Central limit theorems for associated possibly moving partial sums and application to the non-stationary invariance principle

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