A negative correlation property in a random matrix
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I am trying to prove the following negative correlation property. (where neither FKG or the BK inequality apply) Any input/idea is much appreciated:
Suppose each row of an $ntimes n$ matrix is filled by a permutation over $1..n$ drawn independently uniformly at random. Fix the integers $d,k$ with $1leq d < kleq n$.
We say Event $E_i$ happens if all the first $d$ appearances of number $i$ in the matrix that occur in columns larger than $d$ also occur in rows larger than $k$ (where by first we mean smallest column index). The goal is proving the following negative correlation property:
$$Pr{E_1wedgeldotswedge E_m}leq prod_{i=1}^mPr{E_i}$$
(There's an special case in the definition of $E_i$ that can be defined in two ways. In case there are fewer than $d$ appearances of number $i$ after column $d$, then the condition has to hold just on those appearances. Alternatively, you can exclude such matrices from $E_i$.)
Thanks a lot!
probability combinatorics probability-theory correlation probabilistic-method
asked Nov 24 at 0:51
afshi7n
14710
add a comment |
up vote
1
down vote
favorite
I am trying to prove the following negative correlation property. (where neither FKG or the BK inequality apply) Any input/idea is much appreciated:
Suppose each row of an $ntimes n$ matrix is filled by a permutation over $1..n$ drawn independently uniformly at random. Fix the integers $d,k$ with $1leq d < kleq n$.
We say Event $E_i$ happens if all the first $d$ appearances of number $i$ in the matrix that occur in columns larger than $d$ also occur in rows larger than $k$ (where by first we mean smallest column index). The goal is proving the following negative correlation property:
$$Pr{E_1wedgeldotswedge E_m}leq prod_{i=1}^mPr{E_i}$$
(There's an special case in the definition of $E_i$ that can be defined in two ways. In case there are fewer than $d$ appearances of number $i$ after column $d$, then the condition has to hold just on those appearances. Alternatively, you can exclude such matrices from $E_i$.)
Thanks a lot!
probability combinatorics probability-theory correlation probabilistic-method
asked Nov 24 at 0:51
afshi7n
14710
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am trying to prove the following negative correlation property. (where neither FKG or the BK inequality apply) Any input/idea is much appreciated:
Suppose each row of an $ntimes n$ matrix is filled by a permutation over $1..n$ drawn independently uniformly at random. Fix the integers $d,k$ with $1leq d < kleq n$.
We say Event $E_i$ happens if all the first $d$ appearances of number $i$ in the matrix that occur in columns larger than $d$ also occur in rows larger than $k$ (where by first we mean smallest column index). The goal is proving the following negative correlation property:
$$Pr{E_1wedgeldotswedge E_m}leq prod_{i=1}^mPr{E_i}$$
(There's an special case in the definition of $E_i$ that can be defined in two ways. In case there are fewer than $d$ appearances of number $i$ after column $d$, then the condition has to hold just on those appearances. Alternatively, you can exclude such matrices from $E_i$.)
Thanks a lot!
probability combinatorics probability-theory correlation probabilistic-method
asked Nov 24 at 0:51
afshi7n
14710
I am trying to prove the following negative correlation property. (where neither FKG or the BK inequality apply) Any input/idea is much appreciated:
Suppose each row of an $ntimes n$ matrix is filled by a permutation over $1..n$ drawn independently uniformly at random. Fix the integers $d,k$ with $1leq d < kleq n$.
We say Event $E_i$ happens if all the first $d$ appearances of number $i$ in the matrix that occur in columns larger than $d$ also occur in rows larger than $k$ (where by first we mean smallest column index). The goal is proving the following negative correlation property:
$$Pr{E_1wedgeldotswedge E_m}leq prod_{i=1}^mPr{E_i}$$
(There's an special case in the definition of $E_i$ that can be defined in two ways. In case there are fewer than $d$ appearances of number $i$ after column $d$, then the condition has to hold just on those appearances. Alternatively, you can exclude such matrices from $E_i$.)
Thanks a lot!
probability combinatorics probability-theory correlation probabilistic-method
probability combinatorics probability-theory correlation probabilistic-method
asked Nov 24 at 0:51
afshi7n
14710
asked Nov 24 at 0:51
afshi7n
14710
edited Nov 24 at 2:56
asked Nov 24 at 0:51
afshi7n
14710
asked Nov 24 at 0:51
afshi7n
14710
asked Nov 24 at 0:51
afshi7n
14710
14710
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