Minimal conditions in p-stable limit theorems

“From independence to dependence”: the head of Chapter IX in M. Lokve’s book [ 181 must be a program of any attempt to build a limit theory extending the classical one for sums of independent random variables. Lo&e himself suggested replacing dependent summands by their independent copies, but this could not bring much success. The next step in complication consists in dividing the sum into almost independent segments. It is possible, if summands possess ‘mixing’ properties, describable in various ways. Such approach, known as ‘Bernstein’s method’ (see [13,14]) proved to be very fruitful. We refer to [S] and [23] for the nearly up-to-date survey on the present stage of the theory. Some of results obtained on the base of Bernstein’s method can be ‘visualized’ in the form of an almost sure invariance principle (ASIP): the original (dependent) sequence can be redefined on another probability space, on which an accompanied independent sequence exists, with sums of both sequences being close in a strong sense (see [25] for the survey). As a rule, ASIP implies functional convergence of the corresponding partial-sum process. On the other hand, it is easy to find examples of l-dependent sequences with partial sums weakly convergent, when properly normalized, to a p-stable distribution (p < 2), but not convergent in the functional manner. It follows that in the general case we cannot expect results like ASIP (or even invariance principle in probability).