A study in lightly trimmed Levy processes

Abstract

In this dissertation, we study Levy processes with a bounded number of largest jumps removed. The resulting processes are termed lightly trimmed Levy processes. Motivated by the classical precedents in the random walks literature on trimmed sums, we aim to solve completely the domain of attraction problem for trimmed Levy processes at small times. We show that a Levy process is in the domain of attraction at small times, that is there exist appropriate norming and centring functions such that the centred and normed Levy process converges to an a.s. finite random variable as t goes down to 0, if and only if the centred and normed trimmed processes also converges. Despite its simple statement, the problem is rather complex. The necessity is treated first. Distributional representation formulae of trimmed processes at each fixed time t > 0 is derived. This allows us to write a lightly trimmed process as a mixture of a truncated Levy process with a Poisson number of ties randomised by an independent gamma random variable. Further, various trimming functionals are defined in the space of point measures as well as in the functional space. By a continuous mapping argument, we derive the functional convergence of the trimmed processes with respect to the Skorokhod topology by proving the continuity properties of the corresponding functionals. This solves the necessity. In the converse direction, we suppose the trimmed process, after appropriate centring and norming, has a limit at 0. Then the untrimmed Levy process is relatively compact. Light trimming does not have any effect, in the weak convergence, when the limit distribution is a normal or degenerate distribution. However, when the limit of the trimmed process is non-normal and non-degenerate, we show that the untrimmed process is in the domain of attraction of a stable law at 0. This leads to the interesting observation that each trimmed stable law possesses a domain of attraction at 0, which consists of all the lightly trimmed Levy processes. Show more