This estimated error would again have an error, ad infinitum. The probability function can be either estimated from the data or chosen theoretically. In practice, for a reason given in a later section, the gaussian function often occurs. The inequality applies if the random numbers x t are not totally unpredictable random numbers. If x t is an approximation to a continuous function, then it is highly predictable and there will be a lot of slack in the inequality.
Correlation is a concept similar to cosine.
A cosine measures the angle between two vectors. It is given by the dot product of the two vectors divided by their magnitudes Correlation is the same sort of thing, except x and y are scalar random variables, so instead of having a vector subscript their subscript is the implicit ensemble subscript. There is a practical problem when the ensemble average is simulated by averaging over a sample. The problem arises with small samples and is most dramatically illustrated for a sample with only one element. Then the sample correlation is regardless of what value the random number x or the random number y should take.