Fragmentation processes describe phenomena of random splitting, with possibly infinite activity, according to certain rules that give rise to a close relation of these processes to branching processes and L´evy processes. In this thesis we study some asymptotic properties of fragmentation processes. More specifically, we prove certain strong laws of large numbers for self–similar fragmentations and we deal with the existence and uniqueness of solutions ofthe one–sided FKPP travelling wave equation for homogenous fragmentation processes. In addition to being concerned with standard fragmentation processes we also consider fragmentation processes with immigration, fragmentations stopped at a stopping line as well as killed fragmentation processes.

Let $mathcal{B}$ be a separable Banach space. In this thesis, it is shown that the Chung's strong law of large numbers holds for a sequence of independent $mathcal{B}$-valued random elements and an array of rowwise independent $mathcal{B}$-valued random elements under some weaker assumptions by using more generalized functions $phi_{n}$'s.