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 Erdos-Rényi law and Mason's extension of this law, the Shepp law, the Csörgo- 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 α ∈ (1,2] and the parameter of symmetry β = -1.