On Random Rearrangements
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Let $text{Sym}(mathbb{N})$ denote the group of bijections $mathbb{N} to mathbb{N}$. It is well-known that this has cardinality $mathfrak{c}$. Suppose $a_n$ is a conditionally convergent series with $sum a_n = L$. From a a result of Riemann, we have that for any real number $c$ there exists a $pi in text{Sym}(mathbb{N})$ such that $sum a_{pi(n)} = c$.
We want to consider random permutations $pi(n)$, and study the convergence of $sum a_{pi(n)}$ from a probabilistic perspective. We can define a uniform probability measure on $text{Sym}(mathbb{N})$ by $P(E) := mu(f(E))$ where $mu$ is the Lebesgue measure and $f$ is a fixed bijection from $text{Sym}(mathbb{N})$ to $[0,1]$. Note: if this doesn't work, please feel free to suggest some other measures on $text{Sym}(mathbb{N})$.
What can we say about the expected value of $sum a_{pi(n)}$? Do we expect it to diverge? Converge to $L$? Does it depend on the sequence $a_n$? If this is the case, is there nevertheless something meaningful that can be said, specifically for "common" conditionally convergent series like $sum n^{-1}(-1)^n$
Intuitively, I feel that we should expect it to diverge, because it's more likely for a given permutation to be "bad" than it is for it to be "good".
real-analysis probability sequences-and-series
asked Dec 13 '18 at 2:03
MathematicsStudent1122MathematicsStudent1122
8,63622467
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add a comment |
$begingroup$
Let $text{Sym}(mathbb{N})$ denote the group of bijections $mathbb{N} to mathbb{N}$. It is well-known that this has cardinality $mathfrak{c}$. Suppose $a_n$ is a conditionally convergent series with $sum a_n = L$. From a a result of Riemann, we have that for any real number $c$ there exists a $pi in text{Sym}(mathbb{N})$ such that $sum a_{pi(n)} = c$.
We want to consider random permutations $pi(n)$, and study the convergence of $sum a_{pi(n)}$ from a probabilistic perspective. We can define a uniform probability measure on $text{Sym}(mathbb{N})$ by $P(E) := mu(f(E))$ where $mu$ is the Lebesgue measure and $f$ is a fixed bijection from $text{Sym}(mathbb{N})$ to $[0,1]$. Note: if this doesn't work, please feel free to suggest some other measures on $text{Sym}(mathbb{N})$.
What can we say about the expected value of $sum a_{pi(n)}$? Do we expect it to diverge? Converge to $L$? Does it depend on the sequence $a_n$? If this is the case, is there nevertheless something meaningful that can be said, specifically for "common" conditionally convergent series like $sum n^{-1}(-1)^n$
Intuitively, I feel that we should expect it to diverge, because it's more likely for a given permutation to be "bad" than it is for it to be "good".
real-analysis probability sequences-and-series
asked Dec 13 '18 at 2:03
MathematicsStudent1122MathematicsStudent1122
8,63622467
$endgroup$
$begingroup$
A more meaningful thing would be to take finite sums, take a random permutation in the standard sense (uniform distribution on the group), and ask what happens as the number of terms in the sum gets really large. I would be surprised if it hasn't been studied yet.
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– fedja
Dec 13 '18 at 2:45
2
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I'm not sure how you're going to get a uniform measure on $text{Sym}(mathbb{N})$ since there is no uniform measure on $mathbb{N}$. This master's thesis does consider the problem for various measures on $text{Sym}(mathbb{N})$ though: web.cs.elte.hu/szakdolg/gebaboy.pdf
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– zoidberg
Dec 13 '18 at 2:50
$begingroup$
@norfair I meant $text{Sym}({1,2,dots,n})$, of course. +1 for the reference :-)
$endgroup$
– fedja
Dec 13 '18 at 15:25
add a comment |
$begingroup$
Let $text{Sym}(mathbb{N})$ denote the group of bijections $mathbb{N} to mathbb{N}$. It is well-known that this has cardinality $mathfrak{c}$. Suppose $a_n$ is a conditionally convergent series with $sum a_n = L$. From a a result of Riemann, we have that for any real number $c$ there exists a $pi in text{Sym}(mathbb{N})$ such that $sum a_{pi(n)} = c$.
