What implies correlation?

  • #1

Main Question or Discussion Point

If I have three scalar random variables: [itex]a[/itex], [itex]b[/itex] and [itex]c[/itex], which are each zero-mean and have some nonzero variances, and I know:

1) The correlation between [itex]a[/itex] and [itex]b[/itex] is nonzero.

2) The correlation between [itex]b[/itex] and [itex]c[/itex] is nonzero.

Does this imply that the correlation between [itex]a[/itex] and [itex]c[/itex] is nonzero?

I feel like the answer must be yes, but I don't have any sound mathematical reasoning for it. Any advice would be greatly appreciated!
 

Answers and Replies

  • #2
D H
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Simple counterexample: Suppose that a and c are uncorrelated random variables each with zero mean and nonzero variance and suppose that bac, with α and γ non-zero constants. By construction, b is correlated with each of a and c, but a and c are (by construction) uncorrelated.
 
  • #3
Simple counterexample: Suppose that a and c are uncorrelated random variables each with zero mean and nonzero variance and suppose that bac, with α and γ non-zero constants. By construction, b is correlated with each of a and c, but a and c are (by construction) uncorrelated.
Many thanks for clearing that up so elegantly. It's easy when you know how!
 
  • #4
Perhaps I can develop my understanding of a similar problem here without starting a new topic:

If, again, I have three scalar random variables [itex]a[/itex], [itex]b[/itex] and [itex]c[/itex] which are each zero-mean and have some nonzero variances... and in this case [itex]a[/itex] and [itex]b[/itex] are uncorrelated:

[itex]\mathcal{E} \left\{ ab^*\right\} = 0[/itex]

where [itex]\mathcal{E}\left\{\right\}[/itex] denotes expectation and [itex]*[/itex] denotes complex conjugate (although the variables probably need not be complex for this example).

What I'd like to know is whether, in general, we can find a [itex]c[/itex] which can sort of 'recorrelate' [itex]a[/itex] and [itex]b[/itex]:

[itex]\mathcal{E}\left\{ cab^*\right\} > 0[/itex]

I can't seem to find such a case using numerical examples in Matlab, but I'd really like to figure out a proper mathematical approach to this. Any advice or insights would be very much appreciated!
 

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