Hsu and Robbins (1947) introduce the concept complete convergence as follows. A sequence of random variables is said to converge completely a constant if for all The converse true are independent. They also show that arithmetic means independent identically distributed converges expected value variance summands finite. Erdös proved converse. result Hsu-Robbins-Erdös fundamental theorem in proba...

In 1992, Loeb (Adv Math, 91:64–74, 1992) considered a natural extension of the binomial coefficients to negative entries and gave combinatorial interpretation in terms hybrid sets. He showed that many fundamental properties continue hold this extended setting. Recently, Formichella Straub (Ann Comb, 23:725–748, 2019) these results can be q-binomial with arbitrary integer values work further by ...

In this paper we establish some limsup results and a generalized uniform law of the iterated logarithm (LIL) for the increments of partial sums of a strictly stationary and linearly positive quadrant dependent (LPQD) sequence of random variables.