# Regularity for the sum of iid random variables

Let $$(X_i)_{i\in \mathbb{N}}$$ iid random variables such that there exists $$\alpha>0$$ where $$\mathbb{P}\left(X_1\in [x,x+1]\right)\leq \alpha$$ for all $$x\in \mathbb{R}$$. Assume $$\alpha$$ small enough, does there exist a universal constant $$C>0$$ so that $$\mathbb{P}\left(\sum_{i=1}^N X_i\in [x,x+1]\right)\leq \frac{C\alpha}{\sqrt{N}}$$ for all $$x\in\mathbb{R}$$ and $$N\in \mathbb{N}^*$$?

A few remarks :

• From the central limit theorem this is very natural to be expected. However here we do not assume $$\mathbb{E}(X^2)<\infty$$ and it is not so clear to me how to use the characteristic functions with the hypothesis.
• In the case of discrete variable $$X\in \mathbb{Z}$$, one should just do the computation. However I can't see why should it be the "worse case scenario" and how to compare it with the more general case.

In particular, letting $$L=\lambda=:t$$ in that Corollary 1, we get the following:
Let $$X_1,\dots,X_n$$ be independent identically distributed random variables, with $$S_n:=\sum_1^n X_k$$. Then for any positive real $$t$$, $$Q(S_n,t)\le\frac{CQ(X_1,t)}{\sqrt{(1-Q(X_1,t))\, n}},$$ where $$C$$ is a universal positive real constant and $$Q(X,u):=\sup_{x\in\mathbb R}P(x\le X\le x+u)$$ for positive real $$u$$.
• This is a great reference. Thanks! But is it really enough ? I wish one could get $Q(X,t)$ in the numerator ? Sep 24 at 12:58