Based on real data (e.g. spot and futures prices of an index) if two series are correlated in the long run (e.g. strong positive significant correlation) it does not mean that they are cointegrated.
What if two series are cointegrated: can we infer that they are also correlated in the long run? Can we find a case with real data that two series are cointegrated but they are not correlated in the long run?
Correlation and cointegration are different concepts which cannot be compared directly. Here is my post on quant.SE explaining the difference. See also other answers there.
The main problem is that when we talk about cointegration we must assume that time-series in question are unit roots, which means that sample correlation is not meaningful, i.e. its limit is random variable. See this post, where the exact limit in simple case is written down.
