TITLE: Probability that a sum of intependent random variables hits a point
QUESTION [5 upvotes]: Let $X_1,\ldots,X_n$ be independent random elements of a normed space $X$. Suppose that $\sup_{x\in X}\mathbb{P}(X_i=x)=p_i$. What is the best known upper bound for 
$$\sup_{x\in X} \mathbb{P}(X_1+\cdots+X_n=x)?$$
Comments:

If $p_i$ are all bounded away from $0$ and $1$ independently of $i$ and $n$ then, of course, the maximum is at most $C/\sqrt{n}$, where $C$ depends on how far the $p_i$'s are from $0$. Are there bounds that are good when the $p_i$'s are non homogeneous? Like, say, $p_i=1/i$ or even $p_i=c/n$?
Are there any precise results known? Then is, can one in general obtain the exact result in some key situations? Say $p_i=1/2$?

REPLY [2 votes]: What you ask is a generalization of the Littlewood-Offord problem.
(If each $X_i$ can take only two values, with probability $1/2$, then you get their problem.)
This problem has been investigated a lot and has several generalizations, unfortunately I am not an expert on the topic, but I would guess that your problem follows easily from known methods, probably the best is to take each $X_i$ (almost) uniformly distributed on a set of size $\lceil 1/p_i \rceil$.
For a recent paper, see http://annals.math.princeton.edu/wp-content/uploads/annals-v169-n2-p06.pdf.