For a stationary point process, suppose we have the (terrible) bound $$\mathbb{P}\bigl( N(0,\delta) \geq 1 \bigr) \leq \sqrt \delta \quad \quad \text{ for each positive small $\delta$}.$$ Here $N(0,\delta)$ can be thought as the number of points appear in the interval $(0,\delta)$.
Now, fixed a small positive number $\delta_0$, and I would like to bound $$\mathbb{P}\bigl( N(0,\delta_0) \geq k \bigr) \quad \quad \text{ where } k \gg 1,$$ intuitively this probability should tends to zero as $k$ tends to infinity.
What types of additional assumptions I would need to make such estimate possible?
And is there anything in the literature that is related to estimating $$\mathbb{P}\bigl( N(0,\delta_0) \geq k \bigr) \quad \quad \text{ where } k \gg 1?$$