# 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. • Define what you mean by randomness in this context. – Glen_b Mar 20 '14 at 13:36
• By randomness I mean that the variable does not show any autocorrelation. From inspecting lag plots, I find no patterns. – tool.ish Mar 20 '14 at 13:41
• I'm not so much worried yet about the fact that doing a lag plot makes no sense. My issue is far more fundamental. Your data are either observed over time (a time series) or they aren't. You have asserted that they aren't a time series. If your data are not "over time" how does one lag them? One would have to move a series by one in a dimension it doesn't even posses! – Glen_b Mar 21 '14 at 6:57
• Unless the index represents some meaningful ordering, it's no more to the point than any random set of integers. Does the index represent some relevant kind of ordering? – Glen_b Mar 21 '14 at 8:26
• There is no basis for ordering this data that I used for lagging - a random set of integers would be equivalent. Since it is univariate, the only plausible ordering I can think of is a sort on its values - which probably defeats the whole purpose of doing a lag-plot. – tool.ish Mar 21 '14 at 9:05

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.