Understanding the proof of a variant of Chernoff 's bound

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The following paper proves an interesting variant of Chernoff's bound that can be used to bound the probability that a majority in a population will become a minority in a sample.



Please see Lemma 6 here (or Lemma 6.1 in the longer version):
https://pdfs.semanticscholar.org/6ae2/3ac66011aede1a558caaca48cd2c8f3651b8.pdf



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I will appreciate it if someone could help me understand the argument or point me to a better source.



Here goes my partial understanding of it:



The proof begins with a hypothetical experiment in which the variant follows immediately from the classic Chernoff bound.
Then it compares the selection a random subset of $k$ "points" (at once) with another experiment, where the "elements" are chosen one by one, the $i$th element with probability $p_i$.
From this, it concludes the lemma without explaining much.




probability-theory random-variables sampling




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  • I found it on Wikipedia: en.wikipedia.org/wiki/Chernoff_bound#Sampling_variant
    – Parham
    Aug 3 at 16:41















up vote
0
down vote

favorite












The following paper proves an interesting variant of Chernoff's bound that can be used to bound the probability that a majority in a population will become a minority in a sample.



Please see Lemma 6 here (or Lemma 6.1 in the longer version):
https://pdfs.semanticscholar.org/6ae2/3ac66011aede1a558caaca48cd2c8f3651b8.pdf



enter image description here



I will appreciate it if someone could help me understand the argument or point me to a better source.



Here goes my partial understanding of it:



The proof begins with a hypothetical experiment in which the variant follows immediately from the classic Chernoff bound.
Then it compares the selection a random subset of $k$ "points" (at once) with another experiment, where the "elements" are chosen one by one, the $i$th element with probability $p_i$.
From this, it concludes the lemma without explaining much.




probability-theory random-variables sampling




share|cite|improve this question





















  • I found it on Wikipedia: en.wikipedia.org/wiki/Chernoff_bound#Sampling_variant
    – Parham
    Aug 3 at 16:41













up vote
0
down vote

favorite









up vote
0
down vote

favorite











The following paper proves an interesting variant of Chernoff's bound that can be used to bound the probability that a majority in a population will become a minority in a sample.



Please see Lemma 6 here (or Lemma 6.1 in the longer version):
https://pdfs.semanticscholar.org/6ae2/3ac66011aede1a558caaca48cd2c8f3651b8.pdf



enter image description here



I will appreciate it if someone could help me understand the argument or point me to a better source.



Here goes my partial understanding of it:



The proof begins with a hypothetical experiment in which the variant follows immediately from the classic Chernoff bound.
Then it compares the selection a random subset of $k$ "points" (at once) with another experiment, where the "elements" are chosen one by one, the $i$th element with probability $p_i$.
From this, it concludes the lemma without explaining much.




probability-theory random-variables sampling




share|cite|improve this question













The following paper proves an interesting variant of Chernoff's bound that can be used to bound the probability that a majority in a population will become a minority in a sample.



Please see Lemma 6 here (or Lemma 6.1 in the longer version):
https://pdfs.semanticscholar.org/6ae2/3ac66011aede1a558caaca48cd2c8f3651b8.pdf



enter image description here



I will appreciate it if someone could help me understand the argument or point me to a better source.



Here goes my partial understanding of it:



The proof begins with a hypothetical experiment in which the variant follows immediately from the classic Chernoff bound.
Then it compares the selection a random subset of $k$ "points" (at once) with another experiment, where the "elements" are chosen one by one, the $i$th element with probability $p_i$.
From this, it concludes the lemma without explaining much.





probability-theory random-variables sampling





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 3 at 17:07
























asked Aug 3 at 16:40









Parham

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  • I found it on Wikipedia: en.wikipedia.org/wiki/Chernoff_bound#Sampling_variant
    – Parham
    Aug 3 at 16:41

















  • I found it on Wikipedia: en.wikipedia.org/wiki/Chernoff_bound#Sampling_variant
    – Parham
    Aug 3 at 16:41
















I found it on Wikipedia: en.wikipedia.org/wiki/Chernoff_bound#Sampling_variant
– Parham
Aug 3 at 16:41





I found it on Wikipedia: en.wikipedia.org/wiki/Chernoff_bound#Sampling_variant
– Parham
Aug 3 at 16:41
















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