First off, I would read [Examples of Independent and uncorrelated data in real-life, and ways to measure/detect them][1].  
It will likely prove useful to you.

If you are looking for something quick, easy, and useful, I would simply [cross-correlate][2] the two vectors and normalize the result.  A small result (less that 0.05?) indicates, but does not prove, independence.  A large result (more than 0.2?) would indicate some amount of dependence.  If the vectors have a non-zero mean, you may need to subtract out the mean before doing the correlation, depending on why they are non-zero mean.

You could also go a more academic route and use the method pichenettes outlines (estimate the probability distribution using a kernel density estimator) in the thread that I linked to at the beginning of the post.


  [1]: https://dsp.stackexchange.com/questions/1242/examples-of-independent-and-uncorrelated-data-in-real-life-and-ways-to-measure
  [2]: http://en.wikipedia.org/wiki/Cross-correlation