An independent collection, where the pairwise products are also independent
$begingroup$
Can you give me a non-trivial example of a sequence ${X_k}_{k=1}^n$ of independent, and identically distributed random variables, such that, the pairwise products, ${X_kX_{ell}}_{1leq k<ell leq n}$ are all independent?
Of course, if ${k,ell}cap {k',ell'}=varnothing$, we automatically have this, and since $(k,ell)neq {k',ell'}$, it boils down to giving an example where $X_kX_ell$ and $X_kX_{ell'}$ are always independent, whenever $ellneq ell'$ and $ell,ell'neq k$.
probability probability-theory probability-distributions independence conditional-probability
$endgroup$
add a comment |
$begingroup$
Can you give me a non-trivial example of a sequence ${X_k}_{k=1}^n$ of independent, and identically distributed random variables, such that, the pairwise products, ${X_kX_{ell}}_{1leq k<ell leq n}$ are all independent?
Of course, if ${k,ell}cap {k',ell'}=varnothing$, we automatically have this, and since $(k,ell)neq {k',ell'}$, it boils down to giving an example where $X_kX_ell$ and $X_kX_{ell'}$ are always independent, whenever $ellneq ell'$ and $ell,ell'neq k$.
probability probability-theory probability-distributions independence conditional-probability
$endgroup$
1
$begingroup$
When you ask for the pairwise products to be independent, do you mean to ask for the pairwise products be jointly independent or only pair-wise independent? Your "boils down" statement makes it sound like the latter.
$endgroup$
– kimchi lover
Dec 5 '18 at 20:44
$begingroup$
kimchi lover, I means pairwise. However, even the other direction seems interesting.
$endgroup$
– Aaron
Dec 5 '18 at 22:35
add a comment |
$begingroup$
Can you give me a non-trivial example of a sequence ${X_k}_{k=1}^n$ of independent, and identically distributed random variables, such that, the pairwise products, ${X_kX_{ell}}_{1leq k<ell leq n}$ are all independent?
Of course, if ${k,ell}cap {k',ell'}=varnothing$, we automatically have this, and since $(k,ell)neq {k',ell'}$, it boils down to giving an example where $X_kX_ell$ and $X_kX_{ell'}$ are always independent, whenever $ellneq ell'$ and $ell,ell'neq k$.
probability probability-theory probability-distributions independence conditional-probability
$endgroup$
Can you give me a non-trivial example of a sequence ${X_k}_{k=1}^n$ of independent, and identically distributed random variables, such that, the pairwise products, ${X_kX_{ell}}_{1leq k<ell leq n}$ are all independent?
Of course, if ${k,ell}cap {k',ell'}=varnothing$, we automatically have this, and since $(k,ell)neq {k',ell'}$, it boils down to giving an example where $X_kX_ell$ and $X_kX_{ell'}$ are always independent, whenever $ellneq ell'$ and $ell,ell'neq k$.
probability probability-theory probability-distributions independence conditional-probability
probability probability-theory probability-distributions independence conditional-probability
asked Dec 5 '18 at 20:31
AaronAaron
1,828415
1,828415
1
$begingroup$
When you ask for the pairwise products to be independent, do you mean to ask for the pairwise products be jointly independent or only pair-wise independent? Your "boils down" statement makes it sound like the latter.
$endgroup$
– kimchi lover
Dec 5 '18 at 20:44
$begingroup$
kimchi lover, I means pairwise. However, even the other direction seems interesting.
$endgroup$
– Aaron
Dec 5 '18 at 22:35
add a comment |
1
$begingroup$
When you ask for the pairwise products to be independent, do you mean to ask for the pairwise products be jointly independent or only pair-wise independent? Your "boils down" statement makes it sound like the latter.
$endgroup$
– kimchi lover
Dec 5 '18 at 20:44
$begingroup$
kimchi lover, I means pairwise. However, even the other direction seems interesting.
