VERTEX-REINFORCED JUMP PROCESS ON THE INTEGERS WITH NONLINEAR REINFORCEMENT
We consider a nonlinear vertex-reinforced jump process (VRJP(w)) on Z with an increasing measurable weight function w : [1, ∞) → [1, ∞) and initial weights equal to one. Our main goal is to study the asymptotic behaviour of VRJP(w) depending on the integrability of the reciprocal of w. In particular, we prove that if (Equation presented). then the process is recurrent, that is, it visits each vert