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I posted this question on Math Stack Exchange, though I'm not satisfied with the answer I received.

Question:

For a Poisson random variable $Z$ with the parameter $\lambda,\,$ what would be a good upper bound (sub-exponential type perhaps?) for $P(Z \geq \frac{\lambda}{2})$; especially for large $\lambda$?

More generally, is it possible to find a sub-expoential type upper bound for $P(Z \geq \lambda - t)$ for $0\leq t \leq \lambda$?

Any comments (or references) will be appreciated.

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    $\begingroup$ This is the referenced question I think math.stackexchange.com/questions/4065525 $\endgroup$
    – J.J. Green
    Commented Mar 22, 2021 at 10:16
  • $\begingroup$ @J.J.Green Yes it is $\endgroup$
    – Jane
    Commented Mar 22, 2021 at 10:37
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    $\begingroup$ Why are you not satisfied by the answer received there? And why did you not mention what you'd like to get there? $\endgroup$
    – Clement C.
    Commented Mar 22, 2021 at 10:42
  • $\begingroup$ I'm looking for a sub-exponential type bound (if possible) rather than something of the form $1-ke^{-c \lambda}\, , k,c>0$. $\endgroup$
    – Jane
    Commented Mar 22, 2021 at 10:51
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    $\begingroup$ But you won't get that, unfortunately. Look at the lower tail $\Pr[X_n \leq \lambda/2]$ for a Binomial$(n,\lambda/n)$, and take the limit as $n\to \infty$. $\endgroup$
    – Clement C.
    Commented Mar 22, 2021 at 11:10

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