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LSpice
LSpice
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Upper bound for $P(X \geq x)$, where $X \sim Pois\operatorname{Pois}(\lambda)$

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

Question:Question:

Let $X \sim Pois(\lambda).$$X \sim \operatorname{Pois}(\lambda)$. I'm interested in a large deviation bound for $X$; more specifically a sub-exponential type bound for $P(X \geq x)$ for all $x\geq 0.$$x\geq 0$. While it's known that such a bound exists for $x \geq \lambda$ (chernoffChernoff bound), it's not clear whether there exists one for $x< \lambda.$$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. not?

Upper bound for $P(X \geq x)$, where $X \sim Pois(\lambda)$

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.

Upper bound for $P(X \geq x)$, where $X \sim \operatorname{Pois}(\lambda)$

I posted the following question in a comment on CDF of a log-concave discrete random variable. Since it is not related to my main question, I thought of reposting it as separate post.

Question:

Let $X \sim \operatorname{Pois}(\lambda)$. 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, why not?

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SL_MathGuy
SL_MathGuy
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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.

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 for these $x$ values using standard large-deviation techniques? If not, can someone explain why.

Any comments would be appreciated.

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.

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SL_MathGuy
SL_MathGuy
  • 135
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Upper bound for $P(X \geq x)$, where $X \sim Pois(\lambda)$

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 for these $x$ values using standard large-deviation techniques? If not, can someone explain why.

Any comments would be appreciated.