  Let \zeta be the intersection exponent of random walks in Z^3 and \alpha be a
positive real number. We construct a stochastic process from a simple random
walk by erasing loops of length at most N^\alpha. We will prove that for \alpha
< \frac{1}{1+2\zeta}, the limiting distribution is Gaussian. For \alpha > 2 the
limiting distribution will be shown to be equal to the limiting distribution of
the loop erased walk.
