The asymptotic behavior of increments of sums of independent identically distributed random variables with incremental length (logn)p is considered. The laws describing increments of such length are intermediate between the Csogő-Revesz law (for large incremental lengths) and the Erdo-Renyi law (for small incremental lengths). A new result for random variables from the domain of normal attraction of asymmetric stable laws with parameter α e (1, 2) is obtained.