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user108
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Let $X,Y,Z$ be random variables such that

  • $X,Y,Z$ have mean $0$ and variance $1$.
  • $X$ and $Y$ are pairwise independent.
  • $Z$ can be arbitrarily dependent on $X$ and $Y$.

My question is: What can we say about the dependence of $Z$ on $X$ and $Y$? For example, if $\text{Cov}(X,Z)=1$$\operatorname{Cov}(X,Z)=1$ (so they are perfectly correlated), then $Z$ and $Y$ must be independent. In general, can we say anything about how much $Z$ can depend on (or be correlated with) $X$ and $Y$? Maybe something like $|\text{Cov}(X,Z)+\text{Cov}(Y,Z)|\leq 1$$|\operatorname{Cov}(X,Z)+\operatorname{Cov}(Y,Z)|\leq 1$ holds?

Let $X,Y,Z$ be random variables such that

  • $X,Y,Z$ have mean $0$ and variance $1$.
  • $X$ and $Y$ are pairwise independent.
  • $Z$ can be arbitrarily dependent on $X$ and $Y$.

My question is: What can we say about the dependence of $Z$ on $X$ and $Y$? For example, if $\text{Cov}(X,Z)=1$ (so they are perfectly correlated), then $Z$ and $Y$ must be independent. In general, can we say anything about how much $Z$ can depend on (or be correlated with) $X$ and $Y$? Maybe something like $|\text{Cov}(X,Z)+\text{Cov}(Y,Z)|\leq 1$ holds?

Let $X,Y,Z$ be random variables such that

  • $X,Y,Z$ have mean $0$ and variance $1$.
  • $X$ and $Y$ are pairwise independent.
  • $Z$ can be arbitrarily dependent on $X$ and $Y$.

My question is: What can we say about the dependence of $Z$ on $X$ and $Y$? For example, if $\operatorname{Cov}(X,Z)=1$ (so they are perfectly correlated), then $Z$ and $Y$ must be independent. In general, can we say anything about how much $Z$ can depend on (or be correlated with) $X$ and $Y$? Maybe something like $|\operatorname{Cov}(X,Z)+\operatorname{Cov}(Y,Z)|\leq 1$ holds?

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user108
user108
  • 73
  • 8

How does pairwise independence restrict dependence to a third variable?

Let $X,Y,Z$ be random variables such that

  • $X,Y,Z$ have mean $0$ and variance $1$.
  • $X$ and $Y$ are pairwise independent.
  • $Z$ can be arbitrarily dependent on $X$ and $Y$.

My question is: What can we say about the dependence of $Z$ on $X$ and $Y$? For example, if $\text{Cov}(X,Z)=1$ (so they are perfectly correlated), then $Z$ and $Y$ must be independent. In general, can we say anything about how much $Z$ can depend on (or be correlated with) $X$ and $Y$? Maybe something like $|\text{Cov}(X,Z)+\text{Cov}(Y,Z)|\leq 1$ holds?