Bond - True Self - Avoiding Walk on Z

Thètrue' self-avoiding walk with bond repulsion is a nearest neighbour random walk on Z, for which the probability of jumping along a bond of the lattice is proportional to exp(?g number of previous jumps along that bond). First we prove a limit theorem for the distribution of the local time process of this walk. Using this result, later we prove a local limit theorem, as A ! 1, for the distribution of A ?2=3 X s=A , where s=A is a random time of geometric distribution with mean e ?s=A ? 1 ? e ?s=A ?1 = A s + O(1). As a by-product we also obtain an apparently new identity related to Brownian excursions and Bessel bridges.