A sequence of independent non-identically distributed random variables is considered. Asymptotical behavior with probability 1 of maximal increments of theirs sums is studied. Exact convergence rates in the Erdos-Renyi law are found, and the Shepp law is proved. A version of the Erdos-Renyi law is obtained when Cramer's condition is replaced by the Linnik's condition from the probability theory of large deviations. The behavior of increments of different lengths is studied.