Packing of discrete random variables with finite second moment

Question

I am considering a discrete random variable $X \in\mathbb{R}$ with $N$ points (where each point has non-zero probability) and $E[X^2]=1$ and $E[X]=0$.

Let $d_l$ be the the smallest distance between $x_l \in X$ and its closest neighbor in the support of $X$ that is \begin{align} d_{l}=\min_{x_l, x_k: l \neq k} |x_l-x_k| \end{align}

Also, define \begin{align} d_{\max}&=\max_{l} d_l \\ d_{\min}&=\min_{l} d_l \\ \end{align}

I know that amongst random variables with finite second moment uniformly spaced and uniformly described random variable gives the best packing that \begin{align} d_{\min} \le d_{\min \text{Unif}}=\sqrt{\frac{12}{N^2-1}} \end{align}

Here some question that I was thinking about:

1. Can we have also a lower bound on $d_{\min}$ in terms of $N$ other than $d_{\min} \ge 0$.
   I am thinking no. We can always find an $X$ with a smaller and small minimum distance.

2. Can we find upper and lower bounds $d_{\max}$? I am thinking that \begin{align*} d_{\max} \le N d_{\min} \le \sqrt{\frac{12 N^2}{N^2-1}} \end{align*}

3. Can we find lower and upper bounds on $\frac{d_{\min}}{d_{\max}}$?

4. If we know $d_{\min}$ can we say something about the second smallest distance or $d_{\max}$?

5. Finally, where I can find any reference on this subject? And what branch of math studies some thing like this?

Also, it seems to me that we do not have to talk about random variables here and just talk about packing deterministic points.

Please let me know how I can improve the question. Thank you for any help in advance.

Answer 1

The format of this site is not optimally suited for multiple questions. Anyway, here are some answers:

1. At least if $N \ge 3$ there will be no lower bound. Basically, you can start with any such random variable with $N-1$ points and split one of the masses (i.e., probabilities sitting on points) into two points arbitrarily close to the original point without changing the expectation and variance.

2. No, there will be no upper bound on $d_{max}$, because $X$ can have points very far away, as long as their probabilities are sufficiently small.

3. From previous points, there will be no lower bound on $d_{min}/d_{max}$, but trivially the upper bound is $1$.

I don't really have any reference, and I would suppose the general answer to 4. is "no", but I will leave it here for now.