Does Nonzero Correlation Between Pairs Imply Correlation Between All?

[weetabixharry]

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

1) The correlation between a and b is nonzero.

2) The correlation between b and c is nonzero.

Does this imply that the correlation between a and c 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!

 

[D H]

Simple counterexample: Suppose that a and c are uncorrelated random variables each with zero mean and nonzero variance and suppose that b=αa+γc, 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.

 

[weetabixharry]

Simple counterexample: Suppose that a and c are uncorrelated random variables each with zero mean and nonzero variance and suppose that b=αa+γc, 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!

 

[weetabixharry]

Perhaps I can develop my understanding of a similar problem here without starting a new topic:

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

\mathcal{E} \left\{ ab^*\right\} = 0

where \mathcal{E}\left\{\right\} denotes expectation and * 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 c which can sort of 'recorrelate' a and b:

\mathcal{E}\left\{ cab^*\right\} > 0

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!