Does randomness imply independence? By looking at a lag plot, it seems that my dataset is random (no autocorrelation whatsoever). Should this be sufficient to infer independence of observations?
As a matter of fact, I know that the observations are dependent and hence randomness should not guarantee that. But intuitively it feels otherwise.
Given that randomness does not determine independence. Is there some statistical measure to test independence of observations?
edit: Please note that the data is not of timeseries nature, it is the average bandwidth usage of a customer across the month.

 A: Randomness is not the same as independence. There may be a small dependency in there but it appears to be lost in the noise. You can try various measures of independence (googleable). Pearson's correlation coefficient is the simplest one and will work best if there is a small linear correlation. But by looking at that, I can tell that you won't find a value that inconsistent with zero but you can still go ahead and try. 
There are other ones such as Brownian Covariance Distance that works for any kind of relation, not just linear, and it is good when you have to blindly check correlation of a large number of pairs of variables (for example in automated financial trading) especially when there is obvious non-linear structure.    
A: there is no one statistical measure of independence, it's impossible to test. the way people go about is to make assumptions about certain aspect of independence and test it. for instance, look at Engle's ARCH test, it tests for heteroscedasticity in time series, i.e. that variance doesn't change.
another example of dependence: $var(y_t)\sim y_{t-1}$, which would be difficult to detect visually in the scatter plot you have shown.
