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
$$ liminf_ {x rightarrow { infty}} frac {Var ; Z (0, x)} {x}> 0? $$

more information:
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]$