We want to consider random permutations $pi(n)$, and study the convergence of $sum a_{pi(n)}$ from a probabilistic perspective. We can define a uniform probability measure on $text{Sym}(mathbb{N})$ by $P(E) := mu(f(E))$ where $mu$ is the Lebesgue measure and $f$ is a fixed bijection from $text{Sym}(mathbb{N})$ to $[0,1]$. Note: if this doesn't work, please feel free to suggest some other measures on $text{Sym}(mathbb{N})$.
What can we say about the expected value of $sum a_{pi(n)}$? Do we expect it to diverge? Converge to $L$? Does it depend on the sequence $a_n$? If this is the case, is there nevertheless something meaningful that can be said, specifically for "common" conditionally convergent series like $sum n^{-1}(-1)^n$
Intuitively, I feel that we should expect it to diverge, because it's more likely for a given permutation to be "bad" than it is for it to be "good".
real-analysis probability sequences-and-series
asked Dec 13 '18 at 2:03
MathematicsStudent1122MathematicsStudent1122
8,63622467
$endgroup$
Let $text{Sym}(mathbb{N})$ denote the group of bijections $mathbb{N} to mathbb{N}$. It is well-known that this has cardinality $mathfrak{c}$. Suppose $a_n$ is a conditionally convergent series with $sum a_n = L$. From a a result of Riemann, we have that for any real number $c$ there exists a $pi in text{Sym}(mathbb{N})$ such that $sum a_{pi(n)} = c$.
We want to consider random permutations $pi(n)$, and study the convergence of $sum a_{pi(n)}$ from a probabilistic perspective. We can define a uniform probability measure on $text{Sym}(mathbb{N})$ by $P(E) := mu(f(E))$ where $mu$ is the Lebesgue measure and $f$ is a fixed bijection from $text{Sym}(mathbb{N})$ to $[0,1]$. Note: if this doesn't work, please feel free to suggest some other measures on $text{Sym}(mathbb{N})$.
What can we say about the expected value of $sum a_{pi(n)}$? Do we expect it to diverge? Converge to $L$? Does it depend on the sequence $a_n$? If this is the case, is there nevertheless something meaningful that can be said, specifically for "common" conditionally convergent series like $sum n^{-1}(-1)^n$
Intuitively, I feel that we should expect it to diverge, because it's more likely for a given permutation to be "bad" than it is for it to be "good".
real-analysis probability sequences-and-series
real-analysis probability sequences-and-series
asked Dec 13 '18 at 2:03
MathematicsStudent1122MathematicsStudent1122
8,63622467
asked Dec 13 '18 at 2:03
MathematicsStudent1122MathematicsStudent1122
8,63622467
asked Dec 13 '18 at 2:03
MathematicsStudent1122MathematicsStudent1122
8,63622467
asked Dec 13 '18 at 2:03
MathematicsStudent1122MathematicsStudent1122
8,63622467
asked Dec 13 '18 at 2:03
MathematicsStudent1122MathematicsStudent1122
8,63622467
8,63622467
$begingroup$
A more meaningful thing would be to take finite sums, take a random permutation in the standard sense (uniform distribution on the group), and ask what happens as the number of terms in the sum gets really large. I would be surprised if it hasn't been studied yet.
