Summary An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.
A certain class of stochastic summability methods of mantissa type is introduced and its connection to almost sure limit theorems
is discussed. The summability methods serve as suitable weights in almost sure limit theory, covering all relevant known examples
for, e.g., normalized sums or maxima of i.i.d. random variables. In the context of semistable domains of attraction the methods
lead to previously unknown versions of semistable almost sure limit theorems.