The process in question is a point process. I am mostly interested in the 2-D case but I will accept answers even if it applies to the 1-D case only. The process is defined as follows, on the real line (for the 1-D case). For each $k\in\mathbb{Z}$, generate a random variable $X_k$ with cumulative distribution
$$P(X_k < x) = F\Big(\frac{x-k/\lambda}{s}\Big), $$
where $\lambda, s>0$ and $F$ is a continuous symmetric probability distribution, centered at zero, with a continuous density $f$ (its derivative). It does not really matter what $F$ is, you can take a logistic distribution, but even a Cauchy distribution works well too. For simplicity, you can assume that $\lambda$, the "intensity" of the process, is equal to $1$.
The process consists of all the independently distributed $X_k$'s (the points of the process). These points are not identically distributed as the distribution of $X_k$ depends on the location parameter $k$. When $s\rightarrow\infty$, it is not that hard to prove that the process converges to a stationary Poisson process of intensity $\lambda$, under mild conditions. In practice the approximation is very good even with $s=20$. Thus I am interested in small values of $s$, say $0<s<5$. I call this process Poisson-binomial for obvious reasons: its counting measure has a Poisson-binomial distribution, see here.
So far, I have assumed that this process is stationary. There are various definitions of stationarity (e.g. weak or strong stationarity), but my question boils down to whether the following operations are allowed:
- To compute the theoretical distribution of the distances between two successive points (once the $X_k$'s are ordered on the real line), I can compute the distribution between $X_0$ and the next point. It does not matter if I use $X_0$ or $X_5$, this is location-independent. This would be true if the process is stationary.
- The number of points in $[x, x+t]$ has a theoretical distribution that depends only on $t$, not on $x$.
Are these assumptions legit for this type of process? They seem intuitive, but intuition failed me many times in the past.
Other questions of interest are:
Are the increments (also called inter-arrival times or distances between two successive points once ordered) independent? Increments can't be independent if the process is stationary, otherwise it would be a Poisson process, and this is not a Poisson process.
Is this process a particular case of a well known, much more general family of point processes? Or has this process been investigated before?
Finally, is this process ergodic? That is, if I want to estimate some statistics, I can either generate one long realization -- a single instance of the process -- and base my empirical computations on that single realization. Or I can generate thousands of smaller realizations, each with much fewer points, and compute my statistics by averaging across these realizations. If the process is ergodic, both methods yield the same results, at the limit (when the total number of points approaches $\infty$). Is it the case here?