Self-Stabilizing Processes Based on Random Signs

A self-stabilizing process $$\{Z(t), t\in [t_0,t_1)\}$${Z(t),t∈[t0,t1)} is, a random process which when localised, that is, scaled to a fine limit near a given $$t\in [t_0,t_1)$$t∈[t0,t1), has the distribution of an $$\alpha (Z(t))$$α(Z(t))-stable process, where $$\alpha : {\mathbb {R}}\rightarrow (0,2)$$α:R→(0,2) is a given continuous function. Thus, the stability index near t depends on the value of the process at t. In another paper [5] we constructed self-stabilizing processes using sums over plane Poisson point processes in the case of $$\alpha : {\mathbb {R}}\rightarrow (0,1)$$α:R→(0,1) which depended on the almost sure absolute convergence of the sums. Here we construct pure jump self-stabilizing processes when $$\alpha $$α may take values greater than 1 when convergence may no longer be absolute. We do this in two stages, firstly by setting up a process based on a fixed point set but taking random signs of the summands, and then randomising the point set to get a process with the desired local properties.