  We test the multiscaling issue of DLA clusters using a modified algorithm.
This algorithm eliminates killing the particles at the death circle. Instead,
we return them to the birth circle at a random relative angle taken from the
evaluated distribution. In addition, we use a two-level hierarchical memory
model that allows using large steps in conjunction with an off-lattice
realization of the model. Our algorithm still seems to stay in the framework of
the original DLA model. We present an accurate estimate of the fractal
dimensions based on the data for a hundred clusters with 50 million particles
each. We find that multiscaling cannot be ruled out. We also find that the
fractal dimension is a weak self-averaging quantity. In addition, the fractal
dimension, if calculated using the harmonic measure, is a nonmonotonic function
of the cluster radius. We argue that the controversies in the data
interpretation can be due to the weak self-averaging and the influence of
intrinsic noise.
