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What implies correlation?

  1. Nov 10, 2011 #1
    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!
     
  2. jcsd
  3. Nov 10, 2011 #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.
     
  4. Nov 10, 2011 #3
    Many thanks for clearing that up so elegantly. It's easy when you know how!
     
  5. Nov 10, 2011 #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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