I have data $\vec{X}=(x_{1}, ..., x_{n})$ with size $N$, sampled from a physical phenomenon. The data takes discrete values in the range $[-N,N]$. As $N$ is increased, the sample variance and fourth order moments explode. I obtained a theoretical result about the problem that implies as $N\rightarrow +\infty$, then $Variance(\vec{X})\rightarrow \infty$. The mean remains finite. If I am not mistaken, this means that as the distribution converges to be continuous, then the variance tends to $+\infty$. I know that a discrete distribution cannot have infinite variance, of course.
The following figure (1) shows one example sample obtained with $N=100$. The y-axis shows the relative frequencies and the x-axis the value of $\vec{X}$. This figure shows another sample with a kernel estimate 2.
I am looking for a distribution that could model this type of data, whether in its continuous limit or discrete. Distributions with infinite variance I am aware of are continuous and exhibit some type of power law, which my estimated samples PMF do not appear show. My question is: Do you know any distributions that could model this type of data? Please tell me if I am missing any important information. Any reference or hint would be extremely helpful. Thank you!
Given the information you have provided it is hard to tell which limiting distribution best describes your data. However, e.g. the student's t-distribution with $1<v\leq 2$ degrees of freedom has a finite mean yet inifite variance. Its support is also the whole real line.