# pr.probability – Lower limit of the variance function for a point process

To let $$Z$$ a stationary random measurement $$mathbb {R}$$ i.e.
$$Z (a, b) stackrel {D} {=} Z (a + u, b + u) quad for all u in mathbb {R}$$ and so that $$Z$$ is simple and tidy. What can you impose? $$Z$$ to ensure
$$liminf_ {x rightarrow { infty}} frac {Var ; Z (0, x)} {x}> 0?$$

This applies to many processes such as the stationary Poisson process, renewal processes, compound Poisson, etc., and to processes with a high degree of determinism. e.g. it fails for $$Z (a, b): = # {n: n + U in (a, b) }$$
from where $$U sim unif[0,1]$$