Abstract:
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.
Keywords:strong law of large numbers, maximal inequality, quasistationary random sequence, Banach space, Bochner measurability, Jensen's inequality.