Let $X$ be a a discrete RV with $\mathbb{P}(X=k)<p$ for every $k$ (that's all we know). Taking the independent sum $S=X\_1+X\_2+\cdots+X\_n$, with each $X\_i$ distributed like $X$, what can we say about an upper bound for
$$
\max\_k \mathbb{P}(S=k) \text{ ?}
$$
If it helps, $n$ is large, but $n\ll \frac1p$. Is it true that
$$
\max\_k \mathbb{P}(S=k)\le \frac{cp}{\sqrt{n}}
$$


for some constant $c$?


Addendum: The last line is what you get for fixed $p$ as $n\to\infty$ with CLT. In a sense, this question is asking if we know enough about the convergence of CLT to make this work for $n\ll \frac1p$.



Bounding maximum probabilities in sum of i.i.d discrete RVs
The requested estimate follows from a theorem of Kesten [2] about concentration functions.
See in particular the inequality (1.6) on page 135 of [2]. Taking $L=\lambda<1$ in 
this inequality gives the proposed inequality with some constant $c$. The constant $c$ needed depends on the constant in Rogozin's Theorem [1]. 


[1] Rogozin, B. A. "An estimate for concentration functions." Theory of Probability & Its Applications 6.1 (1961): 94-97.


[2] Kesten, Harry. "A sharper form of the Doeblin–Levy–Kolmogorov–Rogozin inequality for concentration functions." Mathematica scandinavica 25, no. 1 (1970): 133-144.
See <https://www.mscand.dk/article/download/10950/8971>