$endgroup$
– Aaron
Dec 5 '18 at 22:35
1
1
$begingroup$
When you ask for the pairwise products to be independent, do you mean to ask for the pairwise products be jointly independent or only pair-wise independent? Your "boils down" statement makes it sound like the latter.
$endgroup$
– kimchi lover
Dec 5 '18 at 20:44
$begingroup$
When you ask for the pairwise products to be independent, do you mean to ask for the pairwise products be jointly independent or only pair-wise independent? Your "boils down" statement makes it sound like the latter.
$endgroup$
– kimchi lover
Dec 5 '18 at 20:44
$begingroup$
kimchi lover, I means pairwise. However, even the other direction seems interesting.
$endgroup$
– Aaron
Dec 5 '18 at 22:35
$begingroup$
kimchi lover, I means pairwise. However, even the other direction seems interesting.
$endgroup$
– Aaron
Dec 5 '18 at 22:35
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
I don't know what you mean by non-trivial but consider this example:
$P(X = 1) = P(X = -1) = frac 1 2$
Knowing that the product is -1 or 1 doesn't change the probability for other products. $P(X_i = 1|X_i.X_j = 1) = frac 1 2$ and $P(X_i = -1|X_i.X_j = 1) = frac 1 2$ because 1*1 and -1*-1 happen with the same probability.
Then we can write
$$P(X_i.X_k = 1|X_i.X_j = 1) = P(X_k = 1|X_i.X_j = 1, X_i=1 ).P(X_i = 1|X_i.X_j = 1) + P(X_k = -1|X_i.X_j = 1, X_i=-1 ).P(X_i = -1|X_i.X_j = 1) = frac 1 2 $$
$$P(X_i.X_k = 1|X_i.X_j = -1) = P(X_k = 1|X_i.X_j = -1, X_i=1 ).P(X_i = 1|X_i.X_j = -1) + P(X_k = -1|X_i.X_j = -1, X_i=-1 ).P(X_i = -1|X_i.X_j = -1) = frac 1 2 $$
$$P(X_i.X_k = 1) = P(X_k = 1| X_i=1 ).P(X_i = 1) + P(X_k = -1|X_i=-1 ).P(X_i = -1) = frac 1 2 $$
Since the conditional probability is the same as unconditional, you can say they are independent.
Hope it helps
$endgroup$
add a comment |
$begingroup$
@Ofya, thanks for your answer, I was seeking for something where I can control the support, and I guess, I've found my answer.
EDIT (Extra question) Suppose that, $X_i$ are independent, standard normal random variables. Is it true that, $(X_iX_j)_{1leq i <jleq n}$ forms an independent collection of random variables?
Take $X_isim {rm Unif}{1,2,dots,p-1}$ with $p$ being a prime, and consider $X_iX_j$ as $X_iX_jpmod{p}$. We have,
$$
mathbb{P}(X_iX_j = k,X_iX_kell)=sum_{t=1}^{p-1}mathbb{P}(X_iX_j=k,X_iX_k=ell|X_i=t)mathbb{P}(X_i=t)=sum_{t=1}^{p-1}mathbb{P}(X_j=t^{-1}k,X_k=t^{-1}ell)mathbb{P}(X_i=t),
$$
which evaluates as, $frac{1}{(p-1)^2}$. Finally, it is easy to see that, for any pair $(i ,j)$ and any $kin{1,2,dots,p-1}$, $mathbb{P}(X_iX_j=k)=frac{1}{p-1}$, proving the independence.
$endgroup$
$begingroup$
As for your extra credit problem: take the product of $X_1X_2, X_2X_3,$ and $X_3X_1.$ The product is positive with probability 1, so the 3 pairwise products cannot be jointly independent.
$endgroup$
– kimchi lover
Dec 5 '18 at 22:46
$begingroup$
How about pairwise independence of pairwise products?
