I posted the following question here ([https://mathoverflow.net/questions/383007/cdf-of-a-log-concave-discrete-random-variable/383016?noredirect=1#comment974227_383016][1] ).Since it is not related to my main question, I thought of reposting it as separate post.

Question:

Let $X \sim Pois(\lambda).$ I'm interested in a large deviation bound for $X$; more specifically a sub-exponential type bound for $P(X \geq x)$ for all $x\geq 0.$ While it's known that such a bound exists for $x \geq \lambda$ (chernoff bound), it's not clear whether there exists one for $x< \lambda.$ Is it possible to obtain a bound (or extend the bound obtained for $x \geq \lambda$) for these $x$ values using standard large-deviation techniques? If not, can someone explain why.

Any comments would be appreciated.