$endgroup$
– fedja
Dec 13 '18 at 2:45
2
$begingroup$
I'm not sure how you're going to get a uniform measure on $text{Sym}(mathbb{N})$ since there is no uniform measure on $mathbb{N}$. This master's thesis does consider the problem for various measures on $text{Sym}(mathbb{N})$ though: web.cs.elte.hu/szakdolg/gebaboy.pdf
$endgroup$
– zoidberg
Dec 13 '18 at 2:50
$begingroup$
@norfair I meant $text{Sym}({1,2,dots,n})$, of course. +1 for the reference :-)
$endgroup$
– fedja
Dec 13 '18 at 15:25
add a comment |
$begingroup$
A more meaningful thing would be to take finite sums, take a random permutation in the standard sense (uniform distribution on the group), and ask what happens as the number of terms in the sum gets really large. I would be surprised if it hasn't been studied yet.
$endgroup$
– fedja
Dec 13 '18 at 2:45
2
$begingroup$
I'm not sure how you're going to get a uniform measure on $text{Sym}(mathbb{N})$ since there is no uniform measure on $mathbb{N}$. This master's thesis does consider the problem for various measures on $text{Sym}(mathbb{N})$ though: web.cs.elte.hu/szakdolg/gebaboy.pdf
$endgroup$
– zoidberg
Dec 13 '18 at 2:50
$begingroup$
@norfair I meant $text{Sym}({1,2,dots,n})$, of course. +1 for the reference :-)
$endgroup$
– fedja
Dec 13 '18 at 15:25
$begingroup$
A more meaningful thing would be to take finite sums, take a random permutation in the standard sense (uniform distribution on the group), and ask what happens as the number of terms in the sum gets really large. I would be surprised if it hasn't been studied yet.
$endgroup$
– fedja
Dec 13 '18 at 2:45
$begingroup$
A more meaningful thing would be to take finite sums, take a random permutation in the standard sense (uniform distribution on the group), and ask what happens as the number of terms in the sum gets really large. I would be surprised if it hasn't been studied yet.
$endgroup$
– fedja
Dec 13 '18 at 2:45
2
2
$begingroup$
I'm not sure how you're going to get a uniform measure on $text{Sym}(mathbb{N})$ since there is no uniform measure on $mathbb{N}$. This master's thesis does consider the problem for various measures on $text{Sym}(mathbb{N})$ though: web.cs.elte.hu/szakdolg/gebaboy.pdf
$endgroup$
– zoidberg
Dec 13 '18 at 2:50
$begingroup$
I'm not sure how you're going to get a uniform measure on $text{Sym}(mathbb{N})$ since there is no uniform measure on $mathbb{N}$. This master's thesis does consider the problem for various measures on $text{Sym}(mathbb{N})$ though: web.cs.elte.hu/szakdolg/gebaboy.pdf
$endgroup$
– zoidberg
Dec 13 '18 at 2:50
$begingroup$
@norfair I meant $text{Sym}({1,2,dots,n})$, of course. +1 for the reference :-)
$endgroup$
– fedja
Dec 13 '18 at 15:25
$begingroup$
@norfair I meant $text{Sym}({1,2,dots,n})$, of course. +1 for the reference :-)
$endgroup$
– fedja
Dec 13 '18 at 15:25
add a comment |
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$begingroup$
A more meaningful thing would be to take finite sums, take a random permutation in the standard sense (uniform distribution on the group), and ask what happens as the number of terms in the sum gets really large. I would be surprised if it hasn't been studied yet.
$endgroup$
– fedja
Dec 13 '18 at 2:45
2
$begingroup$
I'm not sure how you're going to get a uniform measure on $text{Sym}(mathbb{N})$ since there is no uniform measure on $mathbb{N}$. This master's thesis does consider the problem for various measures on $text{Sym}(mathbb{N})$ though: web.cs.elte.hu/szakdolg/gebaboy.pdf
$endgroup$
– zoidberg
Dec 13 '18 at 2:50
$begingroup$
@norfair I meant $text{Sym}({1,2,dots,n})$, of course. +1 for the reference :-)
$endgroup$
– fedja
Dec 13 '18 at 15:25