$endgroup$
– Aaron
Dec 5 '18 at 23:53
$begingroup$
In general, Ofya's answer puts paid to that one. In the gaussian case, I think one can show that $XY$ and $YZ$ are not independent, if $X,Y,Z$ are iid $N(0,1)$ rv.s.
$endgroup$
– kimchi lover
Dec 6 '18 at 0:17
add a comment |
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2 Answers
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2 Answers
2
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$begingroup$
I don't know what you mean by non-trivial but consider this example:
$P(X = 1) = P(X = -1) = frac 1 2$
Knowing that the product is -1 or 1 doesn't change the probability for other products. $P(X_i = 1|X_i.X_j = 1) = frac 1 2$ and $P(X_i = -1|X_i.X_j = 1) = frac 1 2$ because 1*1 and -1*-1 happen with the same probability.
Then we can write
$$P(X_i.X_k = 1|X_i.X_j = 1) = P(X_k = 1|X_i.X_j = 1, X_i=1 ).P(X_i = 1|X_i.X_j = 1) + P(X_k = -1|X_i.X_j = 1, X_i=-1 ).P(X_i = -1|X_i.X_j = 1) = frac 1 2 $$
$$P(X_i.X_k = 1|X_i.X_j = -1) = P(X_k = 1|X_i.X_j = -1, X_i=1 ).P(X_i = 1|X_i.X_j = -1) + P(X_k = -1|X_i.X_j = -1, X_i=-1 ).P(X_i = -1|X_i.X_j = -1) = frac 1 2 $$
$$P(X_i.X_k = 1) = P(X_k = 1| X_i=1 ).P(X_i = 1) + P(X_k = -1|X_i=-1 ).P(X_i = -1) = frac 1 2 $$
Since the conditional probability is the same as unconditional, you can say they are independent.
Hope it helps
$endgroup$
add a comment |
$begingroup$
I don't know what you mean by non-trivial but consider this example:
$P(X = 1) = P(X = -1) = frac 1 2$
Knowing that the product is -1 or 1 doesn't change the probability for other products. $P(X_i = 1|X_i.X_j = 1) = frac 1 2$ and $P(X_i = -1|X_i.X_j = 1) = frac 1 2$ because 1*1 and -1*-1 happen with the same probability.
Then we can write
$$P(X_i.X_k = 1|X_i.X_j = 1) = P(X_k = 1|X_i.X_j = 1, X_i=1 ).P(X_i = 1|X_i.X_j = 1) + P(X_k = -1|X_i.X_j = 1, X_i=-1 ).P(X_i = -1|X_i.X_j = 1) = frac 1 2 $$
$$P(X_i.X_k = 1|X_i.X_j = -1) = P(X_k = 1|X_i.X_j = -1, X_i=1 ).P(X_i = 1|X_i.X_j = -1) + P(X_k = -1|X_i.X_j = -1, X_i=-1 ).P(X_i = -1|X_i.X_j = -1) = frac 1 2 $$
$$P(X_i.X_k = 1) = P(X_k = 1| X_i=1 ).P(X_i = 1) + P(X_k = -1|X_i=-1 ).P(X_i = -1) = frac 1 2 $$
Since the conditional probability is the same as unconditional, you can say they are independent.
Hope it helps
$endgroup$
add a comment |
$begingroup$
I don't know what you mean by non-trivial but consider this example:
$P(X = 1) = P(X = -1) = frac 1 2$
Knowing that the product is -1 or 1 doesn't change the probability for other products. $P(X_i = 1|X_i.X_j = 1) = frac 1 2$ and $P(X_i = -1|X_i.X_j = 1) = frac 1 2$ because 1*1 and -1*-1 happen with the same probability.
