General Maximal Inequalities Related to the Strong Law of Large Numbers

For a sequence $(\xi_n)$ of random variables, we obtain maximal inequalities from which we can derive conditions for the a.s. convergence to zero of normalized differences
$$ \frac{1}{2^n} \biggl(\max_{2^n\le k<2^{n+1}} \biggl|\sum^k_{i=2^n}\xi_i\biggr|-\biggl|\sum_{i=2^n}^{2^{n+1}-1}\xi_i\biggr|\biggr). $$
The convergence to zero of this sequence leads to the strong law of large numbers (SLLN). In the special case of quasistationary sequences, we obtain a sufficient condition for the SLLN, which is an improvement on the well-known Móricz conditions.