Suppose $X \sim Pois(\lambda)$. I'm interested in an upper bound on the ratio, $$\dfrac{P(X \leq n)}{P(X \leq n-1)}\,,\,\,\text{for $n=1,2,3,...$}$$ Observe that, the ratio is $>1$ & as $n \to \infty,\,$ the ratio $\to 1$. Thus, It is interesting to see if there exists some constant $K>1$ (depending on whether $\lambda \leq n$ or $\lambda \geq n$) so that the ratio is $\leq K^{1/n}.$


Any comments on this?



Upper bound on the ratio of Poisson CDFs
Clearly $\Pr[X\le n-1]\ge \Pr[X=n-1]$ so you have the ratio


$$
1+\frac{\Pr[X=n]}{\Pr[X\le n-1]}\le
1+\frac{\Pr[X=n]}{\Pr[X=n-1]}
=1+\frac{\lambda^n/n!}{\lambda^{n-1}/(n-1)!}
=1+\lambda/n.
$$
Now
$$
1+\lambda/n \le \exp(\lambda/n).
$$
Hence you can take $K=e^\lambda$ in your example.