Then we can write
$$P(X_i.X_k = 1|X_i.X_j = 1) = P(X_k = 1|X_i.X_j = 1, X_i=1 ).P(X_i = 1|X_i.X_j = 1) + P(X_k = -1|X_i.X_j = 1, X_i=-1 ).P(X_i = -1|X_i.X_j = 1) = frac 1 2 $$
$$P(X_i.X_k = 1|X_i.X_j = -1) = P(X_k = 1|X_i.X_j = -1, X_i=1 ).P(X_i = 1|X_i.X_j = -1) + P(X_k = -1|X_i.X_j = -1, X_i=-1 ).P(X_i = -1|X_i.X_j = -1) = frac 1 2 $$
$$P(X_i.X_k = 1) = P(X_k = 1| X_i=1 ).P(X_i = 1) + P(X_k = -1|X_i=-1 ).P(X_i = -1) = frac 1 2 $$
Since the conditional probability is the same as unconditional, you can say they are independent.
Hope it helps
$endgroup$
I don't know what you mean by non-trivial but consider this example:
$P(X = 1) = P(X = -1) = frac 1 2$
Knowing that the product is -1 or 1 doesn't change the probability for other products. $P(X_i = 1|X_i.X_j = 1) = frac 1 2$ and $P(X_i = -1|X_i.X_j = 1) = frac 1 2$ because 1*1 and -1*-1 happen with the same probability.
Then we can write
$$P(X_i.X_k = 1|X_i.X_j = 1) = P(X_k = 1|X_i.X_j = 1, X_i=1 ).P(X_i = 1|X_i.X_j = 1) + P(X_k = -1|X_i.X_j = 1, X_i=-1 ).P(X_i = -1|X_i.X_j = 1) = frac 1 2 $$
$$P(X_i.X_k = 1|X_i.X_j = -1) = P(X_k = 1|X_i.X_j = -1, X_i=1 ).P(X_i = 1|X_i.X_j = -1) + P(X_k = -1|X_i.X_j = -1, X_i=-1 ).P(X_i = -1|X_i.X_j = -1) = frac 1 2 $$
$$P(X_i.X_k = 1) = P(X_k = 1| X_i=1 ).P(X_i = 1) + P(X_k = -1|X_i=-1 ).P(X_i = -1) = frac 1 2 $$
Since the conditional probability is the same as unconditional, you can say they are independent.
Hope it helps
answered Dec 5 '18 at 20:46
OfyaOfya
5048
5048
add a comment |
add a comment |
$begingroup$
@Ofya, thanks for your answer, I was seeking for something where I can control the support, and I guess, I've found my answer.
EDIT (Extra question) Suppose that, $X_i$ are independent, standard normal random variables. Is it true that, $(X_iX_j)_{1leq i <jleq n}$ forms an independent collection of random variables?
Take $X_isim {rm Unif}{1,2,dots,p-1}$ with $p$ being a prime, and consider $X_iX_j$ as $X_iX_jpmod{p}$. We have,
$$
mathbb{P}(X_iX_j = k,X_iX_kell)=sum_{t=1}^{p-1}mathbb{P}(X_iX_j=k,X_iX_k=ell|X_i=t)mathbb{P}(X_i=t)=sum_{t=1}^{p-1}mathbb{P}(X_j=t^{-1}k,X_k=t^{-1}ell)mathbb{P}(X_i=t),
$$
which evaluates as, $frac{1}{(p-1)^2}$. Finally, it is easy to see that, for any pair $(i ,j)$ and any $kin{1,2,dots,p-1}$, $mathbb{P}(X_iX_j=k)=frac{1}{p-1}$, proving the independence.
$endgroup$
$begingroup$
As for your extra credit problem: take the product of $X_1X_2, X_2X_3,$ and $X_3X_1.$ The product is positive with probability 1, so the 3 pairwise products cannot be jointly independent.
$endgroup$
– kimchi lover
Dec 5 '18 at 22:46
$begingroup$
How about pairwise independence of pairwise products?
$endgroup$
– Aaron
Dec 5 '18 at 23:53
$begingroup$
In general, Ofya's answer puts paid to that one. In the gaussian case, I think one can show that $XY$ and $YZ$ are not independent, if $X,Y,Z$ are iid $N(0,1)$ rv.s.
