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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.

  1. 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).

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  • $\begingroup$ The first question makes no sense since the CDF of a Poisson random variable is not continuous. $\endgroup$
    – user64494
    Commented Feb 3, 2021 at 15:25
  • $\begingroup$ @user64494 That's why I mentioned 'discrete'. log-concave in the sense of discrete log-concavity $\endgroup$
    – SL_MathGuy
    Commented Feb 3, 2021 at 15:43
  • $\begingroup$ Like, in the discrete setting, if PMF is log-concave ,then is it true that its CDF is also log-concave $\endgroup$
    – SL_MathGuy
    Commented Feb 3, 2021 at 15:48

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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. This answers your first question.

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.

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  • $\begingroup$ Thank you sir!. I'm referring to theorem 2. In the case of Poisson, we can choose $q(n) = e^{-\lambda} \lambda^n /n!$ and $r=\infty$ . But, which $\alpha$ would make $\mathcal{J}^\alpha q$ the CDF of Poisson? $\endgroup$
    – SL_MathGuy
    Commented Feb 3, 2021 at 18:00
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    $\begingroup$ @SL_MathGuy : To get the CDF of the Poisson distribution (that is, its left tail), take $\alpha=1$ and "read the the probability mass function right-to-left"; that is, take $q(n)=e^{-\lambda}\lambda^{-n}1(n=0,-1,-2,\dots)/(-n)!$. $\endgroup$
    – Iosif Pinelis
    Commented Feb 3, 2021 at 20:20

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