  General Central limit theorem deals with weak limits (in type) of sums of
row-elements of array random variables. In some situations as in the invariance
principle problem, the sums may include only parts of the row-elements. For
strictly stationary arrays (stationary for each row), there is no change to the
asymptotic results. But for non-stationary data, especially for dependent data,
asymptotic laws of partial sums moving in rows may require extra-conditions to
exist. This paper deals with central limit theorems with Gaussian limits for
non-stationary data. Our main focus is on dependent data, particularly on
associated data. But the non-stationary independent data is also studied as a
learning process. The results are applied to finite-distributional invariance
principles for the types of data described above. In Moreover, results for
associated sequences are interesting and innovative. Beyond their own interest,
the results are expected to be applied for random sums of random variables and
next in statistical modeling in many disciplines, in Actuarial sciences for
example
