arXiv subjects
Publications and source records attributed to Shosei Takeda.
Several long-time limit theorems of one-dimensional Lévy processes weighted and normalized by functions of its supremum are studied. The long-time limits are taken via the families of exponential times and that of constant times, called exponential clock and constant clock, respectively.
Takeda-Yano determined the limit of Lévy processes conditioned to avoid zero via various random clocks in terms of Doob's $h$-transform, where the limit processes may differ according to the choice of random clocks. The purpose of this paper is to investigate sample path behaviors of the limit processes in long time and in short time.
Several long-time limit theorems of one-dimensional Lévy processes weighted and normalized by functions of the local time are studied. The long-time limits are taken via certain families of random times, called clocks: exponential clock, hitting time clock, two-point hitting time clock and inverse local time clock. The limit measure can be characterized via a certain martingale expressed by an invariant function for the process killed upon hitting zero. The limit processes may differ according to the choice of the clocks when the original Lévy process is recurrent and of finite variance.