GENERALIZED RAY-KNIGHT THEORY AND LIMIT-THEOREMS FOR SELF-INTERACTINGRANDOM-WALKS ON Z

We consider non-Markovian, self-interacting random walks (SIRW) on the one-dimensional integer lattice. The walk starts from the origin and at each step jumps to a neighboring site. The probability of jumping a long a bond is proportional to w (number of previous jumps along that lattice bond), where w:N --> R(+) is a monotone weight function. Expon ential and subexponential weight functions were considered in earlier papers. In the present paper we consider weight functions w with polyn omial asymptotics. These weight functions define variants of the ''rei nforced random walk.'' We prove functional limit theorems for the loca l time processes of these random walks and local limit theorems for th e position of the random walker at late times. A generalization of the Ray-Knight theory of local time arises.