  The aim of this paper is to represent any continuous local martingale as an
almost sure limit of a nested sequence of simple, symmetric random walks, time
changed by a discrete quadratic variation process. One basis of this is a
similar construction of Brownian motion. The other major tool is a
representation of continuous local martingales given by Dambis, Dubins and
Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic
variation. Rates of convergence (which are conjectured to be nearly optimal in
the given setting) are also supplied. A necessary and sufficient condition for
the independence of the random walks and the discrete time changes or,
equivalently, for the independence of the DDS Brownian motion and the quadratic
variation is proved to be the symmetry of increments of the martingale given
the past, which is a reformulation of an earlier result by Ocone.

Probability