$endgroup$
– kimchi lover
Dec 6 '18 at 0:17
add a comment |
$begingroup$
@Ofya, thanks for your answer, I was seeking for something where I can control the support, and I guess, I've found my answer.
EDIT (Extra question) Suppose that, $X_i$ are independent, standard normal random variables. Is it true that, $(X_iX_j)_{1leq i <jleq n}$ forms an independent collection of random variables?
Take $X_isim {rm Unif}{1,2,dots,p-1}$ with $p$ being a prime, and consider $X_iX_j$ as $X_iX_jpmod{p}$. We have,
$$
mathbb{P}(X_iX_j = k,X_iX_kell)=sum_{t=1}^{p-1}mathbb{P}(X_iX_j=k,X_iX_k=ell|X_i=t)mathbb{P}(X_i=t)=sum_{t=1}^{p-1}mathbb{P}(X_j=t^{-1}k,X_k=t^{-1}ell)mathbb{P}(X_i=t),
$$
which evaluates as, $frac{1}{(p-1)^2}$. Finally, it is easy to see that, for any pair $(i ,j)$ and any $kin{1,2,dots,p-1}$, $mathbb{P}(X_iX_j=k)=frac{1}{p-1}$, proving the independence.
$endgroup$
$begingroup$
As for your extra credit problem: take the product of $X_1X_2, X_2X_3,$ and $X_3X_1.$ The product is positive with probability 1, so the 3 pairwise products cannot be jointly independent.
$endgroup$
– kimchi lover
Dec 5 '18 at 22:46
$begingroup$
How about pairwise independence of pairwise products?
$endgroup$
– Aaron
Dec 5 '18 at 23:53
$begingroup$
In general, Ofya's answer puts paid to that one. In the gaussian case, I think one can show that $XY$ and $YZ$ are not independent, if $X,Y,Z$ are iid $N(0,1)$ rv.s.
$endgroup$
– kimchi lover
Dec 6 '18 at 0:17
add a comment |
$begingroup$
@Ofya, thanks for your answer, I was seeking for something where I can control the support, and I guess, I've found my answer.
EDIT (Extra question) Suppose that, $X_i$ are independent, standard normal random variables. Is it true that, $(X_iX_j)_{1leq i <jleq n}$ forms an independent collection of random variables?
Take $X_isim {rm Unif}{1,2,dots,p-1}$ with $p$ being a prime, and consider $X_iX_j$ as $X_iX_jpmod{p}$. We have,
$$
mathbb{P}(X_iX_j = k,X_iX_kell)=sum_{t=1}^{p-1}mathbb{P}(X_iX_j=k,X_iX_k=ell|X_i=t)mathbb{P}(X_i=t)=sum_{t=1}^{p-1}mathbb{P}(X_j=t^{-1}k,X_k=t^{-1}ell)mathbb{P}(X_i=t),
$$
which evaluates as, $frac{1}{(p-1)^2}$. Finally, it is easy to see that, for any pair $(i ,j)$ and any $kin{1,2,dots,p-1}$, $mathbb{P}(X_iX_j=k)=frac{1}{p-1}$, proving the independence.
$endgroup$
@Ofya, thanks for your answer, I was seeking for something where I can control the support, and I guess, I've found my answer.
EDIT (Extra question) Suppose that, $X_i$ are independent, standard normal random variables. Is it true that, $(X_iX_j)_{1leq i <jleq n}$ forms an independent collection of random variables?
