Deduce upper bound of variance from Chernoff-type tail bound

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For a random variable $X$, I have a large deviation inequality of the form
beginequation
P(|X-mathbb EX|geq r)leq ce^-alpha r,.
endequation



Consider a sample mean $S_n=frac1n(X_1+X_2+dots+X_n)$. I would to obtain a central limit theorem of the form $sqrt n(S_n-mu)equiv N(0,sigma^2)$. Is this possible?



It seems that I need to know the variance of $X$ to apply the Lindeberg–Lévy CLT. I don't see how I can get the variance of $X$ from its tail bound.




central-limit-theorem distribution-tails




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  • Typed a more relevant title.
    – Did
    Aug 28 at 19:09














up vote
1
down vote

favorite












For a random variable $X$, I have a large deviation inequality of the form
beginequation
P(|X-mathbb EX|geq r)leq ce^-alpha r,.
endequation



Consider a sample mean $S_n=frac1n(X_1+X_2+dots+X_n)$. I would to obtain a central limit theorem of the form $sqrt n(S_n-mu)equiv N(0,sigma^2)$. Is this possible?



It seems that I need to know the variance of $X$ to apply the Lindeberg–Lévy CLT. I don't see how I can get the variance of $X$ from its tail bound.




central-limit-theorem distribution-tails




share|cite|improve this question






















  • Typed a more relevant title.
    – Did
    Aug 28 at 19:09












up vote
1
down vote

favorite









up vote
1
down vote

favorite











For a random variable $X$, I have a large deviation inequality of the form
beginequation
P(|X-mathbb EX|geq r)leq ce^-alpha r,.
endequation



Consider a sample mean $S_n=frac1n(X_1+X_2+dots+X_n)$. I would to obtain a central limit theorem of the form $sqrt n(S_n-mu)equiv N(0,sigma^2)$. Is this possible?



It seems that I need to know the variance of $X$ to apply the Lindeberg–Lévy CLT. I don't see how I can get the variance of $X$ from its tail bound.




central-limit-theorem distribution-tails




share|cite|improve this question














For a random variable $X$, I have a large deviation inequality of the form
beginequation
P(|X-mathbb EX|geq r)leq ce^-alpha r,.
endequation



Consider a sample mean $S_n=frac1n(X_1+X_2+dots+X_n)$. I would to obtain a central limit theorem of the form $sqrt n(S_n-mu)equiv N(0,sigma^2)$. Is this possible?



It seems that I need to know the variance of $X$ to apply the Lindeberg–Lévy CLT. I don't see how I can get the variance of $X$ from its tail bound.





central-limit-theorem distribution-tails





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edited Aug 28 at 19:08









Did

243k23209444




243k23209444










asked Aug 28 at 18:39









Alice Schwarze

8713




8713











  • Typed a more relevant title.
    – Did
    Aug 28 at 19:09
















  • Typed a more relevant title.
    – Did
    Aug 28 at 19:09















Typed a more relevant title.
– Did
Aug 28 at 19:09




Typed a more relevant title.
– Did
Aug 28 at 19:09










1 Answer
1






active

oldest

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up vote
1
down vote



accepted










You have, for a non-negative random variable $Z$,
$$
mathbbE[Z] = int_0^infty mathbbPZ geq zdz tag1
$$
from which you can bound the variance of $X$:
$$
operatornameVar X = mathbbE[(X-mathbbE[X])^2]
= int_0^infty mathbbP(X-mathbbE[X])^2 geq zdz
= int_0^infty mathbbPlvert X-mathbbE[X]rvertgeq sqrt z dztag2
$$
and using your concentration bound you thus obtain
$$
operatornameVar X
leq int_0^infty c e^-alpha sqrtzdz = fraccalpha^2 tag3
$$
Does that suffice for your purposes?






