In the continuous setting, it's known that if a density function is log-concave , then its CDF is also log-concave.


My questions:


1. What can we say about this in the discrete setting?. For ex: Is the CDF of a Poisson random variable log-concave?.


My 2nd question could be somewhat relevant to above.


2. If $X \sim Pois(\lambda)$ , then a large deviation bound $P(X \geq x)$ makes sense only when $x\geq \lambda$. Is there a way to come up with a bound for $x< \lambda$ (or extend the bound for this case)?


Any comments on this would be appreciated (including references).



CDF of a log-concave discrete random variable
Indeed, if the probability mass function of an integer-valued random variable is log concave as a function on $\mathbb Z$, then the corresponding cdf is also log concave as a function on $\mathbb Z$.


This is a special case of [Theorem 2, p. 152](https://www.google.com/books/edition/Advances_in_Stochastic_Inequalities/oksbCAAAQBAJ?hl=en&gbpv=1&pg=PA149&printsec=frontcover). This answers your first question.




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As for your second question, it is rather unrelated to the first one. So, it is better to post it separately. Generally, I think posting multiple questions in one post should be avoided.


