It is a frequent mistake to assume that uncorrelated random variables must be independent. Actually that is true only if the random variables are known to be of a multivariate normal distribution. Otherwise, testing that they are independent is much more difficult. Basically it involves breaking the intersections of their two ranges into a meshwork of little boxes. If the probabilities of samples that fall in each box can be approximated by the product of the marginal sample probabilities of each random variable for the two corresponding box ranges, then they are (approximately) independent. The finer the meshwork, and the larger the sample count, the more reliable the independence finding is.
Often independence is established more on a theoretical basis than by such testing. For example, when we see two coins flipped that have no obvious connection we feel that their toss results must be independent.
In the case of your two vectors it will be necessary to have very long vectors so as to have a reasonable chance to fill each "box" with sufficiently many samples to make your judgement of independence or non-independence reliable. There is no shortcut to doing this.