share|cite|improve this answer






















  • yes, that helps. thank you!
    – Alice Schwarze
    Aug 28 at 19:03










  • @AliceSchwarze You're welcome!
    – Clement C.
    Aug 28 at 19:05










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










You have, for a non-negative random variable $Z$,
$$
mathbbE[Z] = int_0^infty mathbbPZ geq zdz tag1
$$
from which you can bound the variance of $X$:
$$
operatornameVar X = mathbbE[(X-mathbbE[X])^2]
= int_0^infty mathbbP(X-mathbbE[X])^2 geq zdz
= int_0^infty mathbbPlvert X-mathbbE[X]rvertgeq sqrt z dztag2
$$
and using your concentration bound you thus obtain
$$
operatornameVar X
leq int_0^infty c e^-alpha sqrtzdz = fraccalpha^2 tag3
$$
Does that suffice for your purposes?






share|cite|improve this answer






















  • yes, that helps. thank you!
    – Alice Schwarze
    Aug 28 at 19:03










  • @AliceSchwarze You're welcome!
    – Clement C.
    Aug 28 at 19:05














up vote
1
down vote



accepted










You have, for a non-negative random variable $Z$,
$$
mathbbE[Z] = int_0^infty mathbbPZ geq zdz tag1
$$
from which you can bound the variance of $X$:
$$
operatornameVar X = mathbbE[(X-mathbbE[X])^2]
= int_0^infty mathbbP(X-mathbbE[X])^2 geq zdz
= int_0^infty mathbbPlvert X-mathbbE[X]rvertgeq sqrt z dztag2
$$
and using your concentration bound you thus obtain
$$
operatornameVar X
leq int_0^infty c e^-alpha sqrtzdz = fraccalpha^2 tag3
$$
Does that suffice for your purposes?






share|cite|improve this answer






















  • yes, that helps. thank you!
    – Alice Schwarze
    Aug 28 at 19:03










  • @AliceSchwarze You're welcome!
    – Clement C.
    Aug 28 at 19:05












up vote
1
down vote



accepted







up vote
1
down vote



accepted






You have, for a non-negative random variable $Z$,
$$
mathbbE[Z] = int_0^infty mathbbPZ geq zdz tag1
$$
from which you can bound the variance of $X$:
$$
operatornameVar X = mathbbE[(X-mathbbE[X])^2]
= int_0^infty mathbbP(X-mathbbE[X])^2 geq zdz
= int_0^infty mathbbPlvert X-mathbbE[X]rvertgeq sqrt z dztag2
$$
and using your concentration bound you thus obtain
$$
operatornameVar X
leq int_0^infty c e^-alpha sqrtzdz = fraccalpha^2 tag3
$$
Does that suffice for your purposes?






share|cite|improve this answer














You have, for a non-negative random variable $Z$,
$$
mathbbE[Z] = int_0^infty mathbbPZ geq zdz tag1
$$
from which you can bound the variance of $X$:
$$
operatornameVar X = mathbbE[(X-mathbbE[X])^2]
= int_0^infty mathbbP(X-mathbbE[X])^2 geq zdz
= int_0^infty mathbbPlvert X-mathbbE[X]rvertgeq sqrt z dztag2
$$
and using your concentration bound you thus obtain
$$
operatornameVar X
leq int_0^infty c e^-alpha sqrtzdz = fraccalpha^2 tag3
$$
Does that suffice for your purposes?







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Aug 28 at 19:05









Alice Schwarze

8713




8713










answered Aug 28 at 18:50









Clement C.

47.4k33783




47.4k33783











  • yes, that helps. thank you!
    – Alice Schwarze
    Aug 28 at 19:03










  • @AliceSchwarze You're welcome!
    – Clement C.
    Aug 28 at 19:05
















  • yes, that helps. thank you!
    – Alice Schwarze
    Aug 28 at 19:03










  • @AliceSchwarze You're welcome!
    – Clement C.
    Aug 28 at 19:05















yes, that helps. thank you!
– Alice Schwarze
Aug 28 at 19:03




yes, that helps. thank you!
– Alice Schwarze
Aug 28 at 19:03












@AliceSchwarze You're welcome!
– Clement C.
Aug 28 at 19:05




@AliceSchwarze You're welcome!
– Clement C.
Aug 28 at 19:05

















 

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