.
One proof of Mcdiarmid's inequality uses Martingales, so perhaps in your application a Martingale CLT will be useful. See <https://en.wikipedia.org/wiki/Martingale_central_limit_theorem> and the references therein.


