Limit theorems for increments of sums of independent random variables

We investigate the almost surely asymptotic behavior of increments of sums of independent identically distributed random variables satisfying the one-sided Cramér condition. We establish that, irrespective of the length of the increments, the norming sequence in strong limit theorems for increments of sums is determined by a behavior of the inverse function to the function of deviations. This allows for unifying the following well-known results for increments of sums: the strong law of large numbers, the Erdős–Rényi law and Mason's extension of this law, the Shepp law, the Csörgő–Révész theorems, and the law of the iterated logarithm. In the case of large increments, we derive new results for random variables from the domain of attraction of a stable law with index $\alpha\in (1,2]$ and the parameter of symmetry $\beta=-1$.