Take $X_isim {rm Unif}{1,2,dots,p-1}$ with $p$ being a prime, and consider $X_iX_j$ as $X_iX_jpmod{p}$. We have,
$$
mathbb{P}(X_iX_j = k,X_iX_kell)=sum_{t=1}^{p-1}mathbb{P}(X_iX_j=k,X_iX_k=ell|X_i=t)mathbb{P}(X_i=t)=sum_{t=1}^{p-1}mathbb{P}(X_j=t^{-1}k,X_k=t^{-1}ell)mathbb{P}(X_i=t),
$$
which evaluates as, $frac{1}{(p-1)^2}$. Finally, it is easy to see that, for any pair $(i ,j)$ and any $kin{1,2,dots,p-1}$, $mathbb{P}(X_iX_j=k)=frac{1}{p-1}$, proving the independence.
edited Dec 5 '18 at 22:40
answered Dec 5 '18 at 21:55
AaronAaron
1,828415
1,828415
$begingroup$
As for your extra credit problem: take the product of $X_1X_2, X_2X_3,$ and $X_3X_1.$ The product is positive with probability 1, so the 3 pairwise products cannot be jointly independent.
$endgroup$
– kimchi lover
Dec 5 '18 at 22:46
$begingroup$
How about pairwise independence of pairwise products?
$endgroup$
– Aaron
Dec 5 '18 at 23:53
$begingroup$
In general, Ofya's answer puts paid to that one. In the gaussian case, I think one can show that $XY$ and $YZ$ are not independent, if $X,Y,Z$ are iid $N(0,1)$ rv.s.
$endgroup$
– kimchi lover
Dec 6 '18 at 0:17
add a comment |
$begingroup$
As for your extra credit problem: take the product of $X_1X_2, X_2X_3,$ and $X_3X_1.$ The product is positive with probability 1, so the 3 pairwise products cannot be jointly independent.
$endgroup$
– kimchi lover
Dec 5 '18 at 22:46
$begingroup$
How about pairwise independence of pairwise products?
$endgroup$
– Aaron
Dec 5 '18 at 23:53
$begingroup$
In general, Ofya's answer puts paid to that one. In the gaussian case, I think one can show that $XY$ and $YZ$ are not independent, if $X,Y,Z$ are iid $N(0,1)$ rv.s.
$endgroup$
– kimchi lover
Dec 6 '18 at 0:17
$begingroup$
As for your extra credit problem: take the product of $X_1X_2, X_2X_3,$ and $X_3X_1.$ The product is positive with probability 1, so the 3 pairwise products cannot be jointly independent.
$endgroup$
– kimchi lover
Dec 5 '18 at 22:46
$begingroup$
As for your extra credit problem: take the product of $X_1X_2, X_2X_3,$ and $X_3X_1.$ The product is positive with probability 1, so the 3 pairwise products cannot be jointly independent.
$endgroup$
– kimchi lover
Dec 5 '18 at 22:46
$begingroup$
How about pairwise independence of pairwise products?
$endgroup$
– Aaron
Dec 5 '18 at 23:53
$begingroup$
How about pairwise independence of pairwise products?
$endgroup$
– Aaron
Dec 5 '18 at 23:53
$begingroup$
In general, Ofya's answer puts paid to that one. In the gaussian case, I think one can show that $XY$ and $YZ$ are not independent, if $X,Y,Z$ are iid $N(0,1)$ rv.s.
$endgroup$
– kimchi lover
Dec 6 '18 at 0:17
$begingroup$
In general, Ofya's answer puts paid to that one. In the gaussian case, I think one can show that $XY$ and $YZ$ are not independent, if $X,Y,Z$ are iid $N(0,1)$ rv.s.
$endgroup$
– kimchi lover
Dec 6 '18 at 0:17
add a comment |
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$begingroup$
When you ask for the pairwise products to be independent, do you mean to ask for the pairwise products be jointly independent or only pair-wise independent? Your "boils down" statement makes it sound like the latter.
$endgroup$
– kimchi lover
Dec 5 '18 at 20:44
$begingroup$
kimchi lover, I means pairwise. However, even the other direction seems interesting.
$endgroup$
– Aaron
Dec 5 '18 at 22:35